PHY4604–Introduction to Quantum Mechanics
Fall 2004
Problem Set 2
Sept. 7, 2004
Due: Sept. 13, 2004
Reading: PH notes
1. Cosmic microwave background radiation. Planck’s expression for the
mean energy per mode of oscillation of the electromagnetic field at temperature
T is given by
E =
hν
.
eβhν − 1
(1)
Following Einstein, suppose this energy consists of photons, each with energy
hν = h̄ω. The number of photons in each mode is then
Nω =
1
eβh̄ω
−1
.
(2)
(a) By repeating the calculation that led to the Stefan-Boltzmann law, find
an expression for the mean number of photons per unit volume in blackbody radiation at temperature T . Reduce the integral you obtain to a
dimensionless constant.
(b) Space is observed to be filled with blackbody radiation at a temperature
T ∼ 3K left over from the big bang. Using the fact that the dimensionless
integral you found is of order unity, estimate the number density of photons
in this 3K radiation.
Historical note: P.J.E. Peebles and Robert Dicke of Princeton predicted the microwave
background radiation as a residue of the big bang in the early 1960s, following early
work by George Gamov. The radiation was discovered serendipitously a few years
later in 1965 by Arno Penzias and Robert Wilson of Bell Laboratories, who were
awarded the Nobel prize for the discovery. Cosmic background radiation research is
still one of the most important probes of the early universe, see
http://www.astro.ubc.ca/people/scott/cmb.html.
2. Normal modes of simple classical system.
(a) Find the two allowed oscillation frequencies of the coupled mass-spring
system shown above. The outer spring constants are k, and ther inner one
is κ.
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(b) Write down the general solution for the mass displacements y1 (t) and y2 (t).
3. Maple problems.
3a). Recursive operations (”do loops”).
i) Print the cube roots of all the integers from 1 to 10 using a for..by..to
structure within a do..od; loop.
ii) Evaluate the sum S from n = 1 to n = 10 of n1/3 .
iii) Write the cube roots from 1 to 10 to an array called ”root3”, and plot
the data.
3b). Functions. Define f (x) = sin(x)/x2
i) using the arrow (”mapping”) notation
ii) using the unapply command
3c). Combining plots.
i) Plot, on the same graph, the Boltzman distr.P (E) = exp(E/T ) (ignore
normalization) for T=.8,1.,1.2.. over the interval E=0..3
ii) Plot the Lyman and Balmer spectral lines on a plot where wavelength
l is the vertical axis, and a line segment of unit length is drawn for
each of 4 lines for each series (total of 8 lines on plot)
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