Orbital Angular Momentum
description
Transcript of Orbital Angular Momentum
P460 - angular momentum 1
Orbital Angular Momentum • In classical mechanics, conservation of angular momentum L is
sometimes treated by an effective (repulsive) potential
• 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))
2
2
2mr
L
0],[0 LHdt
Ld
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Orbital Angular Momentum • Look at the quantum mechanical angular momentum operator
(classically this “causes” a rotation about a given axis)
• look at 3 components
• operators do not necessarily commute
ip
prLz
100
0cossin
0sincos
)(
)(
)(
xyxyz
zxzxy
yzyzx
yxiypxpL
xzixpzpL
zyizpypL
zyx
yzzx
zxyz
yxxyyx
Lixyi
zyxz
xzzy
LLLLLL
)(
)])((
))([(
],[
2
2
P460 - angular momentum 3
Side note Polar Coordinates • Write down angular momentum components in polar coordinates (Supp
7-B on web,E&R App M)
• and with some trig manipulations
• but same equations will be seen when solving angular part of S.E. and so
• and know eigenvalues for L2 and Lz with spherical harmonics being eigenfunctions
iL
iL
iL
z
y
x
)sincotcos(
)coscot(sin
])(sin[ 2
2
2sin1
sin122
L
lm
lmm
lm
lmlmlmzlmz
Yll
YYL
YmLYL
l
2
sinsin122
2222
)1(
])(sin[ 2
2
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Commutation Relationships • Look at all commutation relationships
• since they do not commute only one component of L can be an eigenfunction (be diagonalized) at any given time
differentall
sameindicesanytensor
LiLLor
LLLLLL
LiLL
LiLL
LiLL
ijk
kijkji
zzxxyy
yxz
xzy
zyx
,1
0
],[
0],[],[],[
],[
],[
],[
P460 - angular momentum 5
Commutation Relationships • but there is another operator that can be simultaneously diagonalized
(Casimir operator)
yzyxyyz
yzxyzyy
xzxyxxz
xzyxzxx
yxzzyx
zzz
zyx
LLLLLLL
LLLLLLL
LLLLLLL
LLLLLLL
gu
LLLLLL
LLLLLL
LLLL
)()(
)()(
)()(
)()(
:sin
0)()(
],[2222
222
2222
P460 - angular momentum 6
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)
P460 - angular momentum 7
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 Abelian non-Abelian• Can often represent a group in many ways. A table, a matrix, a definition
of multiplication. They are then “isomorphic” or “homomorphic”
BAAB
BAAB
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Simple example • Discrete group. Properties of group (its “arithmetic”) contained in
Table
• Can represent each term by a number, and group combination is normal multiplication
• or can represent by matrices and use normal matrix multiplication
bacc
acbb
cbaa
cba
cba
1
1
1
11
1
ic
b
biiaaia
1
1
11
01
10,
10
01,
01
10,
10
011 cba
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Continuous (Lie) Group:Rotations • Consider the rotation of a vector
• 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
• assume angle change is small
rrr
'
identitynearrrr
samelengthrr
rRr
'
|||'|
'
anglessmallR
R
xy
xz
yz
z
1
1
1
100
01
01
100
0cossin
0sincos
)(
)()()()( RRRR
P460 - angular momentum 10
Rotations • Also need a Unitary Transformation (doesn’t change “length”) for how
a function is changed to a new function by the rotation
• U is the unitary operator. Do a Taylor expansion
• the angular momentum operator is the “generator” of the infinitesimal rotation
)(
)()()(
)()()()(
)()(1
rr
unitaryrrU
rRrorrrR
rtochangesr
R
LU
rprr
rpri
r
rrrrr
iR
i
1
)()()(
)()()(
)()()()(
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• For the Rotation group O(3) by inspection as:
• one gets a representation for angular momentum (notice none is diagonal; will diagonalize later)
• satisfies Group Algebra
LUR iR
xy
xz
yz
1
1
1
1
000
001
010
001
000
100
010
100
000
iL
iLiL
z
yx
kijkji LiLL ],[
P460 - angular momentum 12
• 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
• Usually use Pauli spin matrices to represent. Note O(3) gives integer solutions, SU(2) half-integer (and integer)
kijkji LiLL ],[
10
01
0
0
01
10
2
22
z
yx
L
i
iLL
1UU
