MODERN PHYSICS
PHY 320
EXAM 2
(March 21, 2003)
(Answer all questions. To get partial credits show all the steps of your calculation
clearly. Good luck.)
h = 6.626 x 10-34 J.s
1 eV = 1.602 x 10-19 J
me = 9.109 x 10-31 kg = 0.5110 MeV/c2
c = 3.0 x 108 m/s
1. Briefly and clearly answer the following questions:
(a) Briefly describe Rutherford’s model of the atom. What are the major
drawbacks of his model? Explain your answer briefly. (5+5 = 10 pts)
(b) The Bohr theory of the hydrogen atom is based on several assumptions.
What are these assumptions? Do any of these assumptions contradict
classical physics? What are some of the drawbacks of Bohr’s model of the
atom?
(4+3+3=10 pts)
(c) How many visible spectral lines do you see for a gas of hydrogen atoms?
What are the different colors of these lines? Using Bohr’s model of the
atom, explain the origin of these spectral lines.
(2+2+6 =10 pts)
(d) If matter has a wave nature, why is the wave-like character not observable
in our daily experiences?
(5 pts)
(e) Why is it impossible to measure the position and speed of a particle
simultaneously with infinite precision?
(5 pts)
(f) What is a wave-group? What is the difference between group velocity and
phase velocity?
(5 pts)
Problems: (Points will be taken off for not showing the steps of your
calculation and for not getting the right numbers.)
2. Derive the expression for the energy levels for a Bohr atom with atomic
number Z=2 and show that the energy levels are quantized. Make sure to
explain all the symbols used. Draw an energy level diagram showing the
ground state energy and first two excited state energies for Z=2. Define
ionization energy. For Z=2 atom what is the ionization energy? Will photon be
emitted or absorbed if an electron jumps from the third orbit to the second
orbit and what is the frequency of that photon?
(15+5+2+2+1+5=30 pts)
Sol:
3. Newton showed that the phase velocity of deep-water waves having
wavelength  is given by vp = (g /2)1/2 where g is the free-fall acceleration.
Show that in this case the velocity of a group of these waves is one half of the
phase velocity of the central wavelength 0.
(10 pts)
Sol:
4. Calculate the de Broglie wavelength for an electron with kinetic energy 50
eV.
(7 pts)
Sol:
5. We wish to measure simultaneously the wavelength and position of a
photon. Assume that the wavelength measurement gives  = 6000 Angstroms
with an accuracy of one part in a million, i.e.,  = 10-6. What is the
minimum uncertainty in the position of the photon?
(8 pts)