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Units, Constants and Useful Formulas

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Pauli Spin Matrices
-------------------

The Pauli matrices are related to the angular momentum operator
that corresponds to an observable describing the spin of a spin 1/2
particle.  They are Hermitian and unitary.  The Pauli matrices
(after multiplication by i to make them anti-Hermitian), also
generate transformations in the sense of Lie algebras

The Pauli spin matrices have eigenvalues of +1 and -1.  Thus,

sx,y,z|χ> = ±|χ> where χ is the spin wavefunction.

(h/2)σx,y,z|χ> = ±|χ>

We can represent χ as a linear combination of 'up' and 'down' states
as follows:

χ = α|up> + β|down>

z-axis component:

-  -  - -      - -
|1  0|| α | = ±| α |
|0 -1|| β |    | β |
-  -  - -      - -

Positive:

-  -      - -
|  α | = +| α |
| -β |    | β |
-  -      - -

∴ β = -β so β must equal 0 and

-  -      - -
|  α | = +| α |
| -β |    | 0 |
-  -      - -

Negative:

-  -      - -
|  α | = -| α |
| -β |    | β |
-  -      - -

∴ α = -α so α must equal 0 and

-  -      - -
|  α | = -| 0 |
| -β |    | β |
-  -      - -

Therefore, the eigenvectors are,

- -       - -
| 1 | and | 0 |
| 0 |     | 1 |
- -       - -

x-axis component:

-  -  - -      - -
|0  1|| α | = ±| α |
|1  0|| β |    | β |
-  -  - -      - -

Positive:

- -      - -
| β | = +| α |
| α |    | β |
- -      - -

∴ β = α so

- -      - -
| β | = +| α |
| α |    | α |
- -      - -

Negative:

- -      - -
| β | = -| β |
| α |    | α |
- -      - -

∴ β = -α so

- -      -  -
| β | = -|  α |
| α |    | -α |
- -      -  -

Now,

|α|2 + |β|2 = 1

∴ 2α2 = 1 so

α = β = 1/√2

Therefore, the eigenvectors are,

-  -       -   -
|1/√2| and | 1/√2|
|1/√2|     |-1/√2|
-  -       -   -

y-axis component:

-  -  - -      - -
|0 -i|| α | = ±| α |
|i  0|| β |    | β |
-  -  - -      - -

Positive:

-   -      - -
| -iβ | = +| α |
|  iα |    | β |
-   -      - -

∴ -iβ = α so β = iα and

-   -      -  -
| -iβ | = +|  α |
|  iα |    | iα |
-   -      -  -

Negative:

-   -      - -
| -iβ | = -| α |
|  iα |    | β |
-   -      - -

∴ -iβ = -α so β = -iα and

-   -      -   -
| -iβ | = -|   α |
|  iα |    | -iα |
-   -      -   -

Therefore, the eigenvectors are,

-  -       -   -
|1/√2| and | 1/√2|
|i/√2|     |-i/√2|
-  -       -   -

Since Sx, Sy, Sz must have eigenvalues of +/-h/2, the σ matrices
must have eigenvalues of +/-1.

S2 = Sx2 + Sy2 + Sz2

Sx = (h/2)σx

Sy = (h/2)σy

Sz = (h/2)σz

S2 = Sx2 + Sy2 + Sz2

|S|2 = (h2/4){σx2 + σy2 + σz2}

-   -     -   -     -   -
= (h2/4){| 1 0 | + | 1 0 | + | 1 0 |}
| 0 1 |   | 0 1 |   | 0 1 |
-   -     -   -     -   -

- -
= (3h2/4)| I |
- -

- -
= s(s + 1)h2| I | where s = 1/2
- -

This should be compared with the equivalent relation for the orbital angular
momentum operator that is obtained when solving the Schrodinger equation
for the hydrogen atom.

|L|2 = l(l + 1)h2