 # Assignment 4

```PC 242 Assignment 4: Due in class on Wednesday March 18, 2015
1. A wave function can be expressed as a sum over different harmonics as
ψ (x) = 0.7 cos(x) + 0.1cos(3x) +0.2 cos(5x)
(a) What are the constituent wavenumbers and wavelengths ?
(b) What wavelength has the largest amplitude a(k) ?
2. A particle is described by the probability density function
−L
#
% AL + 2Ax, for 2 ≤ x ≤ 0
%
L
P(x) = \$
AL
−
2Ax,
for0
≤
x
≤
%
2
%
0,
otherwise
&
a) Make a sketch of the probability density function, P(x) as a function of x.
b) If the wave function is real, write down a possible expression for the wave function
ψ(x).
c) Normalize the wave function and determine the constant A in terms of L.
d) What is the probability that the particle lies in the region x < L/6?
3. An electron in the n=4 state of a 5nm wide infinite well makes a transition to the
ground state, emitting a photon. What is the wavelength of the photon?
4. An electron is trapped in an infinite well. If the lowest energy transition possible in the
well produces a photon of 450 nm wavelength, what is the width of the well?
5. What is the probability that a particle in the n=2 state of an infinite well will be found
in the middle third of the well?
6. Show that the uncertainty in the momentum of a particle in level n of an infinite well is
given by
nπ
Δp =
L
7. An electron is trapped in a finite square well. How far in energy (eV) is it from being
free if the penetration depth into the walls of the well is 1nm?
8. A 2kg block oscillates with an amplitude of 10cm on a spring with spring constant
120N/m. What energy level is the block in? What is the energy spacing between
successive energy levels?
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