today’s outline - october 03, 2019csrri.iit.edu/~segre/phys405/19f/lecture_13.pdf · today’s...
TRANSCRIPT
![Page 1: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/1.jpg)
Today’s Outline - October 03, 2019
• The 3D Schrodinger equation
• Separation of variables
• The angular solution
• Radial solutions
• Example 4.1 - Particle in a bubble
No lecture on Tuesday, October 8, and Thursday, October 10: Videos willbe prerecorded on Friday, October 4 in room 204 SB at 10:00 and 11:25and posted on blackboard
Homework Assignment #06:Chapter 3: 16,18,26,27,33,38due Tuesday, October 8, 2019, in my mailbox
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 1 / 26
![Page 2: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/2.jpg)
Today’s Outline - October 03, 2019
• The 3D Schrodinger equation
• Separation of variables
• The angular solution
• Radial solutions
• Example 4.1 - Particle in a bubble
No lecture on Tuesday, October 8, and Thursday, October 10: Videos willbe prerecorded on Friday, October 4 in room 204 SB at 10:00 and 11:25and posted on blackboard
Homework Assignment #06:Chapter 3: 16,18,26,27,33,38due Tuesday, October 8, 2019, in my mailbox
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 1 / 26
![Page 3: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/3.jpg)
Today’s Outline - October 03, 2019
• The 3D Schrodinger equation
• Separation of variables
• The angular solution
• Radial solutions
• Example 4.1 - Particle in a bubble
No lecture on Tuesday, October 8, and Thursday, October 10: Videos willbe prerecorded on Friday, October 4 in room 204 SB at 10:00 and 11:25and posted on blackboard
Homework Assignment #06:Chapter 3: 16,18,26,27,33,38due Tuesday, October 8, 2019, in my mailbox
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 1 / 26
![Page 4: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/4.jpg)
Today’s Outline - October 03, 2019
• The 3D Schrodinger equation
• Separation of variables
• The angular solution
• Radial solutions
• Example 4.1 - Particle in a bubble
No lecture on Tuesday, October 8, and Thursday, October 10: Videos willbe prerecorded on Friday, October 4 in room 204 SB at 10:00 and 11:25and posted on blackboard
Homework Assignment #06:Chapter 3: 16,18,26,27,33,38due Tuesday, October 8, 2019, in my mailbox
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 1 / 26
![Page 5: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/5.jpg)
Today’s Outline - October 03, 2019
• The 3D Schrodinger equation
• Separation of variables
• The angular solution
• Radial solutions
• Example 4.1 - Particle in a bubble
No lecture on Tuesday, October 8, and Thursday, October 10: Videos willbe prerecorded on Friday, October 4 in room 204 SB at 10:00 and 11:25and posted on blackboard
Homework Assignment #06:Chapter 3: 16,18,26,27,33,38due Tuesday, October 8, 2019, in my mailbox
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 1 / 26
![Page 6: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/6.jpg)
Today’s Outline - October 03, 2019
• The 3D Schrodinger equation
• Separation of variables
• The angular solution
• Radial solutions
• Example 4.1 - Particle in a bubble
No lecture on Tuesday, October 8, and Thursday, October 10: Videos willbe prerecorded on Friday, October 4 in room 204 SB at 10:00 and 11:25and posted on blackboard
Homework Assignment #06:Chapter 3: 16,18,26,27,33,38due Tuesday, October 8, 2019, in my mailbox
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 1 / 26
![Page 7: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/7.jpg)
Today’s Outline - October 03, 2019
• The 3D Schrodinger equation
• Separation of variables
• The angular solution
• Radial solutions
• Example 4.1 - Particle in a bubble
No lecture on Tuesday, October 8, and Thursday, October 10: Videos willbe prerecorded on Friday, October 4 in room 204 SB at 10:00 and 11:25and posted on blackboard
Homework Assignment #06:Chapter 3: 16,18,26,27,33,38due Tuesday, October 8, 2019, in my mailbox
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 1 / 26
![Page 8: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/8.jpg)
Today’s Outline - October 03, 2019
• The 3D Schrodinger equation
• Separation of variables
• The angular solution
• Radial solutions
• Example 4.1 - Particle in a bubble
No lecture on Tuesday, October 8, and Thursday, October 10: Videos willbe prerecorded on Friday, October 4 in room 204 SB at 10:00 and 11:25and posted on blackboard
Homework Assignment #06:Chapter 3: 16,18,26,27,33,38due Tuesday, October 8, 2019, in my mailbox
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 1 / 26
![Page 9: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/9.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 10: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/10.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 11: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/11.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 12: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/12.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 13: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/13.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 14: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/14.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 15: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/15.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 16: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/16.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 17: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/17.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 18: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/18.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
![Page 19: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/19.jpg)
3D Schrodinger equation
The three-dimensionalSchrodinger equation,written in Cartesiancoordinates is
but
pn =~i
∂
∂n
where the term in paren-theses is simply the Lapla-cian, ∇2
with volume elementd3~r = dx dy dz , we have
if V = V (~r) only
i~∂Ψ
∂t=
p2
2mΨ + VΨ
=1
2m(p2x + p2y + p2z )Ψ + VΨ
= − ~2
2m
(∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)Ψ + VΨ
i~∂Ψ
∂t= − ~2
2m∇2Ψ + VΨ
∫|Ψ(~r , t)|2 d3~r = 1
Ψn(~r , t) = ψn(~r)e−iEnt/~
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 2 / 26
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Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ
, the in-finitesmal volume element can be written
dV = dr
rdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
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Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ
, the in-finitesmal volume element can be written
dV = dr
rdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
