E1 0 0 0 0 . . .H = 0 E2 0 0 0 . . . 0 0 E3 0 0 . . .

0 0 0 E4 0 . . .:

.100

:

·

010

:

·

001

:

·, , ,

...with the “basis set”:

This is not general at all (different electrons, different atoms require different matrices)Awkward because it provides no finite-dimensional representation

That’s why its desirable to abstract the formalism

azrea

z /23

100

1

aZrea

Zr

a

Z 2/23

200 21

2

1

iaZr erea

Z

sin

8

1 2/25

121

cos2

1 2/25

210aZrre

a

Z

iaZr erea

Z sin

8

1 2/25

211

aZrea

rZ

a

Zr

a

Z 3/2

2223

300 21827381

1

Hydrogen Wave Functions 10000 ::

01000 ::

00100 ::

00010 ::

00001 :: 0

0000

Angular Momentum|nlmsms…>

l = 0, 1, 2, 3, ...Lz|lm> = mh|lm> for m = l, l+1, … l1, lL2|lm> = l(l+1)h2|lm>

Sz|lm> = msh|sms> for ms = s, s+1, … s1, sS2|lm> = s(s+1)h2|sms>

Of course |nℓm> is dimensional again

But the sub-space of angular momentum(described by just a subset of the quantum numbesrs)

doesn’t suffer this complication.

can measure all the spatial (x,y,z) components

(and thus L itself) of vmrL

not even possible in principal !

rixyx

irL

,,

ix

yy

xiLzSo, for

example

azimuthalangle inpolar

coordinates

Angular Momentumnlml…

Lz lm(,)R(r) = mħ lm(,)R(r)

for m = l, l+1, … l1, l

L2lm(,)R(r)= l(l+1)ħ2lm(,)R(r)

l = 0, 1, 2, 3, ...

Measuring Lx alters Ly (the operators change the quantum states).The best you can hope to do is measure:

States ARE simultaneously eigenfunctions of BOTH of THESE operators!We can UNAMBIGUOULSY label states with BOTH quantum numbers

ℓ = 2mℓ = 2, 1, 0, 1, 2

L2 = 2(3) = 6|L| = 6 = 2.4495

ℓ = 1mℓ = 1, 0, 1

L2 = 1(2) = 2|L| = 2 = 1.4142

2

1

0

1

0

Note the always odd number of possible orientations:

A “degeneracy” in otherwise identical states!

Spectra of the alkali metals

(here Sodium)all show

lots of doublets

1924: Pauli suggested electrons posses some new, previously un-recognized & non-classical 2-valued property

Perhaps our working definition of angular momentum was too literal…too classical

perhaps the operator relations

yzxxz

xyzzy

zxyyx

LiLLLL

LiLLLL

LiLLLL

may be the more fundamental definition

Such “Commutation Rules”are recognized by mathematicians as

the “defining algebra” of a non-abelian

(non-commuting) group[ Group Theory; Matrix Theory ]

Reserving L to represent orbital angular momentum, introducing the more generic operator J to represent any or all angular momentum

yzxxz

xyzzy

zxyyx

JiJJJJ

JiJJJJ

JiJJJJ

study this as an algebraic group

Uhlenbeck & Goudsmit find actually J=0, ½, 1, 3/2, 2, … are all allowed!

ms = ± 12

spin “up”spin “down”

s = ħ = 0.866 ħ 3 2

sz = ħ 12

| n l m > | > = nlm12

12

10( )

“spinor”

the most general state is a linear expansion in this 2-dimensional basis set

1 0 0 1( ) = + ( ) ( )

with 2 + 2 = 1

spin : 12p, n, e, , , e , , , u, d, c, s, t, b

leptons quarks

the fundamental constituents of all matter!

SPINORBITAL ANGULAR

MOMENTUMfundamental property

of an individual componentrelative motionbetween objects

Earth: orbital angular momentum: rmv plus “spin” angular momentum: I in fact ALSO “spin” angular momentum: Isunsun

but particle spin especially that of truly fundamental particlesof no determinable size (electrons, quarks)

or even mass (neutrinos, photons)

must be an “intrinsic” property of the particle itself

Total Angular Momentumnlmlsmsj… l = 0, 1, 2, 3, ...

Lz|lm> = mħ|lm> for m = l, l+1, … l1, lL2|lm> = l(l+1)ħ2|lm>

Sz|lm> = msħ|sms> for ms = s, s+1, … s1, sS2|lm> = s(s+1)ħ2|sms>

In any coupling between L and S it is the TOTAL J = L + s that is conserved.

ExampleJ/ particle: 2 (spin-1/2) quarks bound in a ground (orbital angular momentum=0) stateExamplespin-1/2 electron in an l=2 orbital. Total J ?

Either3/2 or 5/2possible

BOSONS FERMIONS

spin 1 spin ½ e,p, n,

Nuclei (combinations of p,n) can have

J = 1/2, 1, 3/2, 2, 5/2, …

BOSONS FERMIONS

spin 0 spin ½

spin 1 spin 3/2

spin 2 spin 5/2 : :

“psuedo-scalar” mesons

quarks and leptonse,, u, d, c, s, t, b,

Force mediators“vector”bosons: ,W,Z“vector” mesonsJ

Baryon “octet”p, n,

Baryon “decupltet”

Combining any pair of individual states |j1m1> and |j2m2> forms the final “product state”

|j1m1>|j2m2>

What final state angular momenta are possible?What is the probability of any single one of them?

Involves “measuring” or calculating OVERLAPS (ADMIXTURE contributions)

|j1m1>|j2m2> = j j1 j2;

m m1 m2 | j m >

j=| j1j2 |

j1j2

Clebsch-Gordon coefficients

or forming the DECOMPOSITION into a new basis set of eigenvectors.

