lecture 12 particle on a sphere
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Lecture 12Particle on a sphere
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
The particle on a sphere
Main points: the particle on a sphere leads to a two-dimensional Schrödinger equation and we must use the separation of variables. One of the resulting one-dimensional equations is the particle on a ring. For the other, we seek mathematicians’ help: we introduce associated Legendre polynomials. The product of these and the particle on a ring eigenfunctions are spherical harmonics.
The particle on a sphere
The particle on a sphere
The potential is zero – only kinetic energy:
In the Cartesian (xyz) coordinates, the ‘del squared’ is
What is it in spherical coordinates?
Em
H 22
2ˆ
2
2
2
2
2
22
zyx
The particle on a sphere
The answer is:
The derivation is analogous to that for cylindrical coordinates. You are invited to derive!
sinsin
1
sin
1122
2
222
22
rrrr
The particle on a sphere
The value of r is held fixed. The derivatives with respect to r vanish.
The Schrödinger equation is
sinsin
1
sin
1122
2
222
22
rrrr
Emr
sinsin
1
sin
1
2 2
2
22
2
Two variables θ and φ.
The particle on a sphere
Two variables – let us try the separation of variables technique.
Substituting
For the separation to take place, we must be able to cleanly separate the equation into two parts, each depending on just one variable.
)()(),(
Emr
sinsin
1
sin
1
2 2
2
22
2
The particle on a sphere
The differentiation with respect to θ for example acts on Θ alone. Therefore
Dividing the both sides by ΘΦ
Emr
sinsinsin2 2
2
22
2
The particle on a sphere
Multiplying both by sin2θ
Subtracting the RHS from both sides
Function of φ Function of θ Constant
22
2
2
2
sinsinsin1
2E
mr
Function of φ Function of θ Function of θ
The particle on a sphere
Two independent one-dimensional equations!
We have already solved the first equation.
Emr 2
2
2
2 1
2
EEmr
22
2
sinsinsin
2
liml e
mr
m
mr
;22 2
22
2
2
2
2
These two parts of the equation must be constant.
The particle on a sphere
Emr 2
2
2
2 1
2
EEmr
22
2
sinsinsin
2
liml e
mr
m
mr
;22 2
22
2
2
2
2
Emr
sinsin
1
sin
1
2 2
2
22
2
Separation of variables
2D rotation, ml is introduced
Custom-made solutionsAssociated Legendre polynomials
l and ml are quantum numbers
)()(),( Custom-madeSpherical harmonics
The particle on a sphere
To summarize: the Schrödinger equation is
The eigenfunctions are spherical harmonics specified by two quantum numbers l (= 0, 1, 2, …) and ml (= –l, … l), having the form
Emr
sinsin
1
sin
1
2 2
2
22
2
)()(),(),( lllll mlmlmlmlm NYN
Normalization
Spherical harmonics
AssociatedLegendre eimlφ
The particle on a sphere
Some low-rank spherical hamonics are given on the right.
Spherical harmonics are orthogonal functions. They as fundamental to spherical coordinates as sin and cos to Cartesian coordinates.
Spherical harmonics
Spherical harmonics are the standing waves of a sphere surface (e.g., soap bubble, earthquake).
Imagine a floating bubble. It vibrates – the amplitudes of the vibration is a linear combination of spherical harmonics.
GNU Image from Wikipedia
The particle on a sphere
The total energy is determined by the quantum number l only:
Out of this, the energy arising from the φ rotation is
The latter cannot exceed the former.
,2,1,0 ;2
)1(2
2
lmr
llEl
2
22
2mr
mE lml
The particle on a sphere
Parameter l is called the orbital angular momentum quantum number.
Parameter ml is the magnetic quantum number.
Energy is independent of ml. Therefore, a rotational state with l is (2l +1)-fold degenerate because there are (2l +1) permitted integers ml can take.
The particle on a sphere Let us verify that the associated Legendre polynomial is
indeed the solution for l = 1 and ml = 1.
2
2222
2
2
22
222
2
2
22
2
2
2
22
2
2
2
2)cossin(
2
sin2
2)cos(sin
2
sin)cossin(2
sin)sin(
sinsin
sin
2
mr
m
mr
mrmr
mrmr
mrmr
l
2
222
2
2
2sinsin
sin
2 mr
mE
mrl
)1(2 2
2
llmr
E
Spherical harmonics
This is a breathing mode of a bubble
This is a a-candy-in-mouth mode of a
bubble
This is an accordion mode of a bubble
Summary The spherical harmonics are the most
fundamental functions in a spherical coordinates.
We have encountered a differential equation whose solution involves associated Legendre polynomials.
The eigenfunctions of the particle on a sphere are spherical harmonics and characterized by two quantum numbers l and ml.
The energy is determined by l only and is proportional to l(l + 1).