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Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ
, the in-finitesmal volume element can be written
dV = dr
rdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
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Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ, the in-finitesmal volume element can be written
dV = dr
rdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
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Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ, the in-finitesmal volume element can be written
dV = dr
rdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
![Page 25: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/25.jpg)
Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ, the in-finitesmal volume element can be written
dV = drrdθ
r sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
![Page 26: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/26.jpg)
Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ, the in-finitesmal volume element can be written
dV = drrdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
![Page 27: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/27.jpg)
Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ, the in-finitesmal volume element can be written
dV = drrdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
![Page 28: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/28.jpg)
Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ, the in-finitesmal volume element can be written
dV = drrdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)
+1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
![Page 29: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/29.jpg)
Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ, the in-finitesmal volume element can be written
dV = drrdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)
+1
r2sin2 θ
(∂2
∂φ2
)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
![Page 30: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/30.jpg)
Spherical coordinates
Many common 3-dimensional potentials in physics are central and onlydepend on the distance from the origin. Thus spherical coordinates aremost useful.
r
θ
φ
^
^
^
y
x
z(r,θ,φ)
θ
φ
Using the radial distance r , the polar an-gle θ, and the azimuthal angle φ, the in-finitesmal volume element can be written
dV = drrdθr sin θdφ
and the Laplacian then becomes
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2sin2 θ
(∂2
∂φ2
)C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 3 / 26
![Page 31: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/31.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)
the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:
ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[
Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
![Page 32: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/32.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:
ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[
Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
![Page 33: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/33.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:
ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[
Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
![Page 34: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/34.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:
ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[
Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
![Page 35: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/35.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[
Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
![Page 36: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/36.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[
Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
![Page 37: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/37.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[
Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
![Page 38: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/38.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)
+R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
![Page 39: Today’s Outline - October 03, 2019csrri.iit.edu/~segre/phys405/19F/lecture_13.pdf · Today’s Outline - October 03, 2019 The 3D Schr odinger equation Separation of variables The](https://reader036.vdocuments.net/reader036/viewer/2022062917/5edab17f39be4033970bc00a/html5/thumbnails/39.jpg)
Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)
+R
r2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
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Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]
+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
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Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+ V (r)RY
= E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
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Spherical coordinates
∇2 =1
r2∂
∂r
(r2∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin2 θ
(∂2
∂φ2
)the time-independent Schrodinger equation is thus
− ~2
2m
[1
r2∂
∂r
(r2∂ψ
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂ψ
∂θ
)+
1
r2 sin2 θ
(∂2ψ
∂φ2
)]+V (r)ψ = Eψ
This is difficult to solve unless we can find separable solutions:ψ(r , θ, φ) = R(r)Y (θ, φ) which gives
− ~2
2m
[1
r2∂
∂r
(r2∂RY
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂RY
∂θ
)+
1
r2 sin2 θ
(∂2RY
∂φ2
)]+V (r)RY = ERY
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+ V (r)RY = E RY
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 4 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[
1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[
1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[
1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)
+1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)
+1
Yr2 sin2 θ
(∂2Y
∂φ2
)
]
+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]
+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r)
= E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over
[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[
1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[1
R
∂
∂r
(r2∂R
∂r
)
+1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)
+1
Y sin2 θ
(∂2Y
∂φ2
)
]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)]
− 2mr2
~2[V (r)− E ]
= 0
finally, we regroup the radial and angular parts
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)]− 2mr2
~2[V (r)− E ] = 0
finally, we regroup the radial and angular parts
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)]− 2mr2
