PHY 4604 Fall 2008 – Homework 1
Due at the start of class on Friday, September 5. No credit will be available for
homework submitted after the start of class on Wednesday, September 10.
Answer all four 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.
This assignment is primarily designed to provide practice with standard mathematical techniques encountered in wave mechanics. You may find useful the following integrals: for real
a and n = 0, 1, 2, 3, . . .,
Z ∞
Z ∞
√ (2n)! a 2n+1
π(2n − 1)!!
dx
2n
2
2
x exp(−x /a ) dx = π
=
n!
2
(a2 + x2 )n+1
2n+1 a2n+1 n!
0
0
where n! = n · (n − 1)! with 0! = 1, and n!! = n · (n − 2)!! with 0!! = (−1)!! = 1.
1. We have not yet tackled the task of solving the Schrödinger wave equation for any
specific potential V (x, t). This question addresses the “reverse engineering” process of
working back from a wave function Ψ(x, t) to find the potential that gives rise to it.
(a) Show that
1
∂Ψ(x, t)
~2 ∂ 2 Ψ(x, t)
V (x, t) =
.
i~
+
Ψ(x, t)
∂t
2m ∂x2
(1)
Many wave functions have isolated nodes at points x = xn (n = 1, 2, . . .) where
Ψ(xn , t) = 0. The value of V (xn , t) must be evaluated by taking the limit of Eq.
(1) as x → xn .
We will see in Chapter 2 that in the case of time-independent potentials V (x), quantum
mechanics assigns a special role to stationary-state wave functions of the form Ψ(x, t) =
ψ(x) exp(−iEt/~), where E is the particle’s energy and ψ(x) is its spatial wave function.
(b) Find an expression for V (x) in terms of the spatial wave function.
(c) Consider ψ(x) = Aa−1/2 exp(γx/a), where a is a (real) length scale (e.g., the size
of the region within which the particle is confined), A is a real constant, and γ
can be real or complex. (i) Find V (x) for this case. What is the condition on γ
for V (x) to be real? (ii) Show that this wave function cannot be normalized for
any choice of γ.
(d) Consider ψ(x) = Aa−1/2 exp(γx2 /a2 ), where a, A, and γ have the same interpretations as in part (c). (i) Find V (x) for this case. What is the condition on γ for
V (x) to be real? (ii) Find the condition that γ must satisfy for the wave function
to be normalizable.
2. Consider the wave function
Ψ(x, t) = A exp[−(x2 /2a2 + iEt/~)],
where a is a length scale and E is an energy.
(a) Find the positive, real constant A that normalizes the wave function.
(b) Calculate hxi, hx2 i, and σx .
(c) Calculate hpi, hp2 i, and σp .
(d) Verify that σx and σp satisfy the uncertainty principle.
3. Consider a wave function
Ψ(x, t) =
x2
A
exp(−iEt/~),
+ a2
where a is a length scale and E is an energy.
(a) Find a value of A that normalizes the wave function.
(b) Show that hxi = 0 at any t. Calculate hx2 i and σx .
(c) Show that hpi = 0 at any t. Calculate hp2 i and σp .
(d) Verify that σx and σp satisfy the uncertainty principle.
(e) Use the procedure from Question 1 to find the potential V (x) that gives rise to
Ψ(x, t). Sketch a graph of V (x), marking the minimum, maximum, and |x| → ∞
values of the potential.
4. Suppose that in some region a < x < b, a particle is described by a linear superposition
of right-moving and left-moving wave functions:
Ψ(x, t) = cR ei(kx−ωt+θ) + cL e−i(kx+ωt+θ) ,
where cR , cL , k, ω, and θ are positive, real constants. [This wave function generalizes
an example of linear superpositions that was considered in class. Note that if this form
of Ψ(x, t) were to extend all the way to |x| = ∞, the wave function would not be
normalizable. We will assume that Ψ(x, t) → 0 for x a and for x b.]
(a) Calculate the probability density ρ(x, t) and the probability current j(x, t) within
the region a < x < b. Cast your answers in forms that make clear that both
quantities are real.
(b) Calculate simplified expressions for ρ(x, t) and j(x, t) in the special case cR =
cL = c. You may find it useful to recall that 1 + cos 2φ = 2 cos2 φ.