Orbital Angular Momentum
• In classical mechanics, conservation of angular momentum L is
sometimes treated by an effective (repulsive) potential
L2
2mr 2
• Soon we will solve the 3D Schr. Eqn. The R equation will have an
angular momentum term which arises from the Theta equation’s
separation constant
• eigenvalues and eigenfunctions for this can be found by solving the
differential equation using series solutions
• but also can be solved algebraically. This starts by assuming L is
conserved (true if V(r))


dL
 0  [ H , L]  0
dt
P460 - angular momentum
1
Orbital Angular Momentum
• Look at the quantum mechanical angular momentum operator
(classically this “causes” a rotation about a given axis)
f
  
Lrp


p  i
• look at 3 components
z
 cos f

 sin f
 0

Lx  yp z  zp y  i ( y
L y  zp x  xpz  i ( z
Lz  xpy  yp x  i ( x
• operators do not necessarily commute
 sin f
cos f
0

z

x

y
z
x
 y
0

0
1 

y

z
)
)

x
)
Lx L y  L y Lx  [ Lx , L y ] 
  2 [( y
(z

z

x
i 2 ( y
z
x

x

z

y
)( z
)( y
x

y

z

x
x
z

y

z
)
)] 
)  iLz
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Side note Polar Coordinates
• Write down angular momentum components in polar coordinates
(Supp 7-B on web,E&R App M)
Lx  i (sin f


L y  i (  cos f
Lz  i
 cot  cos f



f
 cot  sin f
)

f
)

f
• and with some trig manipulations
L2   2 [ sin1  (sin   )  sin12  f 2 ]
• but same equations will be seen when solving angular part of S.E. and
so
L2zYlm  L2z lm m   ml2  2Ylm 
2
1
L2Ylm   2 [ sin



(sin 


)
ml2
sin2 
]Ylm
 l (l  1) 2Ylm
• and know eigenvalues for L2 and Lz with spherical harmonics being
eigenfunctions
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Commutation Relationships
• Look at all commutation relationships
[ Lx , L y ]  iLz
[ L y , Lz ]  iLx
[ Lz , Lx ]  iL y
[ L y , L y ]  [ Lx , Lx ]  [ Lz , Lz ]  0
or [ Li , L j ]  i ijk Lk
 ijk  tensor  0any indices same
 1, all different
• since they do not commute only one component of L can be an
eigenfunction (be diagonalized) at any given time
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Commutation Relationships
• but there is another operator that can be simultaneously diagonalized
(Casimir operator)
L2  L2x  L2y  L2z
[ L2 , Lz ]  L2 Lz  Lz L2 
( L2x  L2y ) Lz  Lz ( L2x  L2y )  0
u sin g :
Lx ( Lx Lz )  Lx (  Ly  Lz Lx )
( Lz Lx ) Lx  ( Ly  Lx Lz ) Lx
Ly ( Ly Lz )   Ly (  Lx  Lz Ly )
( Lz Ly ) Ly  ( Lx  Ly Lz ) Ly
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Group Algebra
• The commutation relations, and the recognition that there are two operators that can
both be diagonalized, allows the eigenvalues of angular momentum to be
determined algebraically
• similar to what was done for harmonic oscillator
• an example of a group theory application. Also shows how angular momentum
terms are combined
• the group theory results have applications beyond orbital angular momentum. Also
apply to particle spin (which can have 1/2 integer values)
• Concepts later applied to particle theory: SU(2), SU(3), U(1), SO(10), susy,
strings…..(usually continuous)…..and to solid state physics (often discrete)
• Sometimes group properties point to new physics (SU(2)-spin, SU(3)-gluons). But
sometimes not (nature doesn’t have any particles with that group’s properties)
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Sidenote:Group Theory
• A very simplified introduction
• A set of objects form a group if a “combining” process can be defined
such that
1. If A,B are group members so is AB
2. The group contains the identity AI=IA=A
3. There is an inverse in the group A-1A=I
4. Group is associative (AB)C=A(BC)
• group not necessarily commutative
AB  BA
Abelian
AB  BA
non-Abelian
• Can often represent a group in many ways. A table, a matrix, a
definition of multiplication. They are then “isomorphic” or
“homomorphic”
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Simple example
• Discrete group. Properties of group (its “arithmetic”) contained in
Table
1
a
b
c
1
1
a
b
c
a
a
b
c
1
b
b
c
1
a
c
c
1
a
b
• Can represent each term by a number, and group combination is
normal multiplication
11
a i
a  a  i  i  1  b
b  1
c  i
• or can represent by matrices and use normal matrix multiplication
1
1 
0

