quantum mechanics course zeemansplitting
TRANSCRIPT
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• Hydrogen Atom: 3D Spherical Coordinates
– Ψ = (spherical harmonics)(radial) and probability density P
– E, L2, Lz operators and resulting eigenvalues
• Angular momenta: Orbital L and Spin S
– Addition of angular momenta
– Magnetic moments and Zeeman effect
– Spin-orbit coupling and Stern-Gerlach (Proof of electron
spin s)
• Periodic table
– Relationship to quantum numbers n, l, m
– Trends in radii and ionization energies
Zeeman Effect
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( ) ( ) ( ) ( )xExxVdx
xd
mψψ
ψ=+−
2
22
2
ℏ
Kinetic energyPotential
energyTotal energy
Schrödinger Equation: Coordinate Systems
( ) ( ) ( ) ( )xExzyxVxzyxm
ψψψ =+
∂
∂+
∂
∂+
∂
∂− ,,2 2
2
2
2
2
22ℏ
( )2 2 2
2
2 2 2 2
1 1 1sin
2 2 sin sinr V r
r r r r
E
ψ ψ ψθ ψ
µ µ θ θ θ θ φ
ψ
∂ ∂ ∂ ∂ ∂ − − + + ∂ ∂ ∂ ∂ ∂ =
ℏ ℏ
1D Cartesian
3D Cartesian
3D Spherical
Kinetic energy Potential energy Total energy
Kinetic energy Potential
energy
Total energy
Convert to spherical using: sin cos , sin sin , cosx r y r z rθ φ θ φ θ= = =
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Hydrogen Atom: 3D Spherical Schrödinger Equation
( ) ( ) ( )φθφθψ ,,, lmnnlm YrRr =
Laguerre
Polynomials
Spherical
Harmonics
( ) ( ) ( )QrEQrVQrp
neff ,,,,,,2
ˆ 2θψθψθψ
µ=+
∂∂
∂∂
−=r
rrr
p 2
2
22 1ˆ where ℏ
22
0
2
0
2where
1 13.6
2 n
keE eVE
z
n
Eµ= ≈
− =
ℏEigenvalues:
Eigenfunctions:
“Rewritten” Schrodinger Eqn.:
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Hydrogen Atom: 3D Spherical Schrödinger Equation
3 Quantum Numbers (3-dimensions)
n = energy level value (average radius of orbit)
n = 1, 2, 3 …
l = angular momentum value (shape of orbit)
l = 0, 1, 2, … (n – 1)
m = z component of l (orientation of orbit)
m = -l,(-l + 1) ..0,1,2,..+l
How many quantum states (n, l, m) exist for n = 3? Is there a general
formula?
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Wave Functions : Formulas
n = 1
n = 2
l = 1
l = 0
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m = –2 m = –1 m = 0 m = 1 m = 2
Wave Functions: Angular Component
cosθsinθ sinφ sinθ cosφ
3cos2θ–1sinθcosθsinφ
sinθcosθcosφ
sin2θ sin2φ sin2θ cos2φ
s-orbital
p-orbitals
d-orbitals
l = 0
l = 1
l = 2
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Wave Function: Radial
Component
What is the relationship between the number of zero crossings for the radial
component of the wave function and the quantum numbers n and l?
http://cwx.prenhall.com/bookbind/pubbooks/giancoli3/chapter40/multiple3/deluxe-content.html
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Wave Functions: Angular & Radial Components
200 211 211210
100
300 311 311310 321 321320322 322
400 411 411410 421 421420422 422
431 431430432 432 433433
From: http://pcgate.thch.uni-bonn.de/tc/people/
hanrath.michael/hanrath/HAtomGifs.htmll = 1 p-orbitals
l = 2 d-orbitals
l = 0 s-orbitals
l = 3 f-orbitals
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Probability Density: Formula
2
2
0 0 0
( , , )
sin
P r dV
r d d dr
π π
ψ ψ θ φ
ψ ψ θ ϕ θ
∗
∞∗
=
=
∫
∫ ∫ ∫
ψψψπ symmetricy sphericallfor 4 2
0
drrP ∫∞
∗=
⇒For small ∆r, can use P = P(r) ∆ r (analogous to 1D case)
( ) ( )20
( ) where 4P P r dr P r rπ ψ ψ∞
∗= =∫
Volume Element dV
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Probability Density: “Density” Plots
p-orbitals
200
2ψ 210
2ψ 21 1
2ψ ±
“Dumbbells”“Spherical Shells”
n = 2 n = 2
l = 1
s-orbital
l = 0
m = 0
m = 1
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Probability Density: Cross Sections
Can you draw the radial probability functions for the 2s to 3d wave functions?
http://webphysics.davidson.edu/faculty/dmb/hydrogen/default.html
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Probability Density: Cross Sections
Rank the states (1s to 3d) from smallest to largest for the electron’s most
PROBABLE radial position.
