Problem Set 5
Due: See website for due dates
Matter Waves
Reading: Taylor, Zafiratos, Dubson, Chapter 6; Tipler and Llewellyn, Chapter 5
Question A
(i) Why is the wave function so important to us? (ii) What does the wave function
represent? (iii) What experimental evidence do we have that supports the detailed
nature of the wave function?
Question B
The uncertainty relation would seem to suggest that the quantum mechanical world is
not a precise one. Is there anything imprecise about the rules of quantum mechanics, as
you have seen them up to this point? How about the predictions?
Question C
A classical physicist is determined to find out which slit each electron passes through in
the two-slit experiment (without disrupting the interference pattern). To this end he
places a molecule near one slit, in the hope that electrons passing through this slit will
excite the molecule, causing it to give out a characteristic pulse of light. Show that this
arrangement fares no better than the Heisenberg microscope though experiment using
light.
Question E
(i) Estimate the mass of a quark trio (qqq) entrapment inside the nucleus using a zero
point energy estimate? (ii) The neutron (udd quark combination) is slightly more massive
(1.3 MeV) than the proton (uud quark combination). This difference is the result of two
features, what are they? (iii) Why is an isolated neutron less stable than a proton?
Hint: look at electric charge
Question F
Read the Wikipedia section “Weak nuclear force” for the description of the W-boson
(http://en.wikipedia.org/wiki/W_and_Z_bosons).
(i) Use the uncertainty principle to show that the W boson is an extremely short range
force. (ii) How far can light travel during this brief time of existence? Compare this
distance to the diameter of proton.
Problem 1
Determine the de Broglie wavelength of a golf ball of mass 60 grams with speed 30 m/s.
Does it seem likely that the wave properties of a golf ball could be easily detected? Explain.
Answer: 3.68 x 1034 m
Problem 2
Compute the de Broglie wavelengths of (a) an electron and (b) proton when the KE is
4.5 keV. (c) Does it seem likely that the wave properties of these particles could be
easily detected? Explain? Answer: 0.0183 nm; 4.27 x 104 nm
Problem 3
Use the relativistic relation between E and p to show that electrons and photons with the
same energy have different wavelengths. (Note: Even at relativistic energies the de
Broglie relation is correct.) (b) Show that their wavelengths approach equality as their
common energy E gets much larger than mec2.
Problem 4
The position of a 60-gram golf ball sitting on a tee is determined within 1s. What is its
minimum possible energy? Moving at the speed corresponding to this kinetic energy,
how far would the golf ball move in a year? Answer: 2.3  1056 J, 2.7  1027 m.
Problem 5
In one of George Gamov’s Mr. Tompkins tales, the hero visits a “quantum jungle” where
is very large. Suppose that you are in such a place where ℏ = 50 J.s. A cheetah runs
past you a few meters away. The cheetah is 2 m long from nose to tail tip and its mass is
30 kg. It is moving at 30 m/s. What is the uncertainty in the location of the “mid point” of
the cheetah? Describe how the cheetah would look different to you than when has its
actual value. Answer: 2.8 cm
Problem 6
An unusually long-lived unstable atomic state has a lifetime of 1 ms. (a) Roughly what is
the minimum uncertainty in its energy? (b) Assuming that the photon emitted when this
state decays is visible ( ≈ 550 nm), what are the uncertainty and fractional uncertainty
in its wavelength? Answer: 3.3  1013 eV, 8  1011 nm, 1.5  1013
Problem 7
A particle of mass m moves in a one-dimensional box of length L. (Take the potential
energy of the particle in the box to be zero so that its total energy is its kinetic energy
p2/2m). Its energy is quantized by the standing-wave condition n(/2) = L, where  is the
de Broglie wavelength of the particle and n is an integer. (a) Show that the allowed
energies are given by En = n2E1 where E1 = h2/8mL2. (b) Evaluate En for an electron in a
box of size L = 0.1 nm and make an energy-level diagram for the sate from n = 1 to n =
5. (c) Calculate the wavelength of the radiation emitted when the electron makes a
transition from n = 2 to n = 1. Answer: (c) 11 nm