Transcript Angular momentum

Orbital Angular Momentum
• In classical mechanics, conservation of angular
momentum L is sometimes treated by an effective
(repulsive) potential
2
L
2m r2
• 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
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Orbital Angular Momentum
• Look at the quantum mechanical angular
momentum operator (classically this “causes” a
rotation about a given axis)
  
Lrp


p   i 
f
z
 cosf

 sin f
 0

 sin f
cosf
0
0

0
1 
• look at 3 components
Lx  ypz  zp y  i( y z  z y )
Ly  zp x  xpz  i( z x  x z )
Lz  xpy  ypx  i( x y  y x )
• operators do not necessarily commute
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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Polar Coordinates
• Write down angular momentum components in
polar coordinates (Supp 7-B on web,E&R App M)
Lx  i(sin f   cot cosf f )
Ly  i( cosf   cot sin f f )
Lz  i f
• and with some trig manipulations
L   [
2
2

1
sin  
(sin 


)
2
1
sin 2  f 2
]
• but same equations when solving angular part of
S.E. and so
L2zYlm  L2z lm m  ml2 2Ylm 
L Ylm   [
2
2

1
sin  
(sin


ml2
)  sin 2  ]Ylm
 l (l  1) Ylm
2
• 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
 0, any indices sam e,  1, all different
• since they do not commute only one component of
L can be an eigenfunction (be diagonalized) at any
given time
• but there is another operator that can be
simultaneously diagonalized
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 )
Ly ( Ly Lz )   Ly (  Lx  Lz Ly )
( Lz Lx ) Lx  ( Ly  Lx Lz ) Lx
( 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
non-Abelian
AB  BA
• 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
• If represent each term by a number, and group
combination is normal multiplication
11
ai
b  1
a  a  i  i  1  b
c  i
• or can represent by matrices and use normal matrix
multiplication
 1 0
 0  1
 1 0 
 0 1
, a  
, b  
, c  

1  
 0 1
1 0 
 0  1
 1 0
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Continuous (Lie) Group:Rotations
• Consider the rotation of a vector
    r '  Rr
r' f r r
 
| r ' || r | length sam e
   
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)
• O(3) is non-Abelian R(f ) R( )  R( ) R(f )
• assume angle change is small
 cosf

Rz (f )   sin f
 0

 1

 R   fz
 f
y

 sin f
cos
0
 fz
1
fx
0  1
 
0   f
1   0
f
1
0
0

0
1 
fy 

 f x  sm all 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 )



1 
 ( Rr )   (r ) or  (r )   ( R r )


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 )

  i 

  (r )  f  (  r  p) (r )
 
i
U R  1  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 

 
i
 fx  U R  1   f  L
1 

• one gets a representation for angular momentum
0

Lx  i   0
0

0
0
1
0 
 0


 1 L y  i 0
 1
0 

0
0
0
1
0


0  Lz  i   1
0
0 

1
0
0
0

0
0 
[Li , L j ]  iijk Lk
• satisfies Group Algebra
• Another group SU(2) also satisfies same Algebra.
2x2 Unitary transformations (matrices) with det=1
(gives S=special). SU(n) has n2-1 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)
 0 1
0  i
1 0 


 Ly  2 
 Lx  2 

Lx  
 1 0
i 0 
 0  1

2
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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 positive-definite. the “length” of the zcomponent 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
• Define raising and lowering operators (ignore
Plank’s constant for now). “Raise” m-eigenvalue
while keeping l-eiganvalue fixed
for SU ( 2) m atrices
L  Lx  iLy
0
L  
1
0
1
L  2 
1
1
2
1 i  0
  2 
0
i
1 i  0
 
0  2  i
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 i 0
  
0  0
 i 0

0   1
1

0
0

0 
11
Eigenvalues “Group Theory”
• operates on a 1x2 “vector” (varying m) raising or
lowering it
0
L  
1
0
1
L  2 
1
1
2
1
 
