PH301
MODERN PHYSICS
FALL 2007
EXAM II – QUANTUM MECHANICS
All work on this exam must be your own. You may use a calculator and a 3"X5" card of equations. Clearly indicate
your final answer for each question.
Section 1: Multiple choice. No partial credit will be given for this section. Circle only one answer for each question.
1.
A metal has a cutoff frequency of 3X1015 Hz. What wavelengths of light will cause the photoelectric effect in this
metal?
a)
all wavelengths greater than 100 nm.
b) all wavelengths greater than 300 nm.
c)
all wavelengths shorter than 100 nm.
d) all wavelengths between 100 nm and 300 nm.
2.
A 1.6 W light source is made up of 1-eV photons. How many photons per second are in the beam?
a)
1.6 photons/s.
b) 0.625 photons/s.
c)
1019 photons/s.
d) 10-19 photons/s.
3.
For a stationary state of a particle-in-a-box (1-D), the average momentum is ___ and the uncertainty in the
momentum is ___.
a)
zero, zero.
b) zero, non-zero.
c)
non-zero, zero.
d) non-zero, non-zero.
4.
If you double the energy of an electron, then
a)
its wavelength is halved.
b) its frequency is doubled.
c)
both of the above.
d) None of the above.
5.
A particle in a box is described by the wavefunction  ( x) 
a)
A sin 2 x . The particle’s momentum is
2.
b) 2h.
6.
c)
h.
d)
a combination of handh.
A blackbody that glows with a spectrum that peaks in the visible part of the EM spectrum might have a
temperature of
a)
30 K.
b) 300 K.
c)
6 X103 K.
d) 6X106 K.
7.
A proton and a photon both have the same wavelength. The proton and photon also have the same ___
a)
momentum and energy.
b) momentum but not energy.
c)
energy but not momentum.
d) none of the above.
8.
The diagram shows the graphs of relative intensity (kinetic energy deposited in detector per unit time per unit
area) for a beam of electrons with wavelength  passing through a slit of width a. If, instead of a beam, a single
electron with wavelength  were sent through a slit of width a, which of the following would be true when we
detect the electron beyond the slit?
a)
Most of the electron’s energy would be deposited within about 10o of =0, but some would be deposited
near =+17o.
b) The electron will be found spread out mostly within about 10o of =0, with some of it found near =+17o.
c)
The electron will be found at only one place, with =0 being most likely, and =+5o being about half as
likely.
d) The electron will be found at only one place, with =0 being most likely, and =+5o being about a quarter
as likely.
9.
The wavefunction of a particle is given by Asin(x) for 0<x<10. The probability density at x = 0.5 is ___ the
probability density at x = 1.5. (Note: argument of sine is radians)
a)
greater than
b) less than
c)
equal to
d) more information is needed.
10. The kinetic energy of ___ is inversely proportional to the wavelength squared.
i. photon
ii. non-relativistic massive particle
iii. relativistic massive particle
a) i, ii, and iii.
b) i and ii only.
c)
ii only.
d) iii only.
Section 2: Problems & questions. Partial credit will be given for this section. Show all work and justify all answers.
15-20 pts each.
Relativistic equations are not needed for any of these problems.
“Set up, but do not solve” solutions should have no undefined constants or variables.
11. An electron is in the third excited state (n=4) of an infinite square well of length 2nm.
a) Find the energy of the electron in eV or J.
b) Sketch the wavefunction of the electron, showing its functional form both inside and outside the box.
c) Find the wavelength of light that would be emitted if this electron dropped down to the ground state.
12. A single hypothetical particle is in a state that does not have definite momentum. The possible momenta of this
particle and their probabilities are given in the table below.
2 eV/c
1 eV/c
-2 eV/c
momentum
0.3
0.5
0.2
probability
a) Find the expectation value of momentum, <p>.
b) Find the uncertainty in momentum, p.
c) Find the minimum uncertainty in the position of this particle.
13. An experimenter finds that the work function of a certain metal is 1.5 eV.
a) Find the energy of photon needed to eject electrons from this metal with a maximum kinetic energy of 3
eV.
b) Find the wavelength of light needed to eject electrons from this metal with a maximum kinetic energy of 3
eV.
c) Find the wavelength of an electron with kinetic energy of 3 eV.
14. In a particular region of space, the potential energy experienced by an electron of mass m is given by
B 2  1 
V ( x) 
  , where B is a constant. The wavefunction of the electron in this region of space is given by
m  x
Bx
the stationary state wavefunction  ( x)  Axe .
a) Use the time independent Schrödinger equation to find the energy of the electron. Leave your answer in
terms of the constants B, m, and .
b) Write down the time-dependent wavefunction of the electron in this region,  ( x, t ) .
1
 A   Ax2 / 2
15. Given the stationary state wavefunction of a particle is  ( x)    e
for all x.
 
4
a) Set up, but do not solve, an integral that equals the expectation value of position of this particle.
b) Set up, but do not solve, an equation that determines the value of A.
Some info you may or may not need:
electron mass = 9.11 X 10-31 kg = 5.49 X 10-4 u = 0.511 MeV/c2
mass of proton = 938 MeV/c2 = 1.67X10-27 kg
1 MeV = 106 eV
1 eV = 1.6X10-19 J
1 nm = 10-9 m
-34
-15
Planck's constant h =6.626 X 10 Js = 4.136 X 10 eVs, Combo hc = 1240 eVnm
his other constant  =1.054 X 10-34 Js =6.582 X 10-16 eVs
Combo c = 197 eVnm