Download Problem Set 2

Introduction to Quantum Mechanics, Fall 2014
Problem Set 2
Due Tuesday, September 30
Problem 1: Calculate the exponential etM for

0
−π
0
π
0
0

0
0
0
by two different methods:
• Diagonalize the matrix M (i.e. write as P DP −1 , for D diagonal), then
show that
−1
etP DP = P etD P −1
and use this to compute etM .
• Calculate etM using the Taylor series expansion for the exponential, as
well as the series expansions for the sine and cosine.
Problem 2: Consider a two-state quantum system, with Hamiltonian
H = −Bx σ1
(this is the sort of thing that occurs for a spin-1/2 system subjected to a magnetic field in the x-direction).
• Find the eigenvectors and eigenvalues of H. What are the possible energies
that can occur in this quantum system?
• If the system starts out at time t = 0 in the state
1
|ψ(0)i =
0
(i.e. spin “up”) find the state at later times.
Problem 3: By using the fact that any unitary matrix can be diagonalized by
conjugation by a unitary matrix, show that all unitary matrices can be written
as eX , for X a skew-adjoint matrix in u(n).
By contrast, show that
−1 1
A=
0 −1
1
is in the group SL(2, C), but is not of the form eX for any X ∈ sl(2, C) (this
Lie algebra is all 2 by 2 matrices with trace zero.
Hint: For 2 by 2 matrices X, one can show (this is the Cayley-Hamilton theorem:
matrices X satisfy their own characteristic equation det(λ1 − X) = 0, and for
2 by 2 matrices, this equation is λ2 − tr(X)λ + det(X) = 0)
X 2 − tr(X)X + det(X)1 = 0
For X ∈ sl(2, C), tr(X) = 0, so here X 2 = −det(X)1. Use this to show that
p
p
sin( det(X))
X
p
X
e = cos( det(X))1 +
det(X)
Try to use this for eX p
= A and derive a contradiction (taking the trace of the
equation, what is cos( det(X))?)
Problem 4:
• Show that M is an orthogonal matrix iff its rows are orthonormal vectors
for the standard inner product (this is also true for the columns).
• Show that M is a unitary matrix iff its columns are orthonormal vectors
for the standard Hermitian inner product (this is also true for the rows).
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