PH 242
Spring 2024
Modern Physics II
Problem Set 1
Due: Wednesday, February 8
Reading: Eisberg and Resnick, Section 3-4 (review) and Sections 5-1 through 5-4
1. Matter Waves. In 1999 Anton Zeilinger (Physics Nobel, 2022) and his research group in Zurich reported
de Broglie wave interference of C60 molecules. These
are the most massive and complex objects for which
particle-wave interference has been observed. A silicon
nitride grating with slits d = 100 nm apart was utilized
as the diffraction grating and interference was observed
D = 1.25 m behind the grating as discussed in class.
The figure to the right (copied from one of their paper in
the American Journal of Physics) shows the interference
pattern produced by C60 molecules (grey circles) and a
fit using a complete diffraction theory (solid line). The
paper also reported that the velocity of the molecules in
the beam was 136 ± 3 m/s. A larger copy of the figure is
available on the Class Moodle page. Note: C60 is called
“Buckminsterfullerene” and is a soccer-ball shaped molecule of 60 carbon atoms; for discovery of this
amazingly stable (and easy to produce) molecule of carbon Robert Curl, Harold Kroto, and Richard
Smalley were awarded the 1996 Nobel Prize in Chemistry.
Determine the distance that should be found between the central maximum and the higher order maxima in the detection plane and compare them to what you see in the figure.
2. The wave equation for electromagnetic fields in a medium with an index of refraction n has the form
∂2E
n2 ∂ 2 E
=
∂x2
c2 ∂t2
What must be the relationship between the wavelength λ and frequency ν so that the complex plane
wave
E(x, t) = E0 ei(kx−ωt)
is be a solution to the wave equation?
3. Eisberg and Resnick, Chapter #5, Problem 1
4. Instead of the wave function that we looked at in class,
suppose a particle has the wave function

|x| ≤ a/2
 C( a2 − |x|)
Ψ(x, t = 0) =

0
|x| > a/2
which is sometimes called a “symmetric tent” wave
function as shown to the right.
Note specifically that this is a tent-shaped wave function,
and not a probability distribution as seen on the Moodle
Questions and in Discussion.
(a) Determine the constant C that properly normalizes the wave function.
(b) What is the probability that the particle will be found in the region −a/4 ≤ x ≤ +a/4?
(c) What is the probability that the particle will be found in the region +a/4 ≤ x ≤ +a/2?