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Units, Constants and Useful Formulas
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Last modified: January 26, 2018
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