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).