Physics 249 Homework 3
Due Sep 28th
From Modern Physics 6th Edition.
1) 5.7
A free proton moves back and forth between rigid walls separated by a distance
L=0.01nm.
a) If the proton is represented by a one-dimensional standing de Broglie wave with a
node at each wall, show that the allowed values of the de Broglie wavelength are give by
𝜆 = 2𝐿/𝑛, where n is a positive integer.
b) Derive a general expression for the allowed kinetic energy of the proton and compute
the values for n=1 and n=2.
2) 5.24
A particle moving in one dimension between ridged wall separated by a distance L has a
𝜋𝑥
wave function 𝜓(𝑥) = 𝐴𝑠𝑖𝑛 ( 𝐿 ). Since the particle must always be located between the
walls, what must the value of A be.
3) 5.33 b
The decay of states in atoms and nuclei often leave the system in another, albeit lowerenergy, excited state. An example is the 𝐻𝛼 line of the hydrogen Balmer series. In this
case the lifetime of both states is about the same, 10-8s. What is the uncertainty in the
energy of the 𝐻𝛼 photon.
4) 5.35
A neutron has a kinetic energy of 10 MeV. What size object is necessary to observe
diffractive effects? Is there anything in nature of this size that could serve as a target to
demonstrate the wave nature of 10 MeV neutrons?
As with a slit diffraction occurs when part of the propagating wave is cut off. This will
happen for any object of order the same size as the wavelength of the diffracting wave.
Note, the object is moving at 14% of the speed of light. However, the gamma factor is
only 1% so solving the problem either relativistic or non relativistic equations gives
similar answers. Especially considering that diffraction will occur for objection “of
order” the same size as the wavelength.
5) 5.43
Using the relativistic expression 𝐸 2 = 𝑝2 𝑐 2 + 𝑚2 𝑐 4
a) Show that the phase velocity of an electron wave is greater than c.
b) Show that the group velocity of the electron equals the particle velocity of the electron.