You may consult your notes, the
textbook, your calculator, an integral table (or integrals.com, etc.), or me,
but nothing and no one else. Explain all your reasoning and show all your work.
If you have any doubt about what is being asked, please ask me about it. When youÕve finished, re-read your work,
cross out whatever you no longer believe to be true, and make sure you have
answered the questions. Put the pages in order, put this sheet on top, staple
them, write and sign the honor code, and please put your name on this sheet.
1) (20 points) The gravitational interaction has the same sort
of distance dependence as the Coulomb interaction. This means you could make a
Bohr model of the earth-sun interaction, in which you find the allowed orbital
energies (and radii) for the earth. Using the masses of the earth and the sun
(on the bookÕs endpapers) and the gravitational constant G=6.67X10-11N
m2/kg2, find the allowed orbital radii and allowed
energies for the earth. Then, considering that the earth is 150 million
kilometers from the sun, determine our present value of n. Then, determine the
energy of the photon that would be emitted if the earth made a transition to
the n-1 energy state. What fraction of the earthÕs kinetic energy would that
be?
2) (25 points) Do you remember that delightful wave function, , we worked with in class? Is its kinetic energy well
defined? Regardless of your answer to that question, determine the expectation
value of the kinetic energy of the particle described by this wave function.
3) (25 points) Solving the time-independent Schrodinger
equation for the infinite well of width L, one finds the eigenfunctions un(x).
LetÕs consider another set of functions, wn(x), where . These functions are of course normalized, and their squared
magnitudes do not differ at all from the squared magnitudes of the unÕs.
So, are these w function also eigenfunctions for this system? Find the
expectation value of the energy for w1(x). Compare it with the value
for u1(x), and comment if you can. Finally, is the expectation value
for the particleÕs position constant in time?
4) (30 points) For the particle in the infinite well, consider
the wave function that at t=0 has the form . (a) Write down the
expression for this wave function at all times t. (b) Verify that this wave function is normalized. (c) Determine <x>, not forgