# Chapter 6: Matter Waves

```Chapter 6: Matter Waves
Questions &amp; Problems
  2f 
2
T
E  hf  
xk 
1
2
&amp; xp 
P  x  dx   dx
E
hc  1240 eV  nm
34
1
2
1
2
2
2
h  6.626  10
2

h
p  k

t  21 &amp; tE 
k
2mL2
c  197.3 eV  nm
J  s  4.14  10 16 eV  s
Conceptual Questions
a. When you check the air pressure in a tire, a little air always escapes; the process of
making the measurement changes the quantity being measured. Think of other
examples of measurements that change or disturb the quantity being measured.
b. The upper half of the electron diffraction pattern is the mirror image of the lower half
of the pattern. Would it be correct to say that the upper half of the pattern is caused
by electrons that pass through the upper half of the slit? Explain.
c. Suppose a two-slit interference experiment is carried out using an electron beam.
Would the same interference pattern result if one slit at a time is uncovered instead
of both at once? If not, why not? Does not each electron go through one slit or the
other? Or does every electron go through both slits? Discuss the latter possibility in
the light of the principle of complementarity.
d. You have been asked to design a magnet system to steer a beam of 54-eV
electrons. The goal is to be able to direct the electron beam to a specific target
location with an accuracy of &plusmn;1.0 mm. In your design, do you need to take the wave
nature of electrons into account? Explain.
e. Laser light results from transitions from long-lived metastable states. Why is it more
monochromatic than ordinary light?
f. Some lasers emit light in pulses that only 1012 s in duration. The length of such a
pulse is 0.3 mm. Can pulsed laser light be as monochromatic as light from a laser
that emits a steady, continuous beam? Explain.
Example 6.1
Calculate the wavelengths of the following objects. (a) 500-g billiard ball moving at 10
m/s (23 mph); (b) Electron accelerated through a 10 volt potential; and (c) 10 MeV
electron
Example 6.2
a. The position x of a 0.01-g pellet has been carefully measured and is known within
&plusmn;0.5 m. According to the uncertainty principle what are the minimum uncertainties
in its momentum and velocity, consistent with our knowledge of x?
b. An electron is known to be somewhere in an interval of total width a  0.1 nm (the
size of a small atom). What is the minimum uncertainty in its velocity, consistent with
this knowledge?
Example 6.3
Many excited states of atoms are unstable and decay by emission of a photon in a time
of order t  10-8 s. What is the minimum uncertainty in the energy of such an atomic
state?
Example 6.4
Suppose that a particle of mass m is constrained to move in a one-dimensional space
between two infinitely high barriers located A apart.
a. Compute the minimum energy of an electron for A = 1010 m and A = 102 m. Interpret
the results.
b. Calculate the minimum energy for a 100-mg bead moving on a thin wire between two
stops located 2 cm apart. Interpret the results.
Example 6.5
It is possible for some fundamental particles to “violate” conservation of energy by creating
and quickly by reabsorbing another particle. For example, a proton can emit a +
according to p  n   where n represents a neutron. The + has a mass of 140 MeV/c2.
The reabsorption must occur within a time t consistent with the uncertainty principle. (a)
By how much E is energy conservation violated? (Ignore KE) (b) For how long t can the
+ exist? (c) Assuming that the is moving at nearly the speed of light, how far from the
nucleus could it get in the time t?
```