Quantum angular-momentum
operators
Vector definitions
Physical Chemistry
Lecture 15
Angular momentum and the rigid
rotor
L  Lx i  L y j  Lz k
L2
 L  L  L2x  L2y  L2z
Expression by correspondence
Lx
L2
 
 
  i y  z 
y 
 z
 L2x  L2y  L2z

 
  i  z  x 
z 
 x
Ly
Lz
 

  i x  y 
x 
 y
Form of operators with a fixed r
L   ir  
L2    2 r     r   
Angular momentum
Vector property that
describes circular
motion of a particle or a
system of particles
Rigid rotor model: A
particle of mass m fixed
to a massless rod
Examples
 Swinging a bucket of
water
 Movement of the
Earth around the Sun
Quantum angular momentum
Commutators of operators
L , L 
L , L 
x
y
 iLz
i
 0
2
and cyclic permutations
Can have common set of eigenstates of L2 and
any one component
L  rp
Classical constant-angularmomentum problem
Solve for trajectories for constant angular momentum
Frequency, , must be constant
r must be constant
Constant L is provided by the fact that r and  are
constant
L  constant  mr 2ω k
r (t )  r i cos t  j sin t 
p(t )  mr i sin t  j cos t 
L2 km
 k 2 km
Lz km
 mkm
Operators in spherical coordinates
Natural system for
describing angular
motion is spherical coordinates
Lz depends only on 

Suggests that the wave
functions may be
written as a product
Lx
Ly


 

 i sin 
 cot  cos 

 



 

  i cos 
 cot  sin 

 



Lz
  i
L2
 2

1 2 

   2  2  cot 

 sin 2   2 
 
km ( ,  )   km ( ) m ( )
1
Grotrian diagram for the rigid
rotor
Differential equations for angularmomentum eigenstates
The z component yields
a simple differential
 m
equation for m
 i
 m m

The square of the
angular momentum
yields an equation for
  2  km

 km
m2
 
 cot 
 2  km  
2
km ( P(cos)
sin 

 



Legendre’s associated
differential equation
Depend on a quantum
number, 
Rigid rotor’s energies determined
by the quantum number, 
k km
Each energy level is degenerate

Ym ( ,  ) 
Am P|m| (cos ) m ( )
where
Solutions are a
complete set called the k    1 and   0, 1, 2, 
spherical harmonic
functions
States with different values of m
have the same energy
 2  1
g
Angular-momentum wave
functions
Functions of  are exponentials
 m ( ) 
1
exp(im )
2
Legendre polynomials

0
1
1
2
2
2
|m|
0
0
1
0
1
2
P|m|
1
cos
sin
3cos2 - 1
sin cos
sin2
Should look familiar, as these are the angular
parts of hydrogenic wave functions
Spin
Goudschmidt and
Uehlenbeck proposed
electronic “intrinsic angular
momentum” to explain
spectroscopic anomalies
Fundamental property of
particle called spin



H

1
L2
2mr02
The Hamiltonian commutes with L2 and Lz

The three operators have a complete set of
eigenstates in common
HYm ( ,  )  EmYm ( ,  )
1
L2Ym ( ,  ) 
2mr02
Em

2
  1
2mr02
1
  1 2Ym ( ,  )
2mr02
Proton
½
Neutron
½
Deuteron
1
Often labeled I or S
Acts like other quantum
angular momenta
Integer or half-integer values

Consequence of relativistic
motion of electron
12C
0
13C
½
23Na
½
27Al
5/2
and 65Cu
3/2
Dirac theory of an electron
Quantum rigid rotor
Hamiltonian
PRINCIPAL SPIN QUANTUM
NUMBERS OF PARTICLES
Electron
½
63Cu
Summary
Angular momentum is quantized



Rotation equation
Legendre’s differential equation
Restricted values of  and m
  must be a positive integer
 |m| must be less than or equal to 
 m must be an integer
Rigid rotor



Hamiltonian is directly proportional to L2
Same set of eigenstates
Degenerate levels
 g = 2 + 1
2