1
Name: ____________________________
Lecture Problem Set #2
Due Friday, Friday, Sep. 11, 2009
(10 points)
1. Suppose a black-body chamber is constructed from 111.6 g of iron and heated to 750 K. Classical
theory and the observations of Dulong and Petit indicate that the total energy of vibration of iron
atoms is 3RT per mole. Assuming this to be true,
a) What is the average energy of vibration per atom?
b) About how many iron atoms would need to donate their vibrational energy to excite an
ultraviolet-frequency oscillator into its first excited state? (Consider a 250 nm photon to be
ultraviolet light.)
c) If the probability that 1000 iron atoms will donate their vibrational energy to an oscillator is X,
would the probability that 2000 iron atoms will donate their vibrational energy to an oscillator
be larger or smaller than X. State your reasoning.
d) According to Planck’s assumption, if an oscillator emits light of frequency
ν
, it must have been
oscillating at frequency
ν
. Use your reasoning from part c) to explain how Planck’s assumption
avoids the ultraviolet catastrophe of classical physics.
2. The following questions regard the postulates of quantum mechanics.
a) Use the quantum mechanical operator, ˆx
d
pi
dx
=−=, to derive the Hamiltonian operator for the
free particle in one dimension. You can let the potential energy equal zero.
b) If the probability of finding a particle between r and r + dr at time t is described by 2
(,)rt dr
ψ
,
state what (,)rt
ψ
represents.
c) Explain why the Born interpretation of the wavefunction requires that the wavefunction be
single-valued.
d) The commutator is defined as, [A,B] = AB – BA, where A and B are operators. A and B are
said to commute if [A, B]f = 0 for all functions, f. Determine whether the operators d/dx and
d2/dx2 commute by operating on a general function, f(x).
e) Determine the eigenvalue for the momentum of a free particle described by the wavefunction,
3ix
Ae
ψ
−
=.
f) Calculate the normalization constant, N, for the wavefunction,
()
() exp abs()
x
Nxk
ψ
=− .
abs(x) = the absolute value to x, and k is a constant.