0
0

i
1 i  0
  2 
0
i
i
2
 i 0
  
0  0
1

0
 i 0
  
0  1
0

0
1
s  , ms       
 0
 0
s  12 , ms   12      
1
1
2
L     
1
2
L    0
 0 1  0   1   0 1  1   0 

     , 
    
 0 0  1   0   0 0  0   0 
L    0
L     
 0 0  0   0   0 0  1   0 

     , 
    
 1 0  1   0   1 0  0   1 
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• Can also look at matrix representation for 3x3
orthogonal (real) matrices
• Choose Z component to be diagonal gives choice of
matrices
 1
 1
 
 
Lz m  m m  Lz  0   1 0 ,
 0
 0
 
 
L  Lx  iLy
1

Lz   0
0

0

0 
 1
0
0
0
 0  0
 0
 0
   
 
 
Lz  1   0 1 , Lz  0   1 0 
 0  0
 1
 1
   
 
 
• can write down (need sqrt(2) to normalize)
 0 1 0


L  2  0 0 1 
 0 0 0


 0 0 0


L  2  1 0 0 
 0 1 0


 0 1 0


Lx  12 ( L  L )  12  1 0 1 , Ly  2i ( L  L ) 
 0 1 0


0  i 0 


1
i
0

i


2
0 i
0 

 1 0 1
 1 0  1  1 0 0 



 

L  L  L  L  12  0 2 0   12  0 2 0    0 0 0 
 1 0 1
 1 0 1   0 0 1



 

 2 0 0


  0 2 0   2  Identity[l (l  1)  1* 2]
 0 0 2


2
2
x
2
y
2
z
• and then work out X and Y components
 1  0  0  1  0  0
           
L  0    0 , L  1    0 , L  0    1 
 0  0  0  0  1  0
           
L  L Li
2
i
T
i
 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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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  iLy L2  L L  L2z  Lz
[ Lz , L ]   L
[ L2 , L ]  0 [ L , L ]  2Lz
• Assume if g is eigenfunction of Lz and L2. ,L+g is
also an eigenfunction
L2 ( L g )  L ( L2 g )  l ( L g ) (com m ute)
Lz ( L g )  ( Lz L  L Lz ) g  L Lz g
 L g  L mg  (m  1)( L g )
• new eigenvalues (and see raises and lowers value)
 m  
for operatorsL
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Eigenvalues
• There must be a highest and lowest value as can’t
have the z-component be greater than the total
L gH  0 L gL  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)
• repeat for the lowest state
Lz g L  l g L  l   2l (l  1)
equatel  l (l  1)  l (l  1)  l  l
• eigenvalues of Lz go from -l to l in integer steps (N
steps)
N
l 
 int eger or half  int eger
2
l  0, 12 ,1, 32 .......(SU (2) only)
m  l ,l  1,l  2.....l  1, l (2l  1 term s)
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Raising and Lowering
Operators
• can also (see Gasior,Schiff) determine eigenvalues
by looking at
L l m  C (l , m) l m  1
L l
m  C ( l , m ) l
m 1
• and show
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
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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
 Ylm
coordinates
LY 
 mY
z lm
i f
LYlm  eif (
• solve first
lm


 i cot )Ylm

f
Ylm  F ( )eimf
• and insert this into the second for the highest m
state (m=l)


if
L l , l  0  0  e (
 i cot

f


 eif (
 i cot
) F ( )eimf

f

 eif eilf (
 i cot (il ))F ( )


 ei ( l 1)f (
 l cot ) F ( )

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)Yll
17
Eigenfunctions in spherical
coordinates
• solving
e
i ( l 1)f

(
 l cot  ) F ( )  0

• gives
F ( )  (sin )l
 Yll  Aeilf (sin )l
• then get other values of m (members of the
multiplet) by using the lowering operator


L  e ( 
 i cot )

f
if
LYll   (l  m)(l  m  1)Yll1
• will obtain Y eigenfunctions (spherical harmonics)
also by solving the associated Legendre equation
• note power of l: l=2 will have
sin  ; cos sin ; cos 
2
2
P460 - angular momentum
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