Quantum Mechanics for Mathematicians

vector representation is a real representation ofSO(3) orSpin(3), the spinor
representation is a complex representation.

7.2 The spin^12 particle in a magnetic field

In chapter 3 we saw that a general quantum system withH=C^2 could be
understood in terms of the action ofU(2) onC^2. The self-adjoint observables
correspond (up to a factor ofi) to the corresponding Lie algebra representation.
TheU(1)⊂U(2) subgroup commutes with everything else and can be analyzed
separately, here we will consider only theSU(2) subgroup. For an arbitrary
such system, the groupSU(2) has no particular geometric significance. When
it occurs in its role as double cover of the rotational group, the quantum system
is said to carry “spin”, in particular “spin^12 ” for the two dimensional irreducible
representation (in chapter 8 we will discuss state spaces of higher spin values).
As before, we take as a standard basis for the Lie algebrasu(2) the operators
Xj,j= 1, 2 ,3, where

Xj=−i

σj

2

which satisfy the commutation relations

[X 1 ,X 2 ] =X 3 ,[X 2 ,X 3 ] =X 1 ,[X 3 ,X 1 ] =X 2

To make contact with the physics formalism, we’ll define self-adjoint operators

Sj=iXj=

σj

2

(7.1)

In general, to a skew-adjoint operator (which is what one gets from a unitary
Lie algebra representation and what exponentiates to unitary operators) we
will associate a self-adjoint operator by multiplying byi. These self-adjoint
operators have real eigenvalues (in this case±^12 ), so are favored by physicists as
observables since experimental results are given by real numbers. In the other
direction, given a physicist’s observable self-adjoint operator, we will multiply
by−ito get a skew-adjoint operator (which may be an operator for a unitary
Lie algebra representation).
Note that the conventional definition of these operators in physics texts
includes a factor of~:

Sjphys=i~Xj=

~σj

2

A compensating factor of 1/~is then introduced when exponentiating to get
group elements

Ω(θ,w) =e−i

θ~w·Sphys

∈SU(2)

which do not depend on~. The reason for this convention has to do with the
action of rotations on functions onR^3 (see chapter 19) and the appearance of