~2[V (r)− E ] = 0
finally, we regroup the radial
and angular parts
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Separation of variables
− ~2
2m
[Y
r2∂
∂r
(r2∂R
∂r
)+
R
r2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
R
r2 sin2 θ
(∂2Y
∂φ2
)]+V (r)RY = E RY
dividing by RY
− ~2
2m
[1
Rr2∂
∂r
(r2∂R
∂r
)+
1
Yr2 sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Yr2 sin2 θ
(∂2Y
∂φ2
)]+ V (r) = E
multiplying by −2mr2/~2 and bringing the E over[1
R
∂
∂r
(r2∂R
∂r
)+
1
Y sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
Y sin2 θ
(∂2Y
∂φ2
)]− 2mr2
~2[V (r)− E ] = 0
finally, we regroup the radial and angular partsC. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 5 / 26
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The radial and angular equations{1
R
∂
∂r
(r2∂R
∂r
)− 2mr2
~2[V (r)− E ]
}+
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= 0
Since the two parts depend on different variables, the only way they cancancel is to each be equal to a separation constant, which we will take tobe l(l + 1). This gives two independent equations:{
1
R
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]
}= l(l + 1)
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
We will first solve the angular equation
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 6 / 26
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The radial and angular equations{1
R
∂
∂r
(r2∂R
∂r
)− 2mr2
~2[V (r)− E ]
}+
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= 0
Since the two parts depend on different variables, the only way they cancancel is to each be equal to a separation constant, which we will take tobe l(l + 1). This gives two independent equations:
{1
R
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]
}= l(l + 1)
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
We will first solve the angular equation
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 6 / 26
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The radial and angular equations{1
R
∂
∂r
(r2∂R
∂r
)− 2mr2
~2[V (r)− E ]
}+
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= 0
Since the two parts depend on different variables, the only way they cancancel is to each be equal to a separation constant, which we will take tobe l(l + 1). This gives two independent equations:{
1
R
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]
}= l(l + 1)
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
We will first solve the angular equation
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 6 / 26
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The radial and angular equations{1
R
∂
∂r
(r2∂R
∂r
)− 2mr2
~2[V (r)− E ]
}+
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= 0
Since the two parts depend on different variables, the only way they cancancel is to each be equal to a separation constant, which we will take tobe l(l + 1). This gives two independent equations:{
1
R
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]
}= l(l + 1)
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
We will first solve the angular equation
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 6 / 26
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The radial and angular equations{1
R
∂
∂r
(r2∂R
∂r
)− 2mr2
~2[V (r)− E ]
}+
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= 0
Since the two parts depend on different variables, the only way they cancancel is to each be equal to a separation constant, which we will take tobe l(l + 1). This gives two independent equations:{
1
R
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]
}= l(l + 1)
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
We will first solve the angular equation
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 6 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)
+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)
= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ)
{Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]
+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ ΘΦl(l + 1) sin2 θ
}
+ Θ
(d2Φ
dφ2
)
= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ ΘΦl(l + 1) sin2 θ
}+ Θ
(d2Φ
dφ2
)= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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The angular solution
1
Y
{1
sin θ
∂
∂θ
(sin θ
∂Y
∂θ
)+
1
sin2 θ
(∂2Y
∂φ2
)}= −l(l + 1)
begin by multiplying by Y sin2 θ
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+
(∂2Y
∂φ2
)= −l(l + 1) sin2 θY
rearranging
sin θ∂
∂θ
(sin θ
∂Y
∂θ
)+ l(l + 1) sin2 θY +
(∂2Y
∂φ2
)= 0
This clearly is separable in the two angular variables by assumingY (θ, φ) = Θ(θ)Φ(φ){
Φ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ ΘΦl(l + 1) sin2 θ
}+ Θ
(d2Φ
dφ2
)= 0
final separation can be achieved by dividing by Θ(θ)Φ(φ)C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 7 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead:
We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead:
We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead:
We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)
A look ahead:
We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead:
We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead: We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead: We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead: We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead: We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Second angular separation constant{1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
}+
1
Φ
(d2Φ
dφ2
)= 0
The separation constant is taken to be m2
m2 =1
Θ
[sin θ
d
dθ
(sin θ
dΘ
dθ
)]+ l(l + 1) sin2 θ
−m2 =1
Φ
(d2Φ
dφ2
)A look ahead: We now have identified 3 separation constants of the 3DSchrodinger equation, each of which corresponds to a quantum numberand operator which commutes with the others.
Separation constant Operator Quantum number
E H n
l(l+1) L2
l
m2 Lz m
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 8 / 26
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 9 / 26
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π)
= Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Φ(φ)
Rearranging the equation for Φgives a straightforward differen-tial equation to solve
if we insist on continuity,Φ(φ) = Φ(φ + 2π) we can putconditions on the value of m
d2Φ
dφ2= −m2Φ
Φ(φ) = Ae±imφ
Ae±imφ = Ae±im(φ+2π) = Ae±imφe±2πim
m = 0,±1,±2, . . .
m is called the “magnetic” quantum number and must be an integer. Wewill have to impose more restrictions later.
The integration constant A can be absorbed into the constant determinedfor the solution of the Θ equation, and we have taken care of the negativesolutions by letting m be negative, so the final solution is
Φ(φ) = e imφ, m = 0,±1,±2,±3, . . .