0
0

,a  

1
1

 1
 1

,b  

 0
0 

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0 
 0

,c  

 1
 1

1

0

8
Continuous (Lie) Group:Rotations
• Consider the rotation of a vector


 
r' f r r


r '  Rr


| r ' || r |
length same




r' r f  r
near identity
• R is an orthogonal matrix (length of vector doesn’t change). All 3x3
real orthogonal matrices form a group O(3). Has 3 parameters (i.e.
Euler angles)
R (f ) R ( )  R ( ) R (f )
• O(3) is non-Abelian
• assume angle change is small
 cos f

Rz (f )   sin f

0

 1

 R   fz
f
y

f
cos 
 sin
0
 fz
1
fx
0
1


0   f
0
1


f
1
0
0

0
1

fy 

 f x  small angles
1


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Rotations
• Also need a Unitary Transformation (doesn’t change “length”) for
how a function is changed to a new function by the rotation


 ( r ) changes to  ( r )




 ( Rr )   ( r ) or  ( r )   ( R 1r )


U R (f ) ( r )   ( r )
unitary



  (r  f  r )
• U is the unitary operator. Do a Taylor expansion








 ( r  f  r )   ( r )  (f  r )   ( r )





i
  (r ) 
(f  r )  p ( r )






  ( r )  f  ( i r  p ) ( r )


U R  1  i f  L
• the angular momentum operator is the “generator” of the infinitesimal
rotation
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• For the Rotation group O(3) by inspection as:

1

R   fz
f
y

 fz
1
fx
fy
 fx
1


 U R  1


i


f

 L
• one gets a representation for angular momentum (notice none is
diagonal; will diagonalize later)
Lx
0

 i 0
0

Lz
0

 i 1
0

0 
 0


 1 L y  i  0
1
0 


0
0
1
1
0
0
0
0
0
1

0
0

0

0
0

• satisfies Group Algebra
[ Li , L j ]  i ijk Lk
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• Group Algebra
[ Li , L j ]  i ijk Lk
• Another group SU(2) also satisfies same Algebra. 2x2 Unitary
transformations (matrices) with det=1 (gives S=special). SU(n) has n21 parameters and so 3 parameters
U U  1
• Usually use Pauli spin matrices to represent. Note O(3) gives integer
solutions, SU(2) half-integer (and integer)
Lx 

2
Lz 

2
0

1

1

0

1
Ly 


0
0 

 1


2
0

i

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 i

0 

12
Eigenvalues “Group Theory”
• Use the group algebra to determine the eigenvalues for the two
diagonalized operators Lz and L2 Already know the answer
• Have constraints from “geometry”. eigenvalues of L2 are positivedefinite. the “length” of the z-component can’t be greater than the
total (and since z is arbitrary, reverse also true)
• The X and Y components aren’t 0 (except if L=0) but can’t be
diagonalized and so ~indeterminate with a range of possible values
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Eigenvalues “Group Theory”
• Define raising and lowering operators (ignore Plank’s constant for
now). “Raise” m-eigenvalue (Lz eigenvalue) while keeping leiganvalue fixed
L  Lx  iL y
for SU ( 2 ) matrices
L 
1
2
0
 
0

L 
0
 
1

1
2
0

1

1

0

0

1

0

0

1



0
i
2
0

i

 i

0 

1



0
i
2
0

i

 i

0 

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Eigenvalues “Group Theory”
• operates on a 1x2 “vector” (varying m) raising or lowering it
L
L
0
 
0

0
 
1

L     
0

0

1

0

0

0

1  0 
1
 









0  1 
0
 
L    0
0

1

0  0 
 0
 









0  1 
 0
s 
1
2
s 
1
2
1

 0
  
 
 0
1
, ms   2  
1
  
 
, ms 
1
2
L    0
0

0

1  1 
 0
 









0  0 
0
 
L     
0

1

P460 - angular momentum
0  1 
 0
 









0  0 
1
15
• Can also look at matrix representation for 3x3 orthogonal (real)
matrices
• Choose Z component to be diagonal gives choice of matrices
L  Lx  iL y
Lz
1

 0
0

0
0
0
0 

0  
 1

1
 1
 
 
Lz m  m m  Lz  0   1 0 ,
 0
 0
 
 
 0
 0
 0
 0
 
 
 
 
Lz  1   0 1 ,
Lz  0   1 0 
 0
 0
 1
1
 
 
 
 
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• Can also look at matrix representation for 3x3 orthogonal (real)
matrices
• can write down L+- (need sqrt(2) to normalize) and then work out X
and Y components
L  Lx  iL y
L 
L 
0

2 0
0

1
0
0

2 1
0

0
0
0
1
0

1
0

0

0
0

Lz
1

 0
0

0
0
0
0 

0  
 1

1
 0
 0
1
 0
 0
 
 
 
 
 
 
L  0    0 , L  1    0 , L  0    1 
 0
 0
 0
 0
1
 0
 
 
 
 
 
 
1
 0
 0
 0
 0
 0
 
 
 
 
 