Matrix Representationfor a selected j

J2|jm> = j(j+1)h2| j m >Jz|jm> = m h| j m > for m = j, j+1, … j1, jJ±|jm> = j(j +1)m(m±1) h | j, m1 >

The raising/lowering operators through which we identifythe 2j+1 degenerate energy states sharing the same j.

J+ = Jx + iJy

J = Jx iJy

2Jx = J+ + J Jx = (J+ + J )/2

Jy = i(J J+)/2

adding

2iJy = J+ J

subtracting

The most common representation of angular

momentum diagonalizes the Jz operator:

<jn| Jz |jm> = mmn

1 0 00 0 00 0 -1

Jz =(j=1)

2 0 0 0 00 1 0 0 00 0 0 0 00 0 0 -1 00 0 0 0 -2

Jz =(j=2)

J | 1 1 > =

J±|jm> = j(j +1)m(m±1) h | j, m1 >

J | 1 0 > =

J | 1 -1 > =

J | 1 0 > =

J | 1 -1 > =

J | 1 1 > =

| 1 0 > 2

| 1 -1 > 2

0

| 1 0 > 2

| 1 1 > 2

0

J =

J =

< 1 0 |0 0 0 0 00 02

2

0 00 0 0 0 0

22

< 1 -1 |

< 1 0 |

< 1 1 |

0210

21021

0210

020

202

020

2

1 xJ

020

202

020

020

202

020

2

1

i

ii

i

i

i

i

iJ y

100

010

001

zJ

For J=1 states a matrix representation of the angular momentum operators

Which you can show conform to the COMMUTATOR relationship

you demonstrated in quantum mechanics for the differential operators

of angular momentum

[Jx, Jy] = iJz

Jx Jy Jy Jx =

100

000

001

0

000

0

0

000

0

22

22

22

22i

ii

ii

ii

ii

= iJz

100

010

0011 z

JJ

2/10

02/12/1 zJJ

2/3000

02/100

002/10

0002/3

2/3 zJJ

20000

01000

00000

00010

00002

2 zJJ

x

y

zz′R(1,2,3) =

11

11

cossin0

sincos0

001

1

y′

1

=x′

x

y

zz′R(1,2,3) =

11

11

cossin0

sincos0

001

1

y′

1

=x′

2

2

2 x′′

z′′

=y′′

22

22

cos0sin

010

sin0cos

x

y

zz′R(1,2,3) =

11

11

cossin0

sincos0

001

1

y′

1

=x′

2

2

2 x′′

z′′

=y′′

22

22

cos0sin

010

sin0cos

3

y′′′

z′′′ =

x′′′

3

3

100

0cossin

0sincos

33

33

R(1,2,3) =

11

11

cossin0

sincos0

001

22

22

cos0sin

010

sin0cos

100

0cossin

0sincos

33

33

These operators DO NOT COMMUTE!

about x-axis

about y′-axis

about z′′-axis1st

2nd3rd

Recall: the “generators” of rotations are angular momentum operators and they don’t commute!

but as nn

Infinitesimal rotations DO commute!!

10

010

01

100

01

01

2

2

3

3

100

01

01

10

010

01

3

3

2

2

10

01

1

2

3

23

10

01

1

2

3

23

),,( 321 R 1 3 0-3 1 0

0 0 1

1 0 -2 0 1 0

2 0 1

1 0 0 0 1 1

0 -1 1R(1, 2, 3 ) =

1 3 -2 -3 1 0

2 0 1

1 0 0 0 1 1

0 -1 1

=

1 0 -2 0 1 1

2 -1 1

= 1 3 0-3 1 0

0 0 1

or

1 3 -2 -3 1 1

2 -1 1

=

R(1, 2, 3 ) = 1 3 -2 -3 1 1

2 -1 1

R(1, 2, 3 ) =

100

010

001

1

010

100

000

2

001

000

100

+ 3

000

001

010

1R(1, 2, 3 ) =If we imagine building up to full rotations by applying this repeatedly N times

ℓim [R(1, 2, 3 )]N

N

NN )/1(

ℓim N

R(1, 2, 3 ) NN )/1(

ℓim

N

e

Which we can re-write in the form / ie

by slightly re-writing the “vector” components:

00

00

000

1

i

i

00

000

00

2

i

i

000

00

00

1 i

i

Check THIS out:

3222

1221

000

001

010

000

000

010

000

001

000

i

We have found a “new” representation of Lx , Ly , Lz!!

Our alternate approach to motivating /

ieRwhere the generator was anhonest-to-goodness matrix!

00

00

000

1

i

i

00

000

00

2

i

i

000

00

00

3 i

i

gave

representing 3-dimensional rotations

with the basis:

0

2/

2/1

,

1

0

0

,

0

2/

2/1

ii

A B C

A

B

C

satisifes all thesame arithmetic

as Lx, Ly, Lz

321, iwhich we argue

000

00

00

3 i

i

Notice can try diagonalizing the zth matrix:

0)1(

00

0

02

i

i

1,0 Eigenvalues of 1, 0, 1

We should be able to diagionalize 3 by a SIMILARITY TRANSFORMATION!

U 3U =†

100

000

001

010

0

0

000

00

00

0

100

0

22

21

21

221

221

ii

i

i

i

i

U 3U =†

100

000

001

000

0

0

0

100

0

22

21

21

221

221

ii

i

i

0210

21021

0210

020

202

020

2

1 xJ

020

202

020

020

202

020

2

1

i

ii

i

i

i

i

iJ y

100

010

001

zJ

For J=1 states a matrix representation of the angular momentum operators