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)?
A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)?
A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)?
A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)
= sin2 θd2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)?
A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2
+ sin θ cos θdΘ
dθ
What must be the form of the function Θ(θ)?
A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)?
A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)?
A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)? A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)? A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)? A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Solution for Θ(θ)
The equation for Θ(θ), after rearranging, is
sin θd
dθ
(sin θ
dΘ
dθ
)+[l(l + 1) sin2 θ −m2
]Θ = 0
This has a more com-plex solution. Lookingat the first term
sin θd
dθ
(sin θ
dΘ
dθ
)= sin2 θ
d2Θ
dθ2+ sin θ cos θ
dΘ
dθ
What must be the form of the function Θ(θ)? A polynomial in cos θ andsin θ will satisfy this differential equation.
The solution is an associated Legendre function, Pml :
Θ(θ) = APml (cos θ)
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
x
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
x
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
P0(x
)
x
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
P1(x
)
x
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
P2(x
)
x
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
P3(x
)
x
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
P4(x
)
x
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
P5(x
)
x
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Legendre polynomials
Pl (x) is the l th Legendre Polyno-mial which is defined by the Ro-drigues formula
Pl (x) ≡ 1
2l l!
(d
dx
)l
(x2 − 1)l
P0 = 1
P1 = x
P2 =1
2(3x2 − 1)
P3 =1
2(5x3 − 3x)
P4 =1
8(35x4 − 30x2 + 3)
P5 =1
8(63x5 − 70x3 + 15x)
-1
0
+1
-1 0 +1
x
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)
=1
2
(cos2 θ − 1
)
P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)=
1
2
(cos2 θ − 1
)P12 = −
(1− x2
)1/2· 3x
= −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)=
1
2
(cos2 θ − 1
)P12 = −
(1− x2
)1/2· 3x = −3 cos θ sin θ
P22 =
(1− x2
)· 3
= 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)=
1
2
(cos2 θ − 1
)P12 = −
(1− x2
)1/2· 3x = −3 cos θ sin θ
P22 =
(1− x2
)· 3 = 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)=
1
2
(cos2 θ − 1
)P12 = −
(1− x2
)1/2· 3x = −3 cos θ sin θ
P22 =
(1− x2
)· 3 = 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Associated Legendre functions
Unlike the Legendre polynomials, the Associated Legendre functions arenot necessarily polynomials
Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
for the specific valueof l = 2, we startwith the Legendrepolynomial
P2 =1
2(3x2 − 1)
imposing the condi-tion |m| ≤ l
P02 =
1
2
(3x2 − 1
)=
1
2
(cos2 θ − 1
)P12 = −
(1− x2
)1/2· 3x = −3 cos θ sin θ
P22 =
(1− x2
)· 3 = 3 sin2 θ
P32 ≡ 0
if x = cos θ, the Pml become polynomials
Thus, the meaningful values of m are integers which range from −l to +land the full solution for both quantum numbers is:
l = 0, 1, 2, . . . ; m = −l ,−(l − 1), . . . ,−1, 0, 1, . . . (l − 1), l
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 12 / 26
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ
, Pml (x) ≡ (−1)m(1− x2)|m|/2
(d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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Angular wave functions
The full angular solutions, Y are
Y (θ, φ) = APml (cos θ)e imφ, Pm
l (x) ≡ (−1)m(1− x2)|m|/2(
d
dx
)|m|Pl (x)
P00 = 1
P01 = cos θ
P11 = − sin θ
P02 = 0.5(3 cos2 θ − 1)
P12 = −3 sin θ cos θ
P22 = 3 sin2 θ
P03 = 0.5(5 cos3 θ − 3 cos θ)
P13 = −1.5 sin θ(5 cos2 θ − 1)
P23 = 15 sin2 θ cos θ
P33 = −15 sin θ(1− cos2 θ)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
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3D angular wavefunctions – l = 1, 0
|Y 00 |
|Y 00 | |Y−11 | |Y 0
1 | |Y+11 |
Re{Y 00 }
