PHY 4604 Fall 2008 – Homework 5
Due at the start of class on Friday, November 7. No credit will be available for
homework submitted after the start of class on Wednesday, November 12.
Answer all three questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
You may find useful the result
Z
0
∞
π
cos(kx)
dx = e−|k|a .
2
2
x +a
2a
1. Quantum mechanics in a two-dimensional Hilbert space. A quantum mechanical system
is described by a two-dimensional vector
space. The Hamiltonian operator for this
system is Ĥ = i∆ |ω1 ihω2 | − |ω2 ihω1 | , where |ω1 i and |ω2 i are the eigenstates of the
observable operator Ω̂ corresponding to eigenvalues ω1 6= ω2 .
(a) What condition(s) must the scalars ω1 , ω2 , and ∆ satisfy?
(b) Provide the matrix representations of Ĥ and Ω̂ in the basis {|ω1 i, |ω2 i}.
(c) Find the eigenvalues E1 and E2 (E1 < E2 ) of Ĥ and their corresponding normalized eigenstates |E1 i and |E2 i. Express the eigenstates as kets, not as column
vectors.
(d) Express |ω1 i and |ω2 i as linear combinations of |E1 i and |E2 i.
(e) Use your answers from (d) to express Ĥ and Ω̂ in the basis {|E1 i, |E2 i}. Give
your answers both in outer-product form and in matrix form.
(f) A system that at time zero is in a state |En i will at time t be in the state
exp(−iEn t/~)|En i. Use this fact to construct the time-evolution operator Û (τ )
that converts the state vector at time t to the state vector at time t + τ , i.e.,
Û (τ )|Ψ(t)i = |Ψ(t + τ )i for any |Ψ(t)i. Give your answer both in outer-product
form and as a matrix in the basis {|E1 i, |E2 i}.
For the remainder of the question, assume that the system is described at time t = 0
by a state vector |Ψ(0)i = |ω1 i.
(g) Find |Ψ(t)i for some later time t > 0. Express your result as a ket in the basis
{|E1 i, |E2 i}.
(h) Express |Ψ(t)i as a ket in the basis {|ω1 i, |ω2 i}.
(i) What are the possible outcomes of a measurement of the energy performed at
time t, and what is the probability of each outcome?
(j) What are the possible outcomes of a measurement of Ω performed at time t, and
what is the probability of each outcome?
(k) Find the expectation values of H and Ω at time t.
This question exemplifies the process changing the basis used to describe vectors in a
Hilbert space. Sections A.4 and A.5 of Griffiths provide a more general discussion of
basis changes using unitary (inner-product-conserving) transformations.
It should be clear from the spectral representation that any Hermitian operator corresponds to a diagonal matrix Ωmn = ωm δm,n when represented in the basis of its own
eigenstates. For this reason, the process of finding the eigenvalues and eigenfunctions
of an operator (or matrix) is often referred to as diagonalizing the operator (or matrix).
2. Position and momentum wave functions. A particle of mass m moving in one dimension
is described at time zero by the normalized spatial wave function
r
2a3
1
,
Ψ(x, 0) =
2
π x + a2
where a is a length scale.
(a) Show that the particle’s momentum wave function can be chosen to be of the
form Φ(p, 0) = |A| exp(−|p|a/~), and evaluate |A|.
(b) What is the probability that a measurement of the particle’s momentum at t = 0
will yield a result p > ~/a?
(c) Use Φ(p, 0) to calculate hpi, hp2 i, and σp .
(d) Use Φ(p, 0) to evaluate hxi, hx2 i, and σx .
(e) Verify that the state obeys the Heisenberg uncertainty principle.
Be careful taking momentum derivatives of Φ(p, t). (Recall Homework 4, Question 1.)
3. Functions of operators.
Consider a function f (x) that can be represented as a power
P∞
k
series: f (x) = k=0 ak x . We define the same function of an operator Ω̂ to be f (Ω̂) =
P∞
k
k=0 ak Ω̂ . Suppose that Ω̂ is a Hermitian operator acting in an infinite-dimensional
Hilbert space, and that Ω̂ has a set {ωn } of discrete eigenvalues as well as one or more
ranges of continuous eigenvalues ω. Assume that the corresponding eigenvectors satisfy
the generalized orthonormality relations.
(a) Show that
f (Ω̂) =
X
Z
f (ωn )|ωn ihωn | +
f (ω)|ωihω| dω,
(1)
n
and
hωm |f (Ω̂)|ωn i = f (ωm ) δm,n ,
hω|f (Ω̂)|ω 0 i = f (ω) δ(ω − ω 0 ),
hω|f (Ω̂)|ωn i = 0.
(b) Use Eq. (1) to show that the expectation value of Ω̂2 in any physical state must
be non-negative.
(c) Apply the results from above to show that the expectation value of the Hamiltonian Ĥ = p̂2 /2m + V̂ in any physical state is no smaller than the minimum value
(over all x) of V (x).