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generates 3 -dimensional rotations. and with the basis:. serve as the angular momentum operators!. And of course we already had. and with the simpler basis:. U s 3 U =. †. U s 1 U =. †. U s 2 U =. †. [  1  2 -  2  1 ] = i 3. U † [  1  2 -  2  1 ] U = U † i 3 U.

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Generates 3 dimensional rotations

generates 3-dimensional rotations

and with the basis:

serve as the angular

momentum operators!

And of course we already had

and with the simpler basis:


Generates 3 dimensional rotations

Us3U =

Us1U =

Us2U =


Generates 3 dimensional rotations

[12 - 21 ] = i3

U†[12 - 21 ]U =U†i3U

U†12U- U†21U = iU†3U

U†1UU†2U- U†2UU†1U = iU†3U

SinceU†U = UU†= I

Jx

Jx Jy-Jy Jx=iJz


Generates 3 dimensional rotations

This 6×6 matrix also satisfies the same algebra:

a

b

c

x

y

z

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

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0

0

0

0

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0

,

3-dimensional transformations (like rotations) are

not limited to 3-dimensional “representations”


Generates 3 dimensional rotations

Besides the infinite number of similarity transformations that could

produce other 3×3 matrix representations of this algebra


Generates 3 dimensional rotations

R ( ) = e- iJ·/ħ

^

can take many forms

The 3-dimensional representation

in the orthonormal basis

that diagonalizes z is the

“DEFINING” representation

of vector rotations


Generates 3 dimensional rotations

ℓ = 2

mℓ= -2, -1, 0, 1, 2

ℓ = 1

mℓ= -1, 0, 1

2

1

0

1

0

L2 = 1(2) = 2

|L| = 2 = 1.4142

L2 = 2(3) = 6

|L| = 6 = 2.4495

Note the always odd number

of possible orientations:

A “degeneracy” in otherwise identical states!


Generates 3 dimensional rotations

Spectra of the

alkali metals

(here Sodium)

all show

lots of doublets

1924:Paulisuggested electrons

posses some new, previously

un-recognized & non-classical

2-valued property


Generates 3 dimensional rotations

Perhaps our working definition of angular momentum was too literal

…too classical

perhaps the operator relations

Such “Commutation Rules”

are recognized by

mathematicians as

the “defining algebra”

of a non-abelian

(non-commuting) group

may be the more fundamental definition

[ Group Theory; Matrix Theory ]

Reserving L to represent orbital angular momentum, introducing the

more generic operator J to represent any or all angular momentum

study this as an

algebraic group

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


Generates 3 dimensional rotations

quarks

3

2

1

2

1

2

1

2

1

2

1

2

leptons

spin :

p, n,e, , , e ,  ,  ,u, d, c, s, t, b

the fundamental constituents of all matter!

spin “up”

spin “down”

s = ħ = 0.866 ħ

ms = ±

sz = ħ

( )

1

0

| n l m > | > = nlm

“spinor”

( )

( )

( )

the most general state is

a linear expansion in this

2-dimensional basis set

 1 0

 0 1

=  + 

with a 2 + b 2 = 1


Generates 3 dimensional rotations

obviously:

eigenvalues of each are alsoħ/2 but

their matrices are not diagonal

in this basis

How about the operatorssx, sy ?

s-

s-

= 1ħ

ħ

s-

s+

= 1ħ

ħ

obviously work on

the basis we’ve defined


Generates 3 dimensional rotations

You already know these as the Pauli matrices

obeying the same commutation rule:

but 2-dimensionally!

What if we used THESE as generators?


Generates 3 dimensional rotations

Should still describe “rotations”.

Its 3-components will still require

3 continuously variable

independent parameters:

x ,y ,z

But this is not the defining representation

and 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.


Generates 3 dimensional rotations

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?

actually can only act on the spinor part


Generates 3 dimensional rotations

The rotations on 3-dim vector space involved ORTHOGONAL operators

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

These carry complex elements,

cannot be unitary!

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)

^


Generates 3 dimensional rotations

and obviously:zzz = z

Notice: zz =

(

)

(

)

+ iz


Generates 3 dimensional rotations

an operator analog to: ei = cos + isin

( )

( )

1 0

0 -1

1 0

0 1

Let’s look at a rotation of2(360o)

= -1 + 0

This means

A 360o rotation does not bring a spinor “full circle”.

Its phase is changed by the rotation.


Generates 3 dimensional rotations

Limitations of Schrödinger’s Equation

1-particle

equation

2-particle

equation:

mutual interaction


Generates 3 dimensional rotations

But in many high energy reactions

the number of particles is not conserved!

np+e++e

n+p  n+p+3

e-+ p  e-+ p + 6 +3g


Generates 3 dimensional rotations

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

i.e.

lowered

since operates on x

and as we’ve argued before,

the starting point for a relativistic

QM equation:

then

The Klein-Gordon

Equation

or if you prefer:


Generates 3 dimensional rotations

But this equation has a drawback:

Look at Schrödinger’s Equation for a free particle

*( )

( )

-

= 0

The Continuity

Equation

probability

density

probability

current density


Generates 3 dimensional rotations

starting from the Klein-Gordon Equation

*( ) -

( )*= 0

not positive definite

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

much more

complicated

time evolution

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


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