Re{Y 00 } Im{Y−11 } Im{Y 0
1 } Im{Y+11 }
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3D angular wavefunctions – l = 1, 0
|Y 00 |
|Y 00 | |Y−11 | |Y 0
1 | |Y+11 |
Im{Y 00 }
Re{Y 00 } Im{Y−11 } Im{Y 0
1 } Im{Y+11 }
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3D angular wavefunctions – l = 1, 0
|Y 00 |
|Y 00 |
|Y−11 | |Y 01 | |Y+1
1 |
Im{Y 00 }
Re{Y 00 }
Re{Y−11 } Re{Y 01 } Re{Y+1
1 }
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3D angular wavefunctions – l = 1, 0
|Y 00 |
|Y 00 |
|Y−11 | |Y 01 | |Y+1
1 |
Im{Y 00 }
Re{Y 00 }
Im{Y−11 } Im{Y 01 } Im{Y+1
1 }
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3D angular wavefunctions – l = 2
|Y−22 | |Y−12 | |Y 02 | |Y+1
2 | |Y+22 |
Re{Y−22 } Re{Y−12 } Re{Y 02 } Re{Y+1
2 } Re{Y+22 }
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3D angular wavefunctions – l = 2
|Y−22 | |Y−12 | |Y 02 | |Y+1
2 | |Y+22 |
Im{Y−22 } Im{Y−12 } Im{Y 02 } Im{Y+1
2 } Im{Y+22 }
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3D angular wavefunctions – l = 3
|Y−33 | |Y−23 | |Y−13 |
|Y 03 |
|Y 03 |
Re{Y−33 } Re{Y−23 } Re{Y−13 }
Re{Y 03 }
Re{Y 03 }
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3D angular wavefunctions – l = 3
|Y−33 | |Y−23 | |Y−13 |
|Y 03 |
|Y 03 |
Im{Y−33 } Im{Y−23 } Im{Y−13 }
Re{Y 03 }
Im{Y 03 }
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3D angular wavefunctions – l = 3
|Y 03 |
|Y 03 |
|Y+13 | |Y+2
3 | |Y+33 |
Re{Y 03 }
Re{Y 03 }
Re{Y=13 } Re{Y+2
3 } Re{Y+33 }
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3D angular wavefunctions – l = 3
|Y 03 |
|Y 03 |
|Y+13 | |Y+2
3 | |Y+33 |
Im{Y 03 }
Re{Y 03 }
Im{Y+13 } Im{Y+2
3 } Im{Y+33 }
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Spherical harmonics
When normalized, these angular functions are called spherical harmonics
Y ml (θ, φ) = ε
√(2l + 1)(l − |m|)!
4π(l + |m|)!e imφPm
l (cos θ), ε =
{(−1)m, m ≥ 0
1, m ≤ 0
∫ 2π
0
∫ π
0[Y m
l (θ, φ)]∗[Y m′l ′ (θ, φ)] sin θ dθ dφ = δll ′δmm′
where l is called the azimuthal quantum number and m is called themagnetic quantum number.
Now there is only the radial equation to be solved, but this depends on thespecifics of the potential.
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Spherical harmonics
When normalized, these angular functions are called spherical harmonics
Y ml (θ, φ) = ε
√(2l + 1)(l − |m|)!
4π(l + |m|)!e imφPm
l (cos θ), ε =
{(−1)m, m ≥ 0
1, m ≤ 0
∫ 2π
0
∫ π
0[Y m
l (θ, φ)]∗[Y m′l ′ (θ, φ)] sin θ dθ dφ = δll ′δmm′
where l is called the azimuthal quantum number and m is called themagnetic quantum number.
Now there is only the radial equation to be solved, but this depends on thespecifics of the potential.
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Spherical harmonics
When normalized, these angular functions are called spherical harmonics
Y ml (θ, φ) = ε
√(2l + 1)(l − |m|)!
4π(l + |m|)!e imφPm
l (cos θ), ε =
{(−1)m, m ≥ 0
1, m ≤ 0
∫ 2π
0
∫ π
0[Y m
l (θ, φ)]∗[Y m′l ′ (θ, φ)] sin θ dθ dφ = δll ′δmm′
where l is called the azimuthal quantum number and m is called themagnetic quantum number.
Now there is only the radial equation to be solved, but this depends on thespecifics of the potential.
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Spherical harmonics
When normalized, these angular functions are called spherical harmonics
Y ml (θ, φ) = ε
√(2l + 1)(l − |m|)!
4π(l + |m|)!e imφPm
l (cos θ), ε =
{(−1)m, m ≥ 0
1, m ≤ 0
∫ 2π
0
∫ π
0[Y m
l (θ, φ)]∗[Y m′l ′ (θ, φ)] sin θ dθ dφ = δll ′δmm′
where l is called the azimuthal quantum number and m is called themagnetic quantum number.
Now there is only the radial equation to be solved, but this depends on thespecifics of the potential.
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Spherical harmonics
When normalized, these angular functions are called spherical harmonics
Y ml (θ, φ) = ε
√(2l + 1)(l − |m|)!
4π(l + |m|)!e imφPm
l (cos θ), ε =
{(−1)m, m ≥ 0
1, m ≤ 0
∫ 2π
0
∫ π
0[Y m
l (θ, φ)]∗[Y m′l ′ (θ, φ)] sin θ dθ dφ = δll ′δmm′
where l is called the azimuthal quantum number and m is called themagnetic quantum number.
