Quantum Mechanics for Mathematicians

In the case of a spin^12 particle, the groupSpin(3) =SU(2) acts on states
by the spinor representation with the element Ω(θ,w)∈SU(2) acting as

|ψ〉→Ω(θ,w)|ψ〉

As we saw in chapter 6, the Ω(θ,w) also act on self-adjoint matrices by conju-
gation, an action that corresponds to rotation of vectors when one makes the
identification
v↔v·σ

(see equation 6.5). Under this identification theSjcorrespond (up to a factor
of 2) to the basis vectorsej. Their transformation rule can be written as

Sj→S′j= Ω(θ,w)SjΩ(θ,w)−^1

and 



S 1 ′

S 2 ′

S 3 ′



=R(θ,w)T





S 1

S 2

S 3





Note that, recalling the discussion in section 4.1, rotations on sets of basis
vectors like this involve the transposeR(θ,w)Tof the matrixR(θ,w) that acts
on coordinates.
Recalling the discussion in section 3.3, the spin degree of freedom that we
are describing byH=C^2 has a dynamics described by the Hamiltonian

H=−μ·B (7.2)

HereBis the vector describing the magnetic field, and

μ=g

−e

2 mc

S

is an operator called the magnetic moment operator. The constants that appear
are:−ethe electric charge,cthe speed of light,mthe mass of the particle, andg,
a dimensionless number called the “gyromagnetic ratio”, which is approximately
2 for an electron, about 5.6 for a proton.
The Schr ̈odinger equation is
d
dt

|ψ(t)〉=−i(−μ·B)|ψ(t)〉

with solution
|ψ(t)〉=U(t)|ψ(0)〉

where
U(t) =eitμ·B=eit

−ge

2 mcS·B=et

ge

2 mcX·B=et

ge|B|

2 mcX·|BB|

The time evolution of a state is thus given at timetby the sameSU(2) element
that, acting on vectors, gives a rotation about the axisw=|BB|by an angle

ge|B|t

2 mc