Download - 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


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


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


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],[],[],[

],[

],[

],[


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


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)


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


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


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


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

)()()(

)()()(

)()()()(


• 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 ],[


• 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


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


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


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


• 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


• 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


• 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


• 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


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


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


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


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


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


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


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