AIN SHAMS UNIVERSITY
Faculty of Engineering
Engineering Physics & Mathematics
Department
Modern Physics & Quantum Prof. Wael Fikry
Mechanics Dr. Michael Gad
Problem Set 3
Matter waves
Constants in SI units: Electron charge (e) = 1.6×10-19, Electron rest mass (mo) = 9.1×10-31,
Speed of light = 3×108, Planck’s constant (h) = 6.62×10-34
Part I: State whether the following statements are true or false, and if false state
the reason:
1. The group velocity is equal to the phase velocity for a wave propagating through a dispersive
medium.
2. The effect of diffraction becomes more manifest as the window of diffraction gets larger.
3. Bragg’s experiment works better for non-crystalline materials than for crystalline materials.
4. De-Broglie wavelength describes the wavelength of the wave associated with a moving particle.
5. An electron exhibits a particle phenomenon in the double slit experiment.
6. The uncertainty principle is a result of the measurement errors.
Part II: Problems
1. Show that the ratio of the Compton wavelength λC to the de Broglie wavelength λ = h/p for a
relativistic electron is ( λc / λ = [( E / m ec 2 ) 2 − 1]0.5 ), where E is the total energy of the electron and me is
its mass.
2. (a) Show that the frequency f and wavelength λ of a freely moving particle are related by the
expression ( (f / c )2 = 1 / λ 2 + 1 / λc 2 ), where λC = h/moc is the Compton wavelength of the particle. (b)
Is it ever possible for a particle having nonzero mass to have the same wavelength and frequency as a
photon? Explain.
3. A narrow beam, of 60 keV, electrons passes
through a thin silver polycrystalline foil as shown in
Fig. 1. The interatomic spacing of silver crystals is Fig. 1
0.408 nm. Calculate the radius of the first-order
diffraction pattern from the principal Bragg planes
on a screen placed 40 cm behind the foil.
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4. Calculate the de-Broglie wavelength of an electron:
(a) Having a kinetic energy of 0.25 MeV.
(b) Accelerated under a potential of 105 V in an electronic microscope.
5. You want to study a biological specimen by means of a wavelength of 10 nm, and you have a
choice of using electromagnetic waves or an electron microscope. (a) Calculate the ratio of the
energy of a 10 nm wavelength photon to the kinetic energy of a 10 nm wavelength electron. (b) In
view of your answer to part (a), which would be less damaging to the specimen you are studying:
photons or electrons?
6. Use the uncertainty principle to show that if an electron was confined inside an atomic nucleus of
diameter 2×10–15m, it would have to be moving relativistically, while a proton confined to the same
nucleus can be moving non-relativistically.
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7. An electron (me = 9.11 × 10 kg) and a bullet (m = 0.020 kg) each have a speed of 500 m/s,
accurate to within 0.010%. Within what limits could we determine the position of the objects along the
direction of the velocity?
8. An atom in an excited state 1.80 eV above the ground state remains in that excited state2.0 μs before
moving to the ground state. Find (a) the frequency and (b) the wavelength of the emitted photon. (c)
Find the approximate uncertainty in energy and frequency of the photon.
9. A typical atomic nucleus is about 5.0 × 10-15 m in radius. Use the uncertainty principle to place a
lower limit on the energy an electron must have if it is to be part of a nucleus. [Hint: From
experiments, typical binding energies of electrons in unstable atoms are measured to be on the order of
a few eV].
10. A particle of mass m is confined to a one-dimensional line of length L. From arguments based on
the wave interpretation of matter:
(a) Assume nonrelativistic relations, show that the kinetic energy (KE) of the particle can have only
discrete values and determine these values.
(b) Assume the box is so small that the particle’s motion is relativistic, so that KE = p2/2m is not valid,
Derive an expression for the kinetic energy levels of the particle.
(c) Let the particle be an electron in a box of length L = 1.0×10-12m. Find its lowest possible kinetic
energy. By what percent is the nonrelativistic equation in error?
Part III: MATLAB exercise (Optional)
Write a MATLAB code that plots the superposition between two electric field waves travelling with the
speed of light. One of the waves has a wavelength of (500݊݉) and the other (520݊݉). Assume that the
two waves have an amplitude of (1 ܸ/݉) each.
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