Quantum Mechanics for Mathematicians

and thus a 3 by 3 real matrix. ThisK′is determined by taking the trace

of the product of two such matrices. How areKandK′related?

Problem 2:
Under the homomorphism

Φ :Sp(1)→SO(3)

of section 6.2.3, what elements ofSO(3) do the quaternionsi,j,k(unit length,
so elements ofSp(1)) correspond to? Note that this is not the same question as
that of evaluating Φ′oni,j,k.

Problem 3:
In special relativity, we consider space and time together asR^4 , with an inner
product such that (v,v) =−v^20 +v 12 +v^22 +v 32 , wherev= (v 0 ,v 1 ,v 2 ,v 3 )∈R^4.
The group of linear transformations of determinant one preserving this inner
product is writtenSO(3,1) and known as the Lorentz group. Show that, just
asSO(4) has a double coverSpin(4) =Sp(1)×Sp(1), the Lorentz group has
a double coverSL(2,C), with action on vectors given by identifyingR^4 with 2
by 2 Hermitian matrices according to

(v 0 ,v 1 ,v 2 ,v 3 )↔

(

v 0 +v 3 v 1 −iv 2

v 1 +iv 2 v 0 −v 3

)

and using the conjugation action ofSL(2,C) on these matrices (hint: use de-
terminants).
Note that the Lorentz group has a spinor representation, but it is not unitary.

Problem 4:
Consider a spin^12 particle, with a state|ψ(t)〉evolving in time under the
influence of a magnetic field of strengthB=|B|in the 3-direction. If the state
is an eigenvector forS 1 att= 0, what are the expectation values

〈ψ(t)|Sj|ψ(t)〉

at later times for the observablesSj(recall thatSj=σ 2 j)?

B.4 Chapter

Problem 1:
Using the definition

〈f,g〉=

1

π^2

∫

C^2

f(z 1 ,z 2 )g(z 1 ,z 2 )e−(|z^1 |

(^2) +|z 2 | (^2) )
dx 1 dy 1 dx 2 dy 2
for an inner product on polynomials on homogeneous polynomials onC^2