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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 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 l-eiganvalue fixed
01
00
0
0
01
10
00
10
0
0
01
10
)2(
221
221
i
iL
i
iL
matricesSUfor
i
i
yx iLLL
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Eigenvalues “Group Theory” • operates on a 1x2 “vector” (varying m) raising or lowering it
01
00
00
10
L
L
1
0
0
1
01
00
0
0
1
0
01
00
0
0
0
0
1
00
10
0
1
1
0
00
10
0
LL
LL
1
0,
0
1,
21
21
21
21
s
s
ms
ms
P460 - angular momentum 16
• Can also look at matrix representation for 3x3 orthogonal (real) matrices
• Choose Z component to be diagonal gives choice of matrices
100
000
001
z
yx
L
iLLL
1
0
0
1
1
0
0
,
0
1
0
0
0
1
0
,
0
0
1
1
0
0
1
zz
zmmz
LL
LmL
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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
010
001
000
2
000
100
010
2
L
L
100
000
001
z
yx
L
iLLL
0
0
0
1
0
0
,
1
0
0
0
1
0
,
0
1
0
0
0
1
0
1
0
1
0
0
,
0
0
1
0
1
0
,
0
0
0
0
0
1
LLL
LLL
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• Can also look at matrix representation for 3x3 orthogonal (real) matrices. Work out X and Y components
100
000
001
z
yx
L
iLLL
00
0
00
)(
010
101
010
)(
21
2
21
21
i
ii
i
LLL
LLL
iy
x
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• Can also look at matrix representation for 3x3 orthogonal (real) matrices. Work out L2
100
000
001
z
yx
L
iLLL
]2*1)1([2
200
020
002
100
000
001
101
020
101
101
020
101
21
21
2222
llIdentity
LLLL zyx
iTii LLL 2
00
0
00
)(
010
101
010
)(
21
2
21
21
i
ii
i
LLL
LLL
iy
x
P460 - angular momentum 20
Eigenvalues • Done in different ways (Gasior,Griffiths,Schiff)
• Start with two diagonalized operators Lz and L2.
• where m and are not yet known
• Define raising and lowering operators (in m) and easy to work out some relations
z
z
zzyx
LLL
LLLLL
LLLLLiLLL
2],[
0],[],[ 2
22
mmll
mmllZ
mlLlm
mmlLlm
22
P460 - angular momentum 21
Eigenvalues
• Assume if g is eigenfunction of Lz and L2. ,L+g is also an eigenfunction
• new eigenvalues (and see raises and lowers value)
)()1(
)()(
),(
)()()(2
22
gLmgmLgL
gLLhbarLgLL
commuteLL
gLgLLgLL
zz
Loperatorsform
P460 - angular momentum 22
Eigenvalues • There must be a highest and lowest value as can’t have the z-
component be greater than the total
• For highest state, let l be the maximum eigenvalue
• can easily show
):min( 2HHHHz ggLderreglgL
)1()0(
)(2222
22
llll
gLLLLgL HzzH
00 LH gLgL
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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
• repeat for the lowest state
• eigenvalues of Lz go from -l to l in integer steps (N steps)
llllllequate
llglgL LLz
)1()1(
)1(2
)12(,1.....2,1,
))2(.......(,1,,0
intint2
23
21
termsllllllm
onlySUl
egerhalforegerN
l
00 LH gLgL
)1()0(
)(2222
22
llll
gLLLLgL LzzL
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Raising and Lowering Operators • can also (see Gasior,Schiff) determine eigenvalues by looking at
• and show
• note values when l=m and l=-m
• very useful when adding together angular momentums and building up eigenfunctions. Gives Clebsch-Gordon coefficients
1),(
1),(
mlmlCmlL
mlmlCmlL
)1)((),(
)1)((),(
mlmlmlC
mlmlmlC
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Eigenfunctions in spherical coordinates • if l=integer can determine eigenfunctions
• knowing the forms of the operators in spherical coordinates
• solve first
• and insert this into the second for the highest m state (m=l)
mlYlm ,,),(
lmi
lm
lmlm
lmz
YieYL
YmY
iYL
)cot(
imlm eFY )(
)()cot(
)())(cot(
)()cot(
)cot(00,
)1(
Fle
Filiee
eFie
YiellL
li
ili
imi
lli
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Eigenfunctions in spherical coordinates • solving
• gives
• then get other values of m (members of the multiplet) by using the lowering operator
• will obtain Y eigenfunctions (spherical harmonics) also by solving the associated Legendre equation
• note power of l: l=2 will have
0)()cot()1(
Fle li
lilll
l AeYF )(sin)(sin)(
1)1)((
)cot(
llll
i
YmlmlYL
ieL
22 cos;sincos;sin