Now there is only the radial equation to be solved, but this depends on thespecifics of the potential.
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)
=1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2
=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)
=d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]
=du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr
= rd2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Radial solution
Returning to the radial equation, where the potential is included
d
dr
(r2dR
dr
)− 2mr2
~2[V (r)− E ]R = l(l + 1)R
A change of variables to u(r) ≡ rR(r) simplifies the equation
dR
dr=
d
dr
(ur
)=
1
r
du
dr− u
r2=
[rdu
dr− u
]1
r2
d
dr
(r2dR
dr
)=
d
dr
[rdu
dr− u
]=
du
dr+ r
d2u
dr2− du
dr= r
d2u
dr2
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
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Equation for u(r)
We can now obtain a differential equation for u(r)
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
r2d2u
dr2− 2mr2
~2[V (r)− E ] u = l(l + 1)u
r2d2u
dr2−[
2mr2
~2V (r)u + l(l + 1)u
]= −2mr2
~2Eu
− ~2
2m
d2u
dr2+
[V (r) +
~2
2m
l(l + 1)
r2
]u = Eu
This looks like a time-independent Schrodinger equation with an effectivepotential Veff (r) whose solution is normalized as
∫∞0 |u(r)|2 dr = 1
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Equation for u(r)
We can now obtain a differential equation for u(r)
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
r2d2u
dr2− 2mr2
~2[V (r)− E ] u = l(l + 1)u
r2d2u
dr2−[
2mr2
~2V (r)u + l(l + 1)u
]= −2mr2
~2Eu
− ~2
2m
d2u
dr2+
[V (r) +
~2
2m
l(l + 1)
r2
]u = Eu
This looks like a time-independent Schrodinger equation with an effectivepotential Veff (r) whose solution is normalized as
∫∞0 |u(r)|2 dr = 1
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Equation for u(r)
We can now obtain a differential equation for u(r)
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
r2d2u
dr2− 2mr2
~2[V (r)− E ] u = l(l + 1)u
r2d2u
dr2−[
2mr2
~2V (r)u + l(l + 1)u
]= −2mr2
~2Eu
− ~2
2m
d2u
dr2+
[V (r) +
~2
2m
l(l + 1)
r2
]u = Eu
This looks like a time-independent Schrodinger equation with an effectivepotential Veff (r) whose solution is normalized as
∫∞0 |u(r)|2 dr = 1
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Equation for u(r)
We can now obtain a differential equation for u(r)
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
r2d2u
dr2− 2mr2
~2[V (r)− E ] u = l(l + 1)u
r2d2u
dr2−[
2mr2
~2V (r)u + l(l + 1)u
]= −2mr2
~2Eu
− ~2
2m
d2u
dr2+
[V (r) +
~2
2m
l(l + 1)
r2
]u = Eu
This looks like a time-independent Schrodinger equation with an effectivepotential Veff (r) whose solution is normalized as
∫∞0 |u(r)|2 dr = 1
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Equation for u(r)
We can now obtain a differential equation for u(r)
rd2u
dr2− 2mr2
~2[V (r)− E ]
u
r= l(l + 1)
u
r
r2d2u
dr2− 2mr2
~2[V (r)− E ] u = l(l + 1)u
r2d2u
dr2−[
2mr2
~2V (r)u + l(l + 1)u
]= −2mr2
~2Eu
− ~2
2m
d2u
dr2+
[V (r) +
~2
2m
l(l + 1)
r2
]u = Eu
This looks like a time-independent Schrodinger equation with an effectivepotential Veff (r) whose solution is normalized as
∫∞0 |u(r)|2 dr = 1
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1
Consider the potential for the infi-nite spherical well
inside the well the wave equation is
dividing by ~2/2m and rearranging
substituting k =√
2mE/~
V (r) =
{0, r ≤ a
∞, r ≥ a
Eu = − ~2
2m
d2u
dr2+
~2
2m
l(l + 1)
r2u
d2u
dr2=
[l(l + 1)
r2− 2mE
~2
]u
d2u
dr2=
[l(l + 1)
r2− k2
]u
We must solve this for each value of l separately. The l = 0 case is thesimplest
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) + B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) + B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Example 4.1 (cont.)
For l = 0, we need to impose theboundary condition u(a) = 0.
this has the usual solution
since R(r) = u/r , the cos(kr) solu-tion would be infinite at the originand thus B = 0
applying the boundary condition
thus, ka = nπ where n is an integerand the energies for the l = 0 caseare
Normalizing (A =√
2/a) andadding the angular part of the wavefunction Y 0
0 = 1/√
4π
d2u
dr2= −k2u
u(r) = A sin(kr) +��B cos(kr)
= A sin(kr)
u(a) = A sin(ka) = 0
En0 =n2π2~2
2ma2, n = 1, 2, 3, . . .