 
L  0    1 , L  1    0 , L  0    0 
 0
 0
 0
1
1
 0
 
 
 
 
 
 
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17
• Can also look at matrix representation for 3x3 orthogonal (real)
matrices. Work out X and Y components
L  Lx  iL y
Lz
1

 0
0

Lx 
Ly 
1
2
i
2
0
0
0
0 

0  
 1

( L  L ) 
( L  L ) 
1
1
2
0

1
0

i
1
2
0

i
0

0
1
0

1
0

0
i
P460 - angular momentum
0 

 i
0 

18
• Can also look at matrix representation for 3x3 orthogonal (real)
matrices. Work out L2
 0 1 0


Lx 
L  Lx  iL y
Lz
1

 0
0

0
0
0
0 

0  
 1

Ly 
1
2
i
2
( L  L ) 
( L  L ) 
1
2
1
2
1
0

0

i
0

0
1
i
0
i
1
0

0 

 i
0 

L2  L2x  L2y  L2z 
1

0
1

0
2
2

0
0

0
2
1
2
0
0
1
0
 1
0
0
 1
1





1
0  2  0
2
0   0
0
0
1

0

1
0
1
0
1





0

0   2  Identity [l ( l  1)  1 * 2 ]
2

L2i  LiT Li
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19
Eigenvalues
• Done in different ways (Gasior,Griffiths,Schiff)
• Start with two diagonalized operators Lz and L2.
lm LZ l m  m ll  mm
lm L2 l m  l 2 ll  mm
• where m and l are not yet known
• Define raising and lowering operators (in m) and easy to work out
some relations
L  Lx  iL y
[ Lz , L ]   L
L2  L L  L2z  Lz
[ L2 , L ]  0
[ L , L ]  2Lz
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20
Eigenvalues
• Assume if g is eigenfunction of Lz and L2. ,L+g is also an
eigenfunction
L2 ( L g )  L ( L2 g )  l ( L g )
( L2 , L commute)
Lz ( L g )  (   hbarL  L Lz ) g
  L g  L mg  ( m  1)( L g )
 m  
for operators L
• new eigenvalues (and see raises and lowers value)
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Eigenvalues
• There must be a highest and lowest value as can’t have the zcomponent be greater than the total
L g H  0
L g L  0
• For highest state, let l be the maximum eigenvalue
Lz g H  lg H (re min der : L2 g H  lg H )
• can easily show
L2 g H  ( L L  L2z  Lz ) g H
 ( 0   2l 2   2l )  l   2l (l  1)
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Eigenvalues
• There must be a highest and lowest value as can’t have the zcomponent be greater than the total
L g H  0
• repeat for the lowest state
L g L  0
L2 g L  ( L L  L2z  Lz ) g L
 ( 0   2l 2   2l )  l   2l (l  1)
Lz g L  l g L  l   2l (l  1)
equate l  l (l  1)  l (l  1)  l  l
• eigenvalues of Lz go from -l to l in integer steps (N steps)
N
 int eger or half  int eger
2
l  0, 12 ,1, 32 .......( SU ( 2) only )
l 
m  l ,l  1,l  2.....l  1, l
( 2l  1 terms)
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Raising and Lowering Operators
• can also (see Gasior,Schiff) determine eigenvalues by looking at
• and show
L l
m
 C ( l , m ) l
m 1
L l
m
 C ( l , m ) l
m 1
C ( l , m )  
( l  m )( l  m  1)
C ( l , m )  
(l  m )( l  m  1)
• note values when l=m and l=-m
• very useful when adding together angular momentums and building
up eigenfunctions. Gives Clebsch-Gordon coefficients
P460 - angular momentum
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Eigenfunctions in spherical coordinates
• if l=integer can determine eigenfunctions
Ylm ( , f )   , f l , m
• knowing the forms of the operators in spherical coordinates
 Ylm
 mYlm
i f


 eif (
 i cot 
)Ylm

f
LzYlm 
LYlm
• solve first
Ylm  F ( ) e imf
• and insert this into the second for the highest m state (m=l)


 i cot 
)Yll

f


 eif (
 i cot 
) F ( )eimf

f

 eif eilf (
 i cot  (il )) F ( )


 ei ( l 1)f (
 l cot  ) F ( )

L l , l
 0  0  eif (
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Eigenfunctions in spherical coordinates
• solving
ei ( l 1)f (

 l cot  ) F ( )  0

F ( )  (sin  ) l

Yll  Aeilf (sin  ) l
• gives
• then get other values of m (members of the multiplet) by using the
lowering operator


if
 i cot 
L  e
(
LYll  
( l  m )( l  m  1)Yll 1

f
)
• will obtain Y eigenfunctions (spherical harmonics) also by solving the
associated Legendre equation
• note power of l: l=2 will have
sin 2  ; cos  sin  ; cos2 
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