Angular momentum
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Transcript Angular momentum
Orbital Angular Momentum
• In classical mechanics, conservation of angular
momentum L is sometimes treated by an effective
(repulsive) potential
L2
2m r2
• Rewriting the R term in Schr. Eqn. We see an
angular momentum term which arises from the
Theta equation’s separation constant
1 d
2 m r 2 dr
2
(
2
r dR
dr
)
l ( l 1) 2 R
2 mr
2
Ze 2 R
4 0 r
ER
• eigenvalues for L2 (its expectation value) are
l (l 1) 2
• the spherical harmonics are also eigenfunctions of
this operator and of Lz
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Orbital Angular Momentum
• Look at the quantum mechanical angular
momentum operator (classically this “causes” a
rotation about a given axis)
Lrp
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
)
)]
) iLz
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Polar Coordinates
• Write down angular momentum components in
polar coordinates (E&R App M or Griffiths 4.3.2)
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 ] iLz , [ L y , Lz ] iLx , [ Lz , Lx ] iL 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
2
2
2
2
L Lx Ly Lz
[ 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 commutaion 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
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ai
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 ] iijk 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.
• 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)
for SU ( 2) m atrices
L Lx iLy
0
L 12
1
0
1
L 2
1
1 i 0
2
0
i
1 i 0
2
0
i
i 0
0 0
i 0
0 1
1
0
0
0
• and operates on a 1x2 “vector” (varying m) raising
or lowering it
1
0
0
0
1
1 0 1 0
,
0 1 0 0
0 0 0 0
,
0 1 0 1
1 1 0
0 0 0
0 1 0
0 0 1
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s 12 , ms 12
0
0
1
1
s 2 , ms 2
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 0 0
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Eigenvalues
• Done in Griffiths 4.3.1. (also Schiff QM)
2.
• Start with two diagonalized
operators
L
and
L
z
lm L2 l m l ll mm
lm Lz l m m ll mm
• where m and l are not yet known
• Define raising and lowering operators and easy to
work out some relations
L Lx Ly L2 L L L2z Lz
[ Lz , L ] L [ L2 , L ] 0 [ L , L ] 2Lz
• Assume if g is eigenfunction of Lz and L2. it is also
eigenfunction of L+L2 ( L g ) L ( L2 g ) l ( L g ) (commute)
Lz ( L g ) ( Lz L L Lz ) g L Lz g
L g L m g (m )(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 g H 0
L g L 0
• For highest state, let l be the maximum eigenvalue
Lz g H lg 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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