For which state(s) do(es) the most probable value(s) of the electron's position
agree with the Bohr model?
http://cwx.prenhall.com/bookbind/pubbooks/giancoli3/chapter40/multiple3/deluxe-content.html
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Probability Density: Problem
[ ]
[ ]
22100 100
22
100 1001.5 3
2100 3
10
10
0
01
( ) (4 ) ( )
( ) ( )
( ) ( ) (0.05 )
, ,
(( ) 0.00.05 8) 1
o o
o o
r a r a
o o
a a
o o
o
o
o
P r r r r r
e ewhere r and r
a a
eP a r a a
a
after substitution o
P a
f r and r
er
π
π π
ππ
− −
−
−
∆ = Ψ ∆
Ψ = Ψ =
∆ =
Ψ ∆
=∆ =
For the ground state n = 1, l = 0, m = 0 of hydrogen, calculate the
probability P(r)∆r of finding the electron in the range ∆r = 0.05aoat r = a
o/2
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Orbital Angular Momentum L:
Related to Orbital “Shape”
• Magnitude of Orbital Angular Momentum L
( ) ( )
( )
2 2 2
2 2
1 1ˆ , , sin 1sin sin
Eigenvalues: 1
L r l l
L l l
ψ θ φ θ ψ ψθ θ θ θ φ
∂ ∂ ∂ = − + = + ∂ ∂ ∂
= +
ℏ ℏ
ℏ
( )ˆ , ,
Eigenvalues:
z
z
L r i m
L m
ψ θ φ ψ ψφ∂
= − =∂
=
ℏ ℏ
ℏ
• Z-component of L
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Orbital Momentum L: Vector Diagram
For l = 2, find the magnitude of the angular momentum L and the possible m
values. Draw a vector diagram showing the orientations of L with the z axis.
( +1) 2(2+1)
6 2.45
2
0, 1
0, 1 , 2
2
z
L l l
L or
l to l
L
m
m
l
= =
=
= − =
= = ± ±
=
± ±
ℏ ℏ
ℏ ℏ
ℏ ℏ ℏ
z
2h
24º
L = √√√√6h = 2.45h
m = 1
m = 0
m = −−−−1
−−−−2h
h
−−−−h
55º
m = 2
m = −−−−2
Can you draw the vector diagram for l = 3? For j = 3/2?
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Spin Angular Momentum S:
Property of Electron
• Magnitude of Spin Angular Momentum S
( )Eigenvalues:
1 1 3For one electron: 1
2 2 4
1
S
S s s
= + =
= +
ℏ ℏ
ℏ
Eigenvalues:
For one electron: 2
z
z sS m
S =
=ℏ
ℏ
• Z-component of SExample for s = 1/2
z s = √√√√0.75h
m = 1/2
m = −−−−1/2
h/2
-h/2
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• Orbital angular momentum L and spin angular momentum s of
electrons result in magnetic moments µl and µs.
Angular Momentum: Link to Magnetic Moments
( )
z-compone
1
nt
l
l
L B
L
z l L
B
B
L
m g
gl l gµ
µµ
µ µ
=
= −
−= +
�
ℏ
�
Orbital l = 0, 1, 2, …
( )
z-component
1s
sz s s B
s B
s B
B
gss
m g
s gµ
µ µ
µ
µ
µ
= ±
−= = +
− ≈ℏ
� �
Spin: s = ½
where eV55.79 10 and , gyromagnetic ratios
2 TB L s
e
eg g
mµ −= = × =
ℏ�
( )2Remember that where
2 2 2
qv q qr vr L mvriA
r mπ
π → = = =
=
Lµ���
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Zeeman Effect: Splits m values
• Orbital magnetic moment µL interacts with an external magnetic field B and separates degenerate energy levels.
ext lzU B U Bµ µ= − ⋅ ⇒ = −��
assume z
direction
Different energies for
different ml values!
NO B
FieldB Field
l = 1
l = 0
m = 0
m = 1
m = –1
m = 0
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“Anomalous” Zeeman Effect: More Lines??
l = 0
ml = 0
ml = 1
ml = –1
l = 1
????Zeeman
ml = 0
Why are there more energy levels than expected from the Zeeman effect?
