00

00

000

1

i

i

00

000

00

2

i

i

000

00

00

3 i

i

generates 3-dimensional rotations

and with the basis:

0

2/

2/1

,

1

0

0

,

0

2/

2/1

ii serve as the angularmomentum operators!

0210

21021

0210

xJ

020

202

020

i

ii

i

J y

100

010

001

zJ

and with the simpler basis:

1

0

0

,

0

1

0

,

0

0

1

And of course we already had

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

U 1U =†

U 2U =†

020

202

020

i

ii

i

0210

21021

0210

[ 12 21 ] = i3

U†[ 12 21 ]U =U†i3U

U†12UU†21U = iU

†3U

U†1UU†2UU

†2UU†1U = iU

†3U

Since U†U = UU

† = IJx

Jx JyJy JxiJz

0210

21021

0210

00

00

000

i

i

000

000

000

000

000

000 00

000

00

i

i

000

000

000

000

000

000

020

202

020

i

ii

i

a

bcxyz,

This 6×6 matrix also satisfies the same algebra:

000

000

000

000

000

000

000

000

000

000

000

000

3-dimensional transformations (like rotations) are not limited to 3-dimensional “representations”

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

Besides the infinite number of similarity transformations that couldproduce other 3×3 matrix representations of this algebra

The 3-dimensional representation

in the orthonormal basis

that diagonalizes z is the

“DEFINING” representation

of vector rotations

R ( ) = eiJ·/ħ^

can take many forms

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!

obviously:

1

1

2

2

0

0

20

0

zs

How about the operators sx, sy ?eigenvalues of each are also ħ/2 but

their matrices are not diagonal in this basis

s- 2

1

2

1

2

1

2

1 = 1ħ

s- 2

1

2

1

2

1

2

1= 1ħ

s-

01

00ħ

s+

00

10ħ

obviously work on the basis we’ve defined

You already know these as the Pauli matrices

01

10

2

xs

0

0

2 i

isy

10

01

2

zs

obeying the same commutation rule: zyx i ],[but 2-dimensionally!

What if we used THESE as generators?

01

10x

0

0

i

iy

10

01z

01

10x

0

0

i

iy

10

01z

Should still describe “rotations”. Its 3-components will still require

3 continuously variable independent parameters:

x , y , z

But this is not the defining representationand can not act on 3-dimensional space vectors.

These are operators that obviously act on the SPINORS(the SPIN space, not the 3-dimensional wave functions.

Spinors are 2-component objects

intermediate between scalars (1-component) and vectors (3-component)

When we rotate the “coordinate system” scalars are unchanged.Vector components are mixed by the prescriptions we’ve outlined.

What happens to SPINORS?

/2

1 ie

actually can only act on the spinor part

// 2

121

)()( imnmn

i erre

/21ie

/2

1 ie

The rotations on 3-dim vector space involved ORTHOGONAL operators

These carry complex elements, cannot be unitary!

Rt = R-1 i.e. RtR = RRt = 1

As we will see later this is a UNITARY MATRIX

of determinate = 1

Let’s stay with the simplified case of rotation = z (about the z-axis)

^

!2

)2/(

!5

)2/(

!4

)2/(

!3

)2/(

!2

)2/(

21

6655443322/2

1 zzzzzzi iiiiiie

!6

)2/(

!5

)2/(

!4

)2/(

!3

)2/(

!2

)2/(

21

6655443322/2

1 zzzzzzi iiiiiie

Notice: zz =

10

01

10

01

110

01

and obviously: zzz = z

!6

)2/(

!4

)2/(

!2

)2/(1

664422 zzz iii( )

!7

)2/(

!5

)2/(

!3

)2/()2/(

753 ( )+ iz

/2

1ie

)()(22

/ sincos21 z

i ie

)()(22

/ sincos21 z

i ie

an operator analog to: ei = = coscos + i + isinsin

)()(22

/ sincos21 iei

1 00 1( ) 1 0

0 -1( )

Let’s look at a rotation of 2 (360o)

)()(2

22

2/)2( sincos21 z

i ie z

= 1 + 0This means

A 360o rotation does not bring a spinor “full circle”.Its phase is changed by the rotation.

Limitations of Schrödinger’s Equation

1-particle equation

),()(),(2

),( 21

22

txxVtxxm

txt

i

),,(),()()(

),,(2

),,(2

),,(

212121

2122

2

2

2

2121

2

2

2

21

txxxxVxVxV

txxxm

txxxm

txxt

i

2-particle equation:

mutual interaction

But in many high energy reactionsthe number of particles is not conserved!

np+e++e

n+p n+p+3

e+ p e+ p + 6 + 3

Let’s expand the DEL operator from 3- to 4-dimensions

x i.e.

tc 1

0

x1 y

2z

3

lowered

since operates on x

then ipand as we’ve argued before,

the starting point for a relativistic QM equation:

022 cmpp

0222 cm

0)( 22 cm The Klein-Gordon

Equation

or if you prefer: 0)(1 22

2

2

2

mc

tc

But this equation has a drawback:Look at Schrödinger’s Equation for a free particle

22

2

mti

02

22

m

i

22

2

mi

*22

*

2

mi

*( ) ( )

0)(2

)( *22*2

** m

i

0)(2

)( **2

*

mti

probabilitydensity

probabilitycurrent density

The ContinuityEquation

starting from the Klein-Gordon Equation 0)(1 22

2

2

2

mc

tc

*( ) ( )* = 022

222

2

mc

ct

2222

2

2

mc

ct

0*22

*22

*222*2*2

2

2

2*

mcmc

cctt

)()( **2**

ct

not positive definite

The Klein-Gordon Equation is 2nd order in t!

Need to know initial (t=0) state as well as (0)

much morecomplicatedtime evolution