130_notes.dvi

S~ = ̄h

2

~σ

σx=

(

0 1

1 0

)

σy =

(

0 −i

i 0

)

σz=

(

1 0

0 − 1

)

[σi,σj] = 2iǫijkσk

σ^2 i = 1

They alsoanti-commute.

σxσy=−σyσx σxσz = −σzσx σzσy=−σyσz

{σi,σj}= 2δij

Theσmatrices are theHermitian, Traceless matricesof dimension 2. Any 2 by 2 matrix can
be written as a linear combination of theσmatrices and the identity.

• See Example 18.10.9:The expectation value ofSx.*


• See Example 18.10.10:The eigenvectors ofSx.*


• See Example 18.10.11:The eigenvectors ofSy.*

The (passive)rotation operators,for rotations of the coordinate axes can becomputed(see
section 18.11.7) from the formulaRi(θi) =eiSiθi/ ̄h.

Rz(θ) =

(

eiθ/^20

0 e−iθ/^2

)

Rx(θ) =

(

cosθ 2 isinθ 2

isinθ 2 cosθ 2

)

Ry(θ) =

(

cosθ 2 sinθ 2

−sinθ 2 cosθ 2

)

Note that the operator for a rotation through 2πradians is minus the identity matrix for any of the
axes (becauseθ 2 appears everywhere). Thesurprising resultis that the sign of the wave function
of all fermions is changed if werotate through 360 degrees.

• See Example 18.10.12:The eigenvectors ofSu.*

As for orbital angular momentum (L~), there is also amagnetic momentassociated with internal
angular momentum (S~).

~μspin=−

eg

2 mc

S~

This formula has an additional factor ofg, thegyromagnetic ratio,compared to the formula
for orbital angular momenta. For point-like particles, like the electron,ghas been computed in
Quantum ElectroDynamics to be a bit over 2,g= 2 +απ+.... For particles with structure, like the
proton or neutron,gis hard to compute, but has been measured. Because the factor of 2 fromg
cancels the factor of 2 froms=^12 , the magnetic moment due to the spin of an electron is almost
exactly equal to the magnetic moment due to the orbital angular momentum in anℓ= 1 state. Both
are 1Bohr Magneton,μB= 2 emch ̄.

H=−~μ·B~=

eg ̄h

4 mc

~σ·B~=μB~σ·B~

If we choose thezaxis to be in the direction ofB, then this reduces to

H=μBBσz.

• See Example 18.10.13:The time development of an arbitrary electron state in a magnetic field.*


• See Example 18.10.14:Nuclear Magnetic Resonance (NMR and MRI).*