MODERN PHYSICS
PHY 320
EXAM 3
(April 22, 2003)
(Answer all questions. To get partial credits show all the steps of your calculation
clearly. Good luck.)
1. Consider a particle of mass m moving in a one-dimensional space under a
potential U(x). Write down the time dependent Schrodinger equation of the
particle. What is the time independent Schrodinger equation of the particle
if the particle has total energy E?
(5 pts)
2. What are the mathematical conditions for an acceptable quantum wave
function? What do you mean by an observable in quantum mechanics.
Give examples of a few observable.
(5+5=10pts)
3. (a) What do we mean by a free particle? Consider a free particle of mass
m moving in one-dimension with a momentum p. Classically what is the
total energy of the particle? Plot the energy as a function of particle’s
momentum. Is the energy continuous or discrete? On your plot shade the
region that shows the possible values of the energy the particle can have.
(5+5=10pts)
(b) Now consider the free particle quantum mechanically. Imagine that the
free particle is moving in a one-dimensional box of length 2L. Write the
time independent Schrodinger equation of the particle. What is the type of
differential equation that you get? What is the wave function of the
particle? How many unknown constants do you have in your equation?
(2+2+4+2=10pts)
(c) What are the boundary conditions? Using the boundary conditions
eliminate some of the unknowns.
(3+4=7pts)
(d) Now normalize the wave function and write the final form of the wave
function. Show all the steps of your integrations.
(8 pts)
(e) What are the energy values of the free particle? Is the energy discrete
or continuous? What is the difference between two successive energy
values?
(10 pts)
(f) Calculate <x>, <x2>, and the uncertainty in the measurement of x.
Show all the integrations explicitly.
(15 pts)
(g) Suppose the free particle is in a three-dimensional cubic box (with
each side of length 2L). What would be the wave function of the particle?
What is the general form of the total energy of the particle? What are the
ground state, first excited state, and second excited state energies? What
do mean by the statement “an energy state is n-fold degenerate”? Is
there any degeneracy in the first excited state of the free particle in the 3D
cube? Explain your answer.
(5+5+5=15 pts)
4. Classically what is the total energy of a one-dimensional harmonic
oscillator? Plot the potential energy of that oscillator as a function of the x.
What is the energy value quantum mechanically? Is the energy
quantized? What is the ground state of the oscillator classically and
quantum mechanically? What is the difference between energy levels of a
quantum one-dimensional oscillator?
(10 pts)