Winter 2011
EE 400B Homework #1 Solution
1. (30%) Light of a wavelength 250 nm falls on an aluminum surface. In aluminum 4.2 eV are
required to remove an electron. What is the kinetic energy of (a) the fastest and (b) the
slowest emitted photoelectrons? (c) What is the stopping potential? (d) What is the cutoff
excitation wavelength for aluminum? (e) If the intensity of the incident light is 2.0 W/m2,
what is the average number of photons per unit time per unit area that strike the surface?
1.24
 4.96 eV
0.25
K max  4.96  4.2  0.76 eV
K min  0
Since K max  0.76 eV, the stopping potential is 0.76 V.
1.24
0 
 0.382 m = 382 nm
4.2
2
 2.5 1018 m-2sec-1
4.96 1.6 1019
(a) E ph 
(b)
(c)
(d)
(e)
2. (10%) The wavelength of the yellow spectral emission of sodium is 589 nm. At what kinetic
energy would an electron have the same de Broglie wavelength?
1 2 p2
1 h

p  mv   K  mv 
 
2
2m 2m   

h
2
2
 6.63 1034 
1
6
25
K 

  6.96 10 joule = 4.35  10 eV
2  9.11031  589 109 
3. (30%) Griffiths, Problem 1.5.
Delta-function potential produces a wave function  ( x, t )  Ae   | x|eit , where A,  and  are
positive real constants. (a) Normalize the wave function. (b) Determine the expectation
values of x and x2. (c) Find the standard deviation of x. Plot |  |2 as a function of x, and
mark the points ( x   ) and ( x   ) to illustrate the sense in which  represents the
“spread” in x. What is the probability that the particle would be found outside this range?
1
Winter 2011
4. (30%) Griffiths, Problem 1.18.
In general, quantum mechanics is relevant when the de Broglie wavelength of the particle in
question ( h / p ) is greater than the characteristic size of the system (d). In thermal
equilibrium at (Kelvin) temperature T, the average kinetic energy of a particle is
p2 3
h
 k BT , so the typical de Broglie wavelength is  
.
2m 2
3mk BT
The purpose of this problem is to anticipate which systems will have to be treated quantum
mechanically, and which can safely be described classically.
(a) Solids. The lattic spacing in a typical solid is around d  0.3 nm . Find the temperature
below which the free electrons in a solid are quantum mechanical. Below what
temperature are the nuclei in a solid quantum mechanical? (Use sodium as a typical case.)
Moral: The free electrons in a solid are always quantum mechanical; the nuclei are
almost never quantum mechanical. The same goes for liquids (for which the interatomic
spacing is roughly the same), with the exception of helium below 4 K.
(b) Gases. For what temperatures are the atoms in an ideal gas at pressure P quantum
mechanical? Hint: Use the ideal gas law ( PV  Nk BT ) to deduce the interatomic spacing.
Answer: T  (1/ k B )(h 2 / 3m)3/ 5 P 2 / 5 . Obviously (for the gas to show quantum behavior)
we want m to be as small as possible, and P as large as possible. Put in the numbers for
helium at atmospheric pressure. Is hydrogen in outer space (where the interatomic
spacing is about 1 cm and temperature is 3 K) quantum mechanical?
Note: (1) The Boltzmann constant k B  1.38 1023 J / K .
(2) The nucleus in sodium has a mass m  23m p  23 1.7  1027  3.9  1026 kg.
(3) For (b), use N = 1 since we want to evaluate the situation when there is only one
particle in the system.
(4) 1 atm = 1.0  105 N / m 2 . The mass of helium is 4m p and the mass of hydrogen is
2m p .
2
Winter 2011
3