Ch. 28 Problems: 1, 8, 11, 15, 21, 22, 25, 57, 69 1.Strategy Find the de Broglie wavelength using Eq. (28-1). Compare the result to the diameter of the
hoop.
Solution
The wavelength is much too small compared to the diameter of the hoop for any appreciable
diffraction to occur―for a diameter of
it’s a factor of
smaller!
8.Strategy Assume that the neutron is nonrelativistic and use p = mv for the momentum. Find the de
Broglie wavelength using Eq. (28-1).
Solution
This speed is small compared to c, so the assumption was reasonable.
11.Strategy Assume that the electrons are nonrelativistic. To produce the same diffraction pattern, the
electrons must have the same wavelength as the x-rays. Find the de Broglie wavelength using Eq.
(28-1) and the wavelength of the x-rays using
Solution Equate the wavelength of an electron and the wavelength of an x‐ray photon. Now substitute
and solve for the speed. Find the kinetic energy. 15. (a) Strategy Assume that the electrons are nonrelativistic. Use the de Broglie wavelength and p =
mv to find the speed of the electrons. Then find the kinetic energy.
Solution Solve for v.
Calculate the kinetic energy.
The kinetic energy is small compared to the rest energy of an electron, so the use of the
nonrelativistic equations for momentum and kinetic energy was valid.
21. Strategy The single-slit diffraction minima are given by
The edge of the central fringe
corresponds to m = 1. Since the width of the central fringe is small compared to the slit-screen
distance, use small angle approximations. According to conservation of energy, the kinetic energy of
the electrons is related to the potential difference by K = eV. Use the de Broglie wavelength with
for the electrons.
Solution Since
and
we have
where x is half the width of the central fringe and D is the
distance from the slit to the screen. Find the width of the slit.
x
θ
D
22.Strategy The uncertainty of the x-component of the electron’s position is 0.05 nm. Use the positionmomentum uncertainty principle, Eq. (28-2), to find the uncertainty in the momentum. Then use the
classical expressions for the momentum and kinetic energy to estimate the electron’s kinetic energy,
and compare it to the ground-state kinetic energy predicted by the Bohr model.
Solution
(a) Find the uncertainty in the x-component of the momentum.
(b) Estimate the kinetic energy.
(c) 13.6/4 = 3.4, so yes, the estimate has the correct order of magnitude.
25. Strategy Use the energy-time uncertainty principle, Eq. (28-3).
Solution Find the uncertainty in the gamma-ray energies.
57. Strategy The kinetic energy of the electrons as they reach the slits is equal to
due to the 15-kV
potential. Use the relationship between momentum and kinetic energy to find the momentum of the
electrons at the slits. Then, find the de Broglie wavelength of the electrons and use it and the doubleslit interference maxima equation to find the distance between the slits.
m=1
Solution Find the momentum.
x
θ
m=0
D
Find the de Broglie wavelength.
Find the slit separation.
If
then
and
69.Strategy The wavelength of the electrons must equal the wavelength of the photons to give the
same diffraction pattern. The wavelength of a photon is related to its energy by
The
wavelength of the electrons is given by the de Broglie wavelength with
electrons are nonrelativistic
Solution Find the kinetic energy of the electrons.
since the