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Lecture 12
Particle 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 onedimensional 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:
2

2
ˆ
H  
   E
2m
In the Cartesian (xyz) coordinates, the ‘del
squared’ is
2
2
2



2
  2 2 2
x y z
What is it in spherical coordinates?
The particle on a sphere

The answer is:
2
2


2

1
1

1 
 
2

  2
 2  2

sin 
2
r
r r r  sin  
sin  
 

The derivation is analogous to
that for cylindrical coordinates.
You are invited to derive!
The particle on a sphere

The value of r is held fixed. The derivatives
with respect to r vanish.
2 2  1  1 2
1 
 

  2
 2  2

sin 
2
r
r r r  sin  
sin  
 
2

The Schrödinger equation is
2  1 2
1 
 
 2
  E


sin 
2 
2
2mr  sin  
sin  
 
Two variables θ and φ.
The particle on a sphere

Two variables – let us try the separation of
variables technique.
 ( ,  )  ( )( )

Substituting
2  1 2
1 
 
 2
  E


sin 
2 
2
2mr  sin  
sin  
 
For the separation to take place, we must be
able to cleanly separate the equation into two
parts, each depending on just one variable.
The particle on a sphere

The differentiation with respect to θ for
example acts on Θ alone. Therefore
 2    2
 
 
 2
  E


sin 
2 
2
2mr  sin  
sin  
 

Dividing the both sides by ΘΦ
æ 1
¶ 2F
1
¶
¶Q ö
+
sin q
=E
÷
2ç
2
2
Qsin q ¶q
¶q ø
2mr è Fsin q ¶f
2
The particle on a sphere

Multiplying both by sin2θ
 2  1  2 sin  
 
2




sin


E
sin

2 
2

2mr   
 
 
Function of φ

Function of θ
Function of θ
Subtracting the RHS from both sides
æ 1 ¶ 2F sin q ¶
¶Q ö
2
+
sin
q
E
sin
q =0
÷
2ç
2
Q ¶q
¶q ø
2mr è F ¶f
2
Function of φ
Function of θ
Constant
The particle on a sphere

Two independent one-dimensional
equations!
These two
parts of the
equation
must be
constant.

 2 1  2

 E
2
2
2mr  
 2 sin  

2

sin


E
sin
   E
2
2mr  

We have already solved the first equation.
 2  2 ml2 2
iml



;


e
2mr 2  2 2mr 2
The particle on a sphere
2  1 2
1 
 
 2
  E


sin 
2 
2
2mr  sin  
sin  
 
Separation of variables
 1 
 2 sin  

2

sin


E
sin
   E


E

2
2
2
2mr  

2mr  
2
2
2D rotation, ml is introduced
2 2
m
 
iml
l



;


e
2mr 2  2 2mr 2
2
2
Custom-made
Spherical harmonics
Custom-made solutions
Associated Legendre polynomials
l and ml are quantum numbers
 ( ,  )  ( )( )
The particle on a sphere

To summarize: the Schrödinger equation is
2  1 2
1 
 
 2
  E


sin 
2 
2
2mr  sin  
sin  
 

The eigenfunctions are spherical harmonics
specified by two quantum numbers l (= 0, 1, 2,
…) and ml (= –l, … l), having the form
 ( ,  )  N lml Ylml ( ,  )  N lml lml ( ) ml ( )
Normalization
Spherical harmonics
Associated
Legendre
eim φ
l
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:
l (l  1) 2
El 
; l  0,1,2,
2
2mr
Out of this, the energy arising from the φ
2 2
rotation is
ml 
Eml 
2mr 2
The latter cannot exceed the former.
ml = l, l -1,… , 0,… , -l +1, -l
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 2
m
 2 sin  

2
l

sin


E
sin



2mr 2  

2mr 2

 2 sin  
 ( sin  )  2
2

sin


sin

2
2
2mr  sin  

mr
2 
2
2

(

sin

cos

)

sin

2
2
2mr 
mr
2
2
2

2
2
2

(sin


cos

)

sin

2
2
2mr
2mr
2 2
m
2
2
2
l 
2

(

sin


cos

)


2
2

2mr
2mr E 
l (l  1)
2
2mr
Spherical harmonics
This is a breathing
mode of a bubble
This is a a-candy-inmouth 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).