Add external
magnetic field
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Spin-Orbit Coupling: Splits j values
BBBU Bszs µµµ ±=−=⋅= int
��BU Bµ2=∆
• Electron’s spin magnetic moment µs interacts with internal B field caused by its orbital magnetic moment µl and separates energy levels.
Spin up:
High Energy
vp+
L, BlS
µs
e-
vp+
L, Bl
µs
e-
Spin down:
Low Energy
S
l = 1
l = 0
j = 3/2
j = 1/2
j = 1/2
s = 1/2
s = 1/2
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Angular Momentum Addition: L + S gives J
• Special Case:
Example: l = 1, s = ½
( )ℏ�
���
1+=
+=
jjJ
SLJ
Quantum Numbers
,
, 1,... 1,j
j l s l s
m j j j j
= + −
= − − + −
Vectors
SL��
+
21
21
23
21
21
23
21
21
23
21
, ,,,
11
and
and
−=−−=
=−==+=
jj mm
jj
j = 3/2 j = 1/2
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“Anomalous” Zeeman Effect: Spin-Orbit + Zeeman
• Quantum numbers mj (j-j coupling) for HIGHER Z elements.
j = 3/2
j = 1/2
j = 1/2
mj = 1/2
mj = 3/2
mj = –1/2
l = 1mj = –3/2
mj = 1/2
mj = –1/2
mj = 1/2
mj = –1/2
Spin-Orbit Zeeman
s = 1/2
l = 0 s = 1/2
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Angular Momentum Addition: General Rules
21 JJ��
+
( ) ( )jjjjm
jjjjjjj
j ,1,...1,
,...1, 212121
−+−−=
−−++=
( )ℏ�
���
1
21
+=
+=
jjJ
JJJ
tot
tot
Example: j1 = 3/2, j2 = 3/2
• General Case:Quantum NumbersVectors
3 3 3 3max min2 2 2 2
and
for
3 0
3, 2, 1, 0
3, 2, 1, 0,1, 2, 3 3j
j j
j
m j
= + = = − =
=
= − − − =
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Stern-Gerlach Experiment: ALSO splits m values
• A magnetic force deflects atoms up or down by an amount that depends on its magnet moment and the B field gradient.
• For hydrogen (mlz = 0), two lines are observed (spin up, spin down).
– Since l = 0, this experiment gave direct evidence for the existence of spin.
=dz
dBF zz µ
Case of ml = -1, 0, 1
What is the “story” of this experiment? Did Stern & Gerlach know about
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The angular momentum of the yttrium atom in the ground state is
characterized by the quantum number j = 5/2. How many lines would
you expect to see if you could do a Stern-Gerlach experiment with
yttrium atoms?
Remember that in the Stern-Gerlach experiment all of the atoms with
different mj values are separated when passing through an
inhomogeneous magnetic field, resulting in the presence of distinct
lines.
How many lines would you expect to see if the beam consisted of
atoms with l = 1 and s = 1/2?
Stern-Gerlach Experiment: Problem
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Multi-electron Atoms
• Schrödinger equation cannot be solved exactly for multi-electron
atoms because Coulombic repulsion “mixes” variables.
– Estimate energies using single-electron wave functions and
“correcting” energies with 1st-order perturbation theory.
• Orbitals “fill” in table as follows: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p . .
– Only ONE electron per state (n, l, ml , ms ) - Pauli Exclusion Rule!
– Why is 4s filled before 3d? ⇒ 4s orbital has a small bump near
origin and “penetrates” shielding of core electrons better than 3d
orbital, resulting in a larger effective nuclear charge and lower
energy.
12
2
intrr
keV ��
−=
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Periodic Table
l = 1 (p)
l = 0 (s)
l = 2 (d)
l = 3 (f)
1
2
3
4
5
6
7
Noble Gas
Halogen
Alkali
Group VI
Group IV
Group V
Group III
n
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• Effective atomic radii decrease
across each row of table.
– Why? Effective nuclear charge
increases and more strongly
attracts outer electrons,
decreasing their radius.
Periodic Table: Trends for Radii and Ionization Energies
• Ionization energies increaseacross each row of table until the complete “shell” is filled.
– Alkali atoms easily give up s-orbital electrons.
– Halogens have strong affinity for outer electrons.
Maxima = noble gases
Maxima = alkali metals
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APPENDIX: Wave Functions Formulas
Complete Wave Function ψn,l,m
l = 1m = ±1
m = ±1
s-orbital
p-orbital
d-orbital
f-orbital
Spherical Component Yl,m
l = 0n = 1
n = 2
n = 3
l = 0,1
l = 0,1,2
l = 1
l = 0
l = 2
l = 3