ψn00 =1√2πa
sin(nπr/a)
r
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Solutions for l = 0
ψn00 =1√2πa
sin(nπr/a)
r
0 a
ψn00
r
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Solutions for l = 0
ψn00 =1√2πa
sin(nπr/a)
r
0 a
ψ100
r
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Solutions for l = 0
ψn00 =1√2πa
sin(nπr/a)
r
0 a
ψ200
r
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Solutions for l = 0
ψn00 =1√2πa
sin(nπr/a)
r
0 a
ψ300
r
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Solutions for l = 0
ψn00 =1√2πa
sin(nπr/a)
r
0 a
ψ400
r
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Solutions for l = 0
ψn00 =1√2πa
sin(nπr/a)
r
0 a
ψn00
r
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l
and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Example 4.1 (cont.)
What about the solutions for l 6= 0?
they involve Bessel functions; thegeneral solution being
jl (x) are spherical Bessel functionsof order l and nl (x) are sphericalNeumann functions of order l .
while spherical Bessel functions arefinite at the origin, spherical Neu-mann functions are infinite andthus we again set Bl = 0
u(r) = Al r jl (kr) + Bl r nl (kr)
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
nl (x) = −(−x)l
(1
x
d
dx
)l cos x
x
R(r) = Al jl (kr)
we still must apply the boundary condition that jl (ka) = 0 but this is a bitmore complex than for the l = 0 case
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Spherical Bessel functions
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
j0(x) =sin x
x
j1(x) = (−x)1
x
(cos x
x− sin x
x2
)=
sin x
x2− cos x
x
j2(x) = (−x)2(
1
x
d
dx
)x cos x − sin x
x3
= x
(−x sin x
x3− 3
x cos x − sin x
x4
)=
(3 sin x − 3x cos x − x2 sin x
x3
)0
+1
0 5 10
j l(x)
x
Clearly the roots are not at nice, simple, locations!
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Spherical Bessel functions
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
j0(x) =sin x
x
j1(x) = (−x)1
x
(cos x
x− sin x
x2
)=
sin x
x2− cos x
x
j2(x) = (−x)2(
1
x
d
dx
)x cos x − sin x
x3
= x
(−x sin x
x3− 3
x cos x − sin x
x4
)=
(3 sin x − 3x cos x − x2 sin x
x3
)
0
+1
0 5 10j 0
(x)
x
Clearly the roots are not at nice, simple, locations!
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Spherical Bessel functions
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
j0(x) =sin x
x
j1(x) = (−x)1
x
(cos x
x− sin x
x2
)
=sin x
x2− cos x
x
j2(x) = (−x)2(
1
x
d
dx
)x cos x − sin x
x3
= x
(−x sin x
x3− 3
x cos x − sin x
x4
)=
(3 sin x − 3x cos x − x2 sin x
x3
)
0
+1
0 5 10j 0
(x)
x
Clearly the roots are not at nice, simple, locations!
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Spherical Bessel functions
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
j0(x) =sin x
x
j1(x) = (−x)1
x
(cos x
x− sin x
x2
)=
sin x
x2− cos x
x
j2(x) = (−x)2(
1
x
d
dx
)x cos x − sin x
x3
= x
(−x sin x
x3− 3
x cos x − sin x
x4
)=
(3 sin x − 3x cos x − x2 sin x
x3
)
0
0 5 10j 1
(x)
x
Clearly the roots are not at nice, simple, locations!
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Spherical Bessel functions
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
j0(x) =sin x
x
j1(x) = (−x)1
x
(cos x
x− sin x
x2
)=
sin x
x2− cos x
x
j2(x) = (−x)2(
1
x
d
dx
)x cos x − sin x
x3
= x
(−x sin x
x3− 3
x cos x − sin x
x4
)=
(3 sin x − 3x cos x − x2 sin x
x3
)
0
0 5 10j 1
(x)
x
Clearly the roots are not at nice, simple, locations!
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Spherical Bessel functions
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
j0(x) =sin x
x
j1(x) = (−x)1
x
(cos x
x− sin x
x2
)=
sin x
x2− cos x
x
j2(x) = (−x)2(
1
x
d
dx
)x cos x − sin x
x3
= x
(−x sin x
x3− 3
x cos x − sin x
x4
)
=
(3 sin x − 3x cos x − x2 sin x
x3
)
0
0 5 10j 1
(x)
x
Clearly the roots are not at nice, simple, locations!
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Spherical Bessel functions
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
j0(x) =sin x
x
j1(x) = (−x)1
x
(cos x
x− sin x
x2
)=
sin x
x2− cos x
x
j2(x) = (−x)2(
1
x
d
dx
)x cos x − sin x
x3
= x
(−x sin x
x3− 3
x cos x − sin x
x4
)=
(3 sin x − 3x cos x − x2 sin x
x3
)0
0 5 10j 2
(x)
x
Clearly the roots are not at nice, simple, locations!
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Spherical Bessel functions
jl (x) = (−x)l
(1
x
d
dx
)l sin x
x
j0(x) =sin x
x
j1(x) = (−x)1
x
(cos x
x− sin x
x2
)=
sin x
x2− cos x
x
j2(x) = (−x)2(
1
x
d
dx
)x cos x − sin x
x3
= x
(−x sin x
x3− 3
x cos x − sin x
x4
)=
(3 sin x − 3x cos x − x2 sin x
x3
)0
+1
0 5 10j l(
x)
x
Clearly the roots are not at nice, simple, locations!
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 25 / 26
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Solutions for all l
jl (ka) = 0
k =1
aβnl , βnl are the roots
Enl =~2
2ma2β2nl
βn1 βn2 βn3
j0 π 2π 3πj1 4.493 7.726 10.904j2 5.762 9.906 12.325j3 6.988 10.420 13.698
The wavefunctions are thus:
ψnlm(r , θ, φ) = Anl jl
(βnl r
a
)Y m
l (θ, φ)
these are (2l + 1)-fold degenerate states, that is, the energy does notdepend on the quantum numbers which give rise to the degeneracy
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Solutions for all l
jl (ka) = 0
k =1
aβnl , βnl are the roots
Enl =~2
2ma2β2nl
βn1 βn2 βn3
j0 π 2π 3πj1 4.493 7.726 10.904j2 5.762 9.906 12.325j3 6.988 10.420 13.698
The wavefunctions are thus:
ψnlm(r , θ, φ) = Anl jl
(βnl r
a
)Y m
l (θ, φ)
these are (2l + 1)-fold degenerate states, that is, the energy does notdepend on the quantum numbers which give rise to the degeneracy
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Solutions for all l
jl (ka) = 0
k =1
aβnl , βnl are the roots
Enl =~2
2ma2β2nl
βn1 βn2 βn3
j0 π 2π 3πj1 4.493 7.726 10.904j2 5.762 9.906 12.325j3 6.988 10.420 13.698
The wavefunctions are thus:
ψnlm(r , θ, φ) = Anl jl
(βnl r
a
)Y m
l (θ, φ)
these are (2l + 1)-fold degenerate states, that is, the energy does notdepend on the quantum numbers which give rise to the degeneracy
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Solutions for all l
jl (ka) = 0
k =1
aβnl , βnl are the roots
Enl =~2
2ma2β2nl
βn1 βn2 βn3
j0 π 2π 3πj1 4.493 7.726 10.904j2 5.762 9.906 12.325j3 6.988 10.420 13.698
The wavefunctions are thus:
ψnlm(r , θ, φ) = Anl jl
(βnl r
a
)Y m
l (θ, φ)
these are (2l + 1)-fold degenerate states, that is, the energy does notdepend on the quantum numbers which give rise to the degeneracy
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 26 / 26
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Solutions for all l
jl (ka) = 0
k =1
aβnl , βnl are the roots
Enl =~2
2ma2β2nl
βn1 βn2 βn3
j0 π 2π 3πj1 4.493 7.726 10.904j2 5.762 9.906 12.325j3 6.988 10.420 13.698
The wavefunctions are thus:
ψnlm(r , θ, φ) = Anl jl
(βnl r
a
)Y m
l (θ, φ)
these are (2l + 1)-fold degenerate states, that is, the energy does notdepend on the quantum numbers which give rise to the degeneracy
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Solutions for all l
jl (ka) = 0
k =1
aβnl , βnl are the roots
Enl =~2
2ma2β2nl
βn1 βn2 βn3
j0 π 2π 3πj1 4.493 7.726 10.904j2 5.762 9.906 12.325j3 6.988 10.420 13.698
The wavefunctions are thus:
ψnlm(r , θ, φ) = Anl jl
(βnl r
a
)Y m
l (θ, φ)
these are (2l + 1)-fold degenerate states, that is, the energy does notdepend on the quantum numbers which give rise to the degeneracy
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Solutions for all l
jl (ka) = 0
k =1
aβnl , βnl are the roots
Enl =~2
2ma2β2nl
βn1 βn2 βn3
j0 π 2π 3πj1 4.493 7.726 10.904j2 5.762 9.906 12.325j3 6.988 10.420 13.698
The wavefunctions are thus:
ψnlm(r , θ, φ) = Anl jl
(βnl r
a
)Y m
l (θ, φ)
these are (2l + 1)-fold degenerate states, that is, the energy does notdepend on the quantum numbers which give rise to the degeneracy
C. Segre (IIT) PHYS 405 - Fall 2019 October 03, 2019 26 / 26