Quantum Mechanics for Mathematicians

subgroups inside the unitary 2 by 2 matrices, but only one of them (the case
j= 3) will act diagonally onH, with theU(1) representation determined by

Q=

(

1 0

0 − 1

)

For the other two casesj= 1 andj= 2, by a change of basis either one could
be put in the same diagonal form, but doing this for one value ofjmakes the
other two no longer diagonal. To understand theSU(2) action onH, one needs
to consider not just theU(1) subgroups, but the full three dimensionalSU(2)
group one gets by exponentiating general linear combinations of Pauli matrices.
To compute such exponentials, one can check that these matrices satisfy the
following relations, useful in general for doing calculations with them instead of
multiplying out explicitly the 2 by 2 matrices:

[σj,σk]+≡σjσk+σkσj= 2δjk 1 (3.2)

Here [·,·]+is called the anticommutator. This relation says that allσjsatisfy
σj^2 = 1 and distinctσjanticommute (e.g.,σjσk=−σkσjforj 6 =k).
Notice that the anticommutation relations imply that, if we take a vector
v= (v 1 ,v 2 ,v 3 )∈R^3 and define a 2 by 2 matrix by

v·σ=v 1 σ 1 +v 2 σ 2 +v 3 σ 3 =

(

v 3 v 1 −iv 2

v 1 +iv 2 −v 3

)

then taking powers of this matrix we find

(v·σ)^2 = (v 12 +v 22 +v^23 ) 1 =|v|^21

Ifvis a unit vector, we have

(v·σ)n=

{

1 neven

v·σ nodd

Replacingσjbyv·σ, the same calculation as for equation 3.1 gives (forv
a unit vector)
eiθv·σ= (cosθ) 1 +i(sinθ)v·σ (3.3)

Notice that the inverse of this matrix can easily be computed by takingθto−θ

(eiθv·σ)−^1 = (cosθ) 1 −i(sinθ)v·σ

We’ll review linear algebra and the notion of a unitary matrix in chapter 4,
but one form of the condition for a matrixMto be unitary is

M†=M−^1