Homework Assignment 02: Basic Formalism II (Due: 02/04/2016)
Problem 1 (24%, 6% each)
The matrix representation of Sx for spin-half particles is
Sx =
! ⎛⎜ 0 1
⎜
2 ⎜⎜⎝ 1 0
⎞⎟
⎟⎟ ,
⎟⎠
using the basis consisting of eigenstates of Sz denoted by ± : Sz ± = ±
!
± .
2
(a) Find the eigenvalues and eigenstates (denoted by x;± ) of Sx .
(b) Find the unitary operator U represented by a 2×2 matrix, which transforms basis from
{ ± } → { x;± } .
{ } → { x;± } , while U { x;± } → { ± } .
(c) Explicitly verify that U ±
†
(d) Explicitly verify that U †SxU = Sx′ , where Sx′ is the same operator Sx but is represented by
eigenstates of Sx , x;± .
Problem 2 (28%)
We have discussed in class that the determinant and trace of a matrix is invariant under a unitary
transformation of basis. For simplicity, let’s focus on a Hermitian operator A , whose matrix
representation is of N-dimension with eigenvalues of a1 , a 2 , …, aN .
N
(a) [8%] Show that det(A) = ∏ai .
i=1
N
(b) [8%] Show that Tr(A) = ∑ ai .
i=1
A
(c) [12%] Show that det(e ) = exp[Tr(A)] .
Problem 3 (48%)
(a) [16%] Show that a unitary operator U can be written as U = e iC , where C is a Hermitian
operator.
(b) [16%] If U = A + iB , show that A and B commute. Then express A and B in terms of C as
defined above. Here both A and B are Hermitian operators.
(c) [16%] If a is an eigenstate of A , such that [A, U ] = λU , with λ a real value. Show that
U a = const. λ + a . (const.: a constant)
[Hints: If U (matrix) is diagonal, it is straightforward to prove (a); then you can generalize this
for a non-diagonal unitary case, using the fact that a unitary matrix can always be diagonalized
by a similarity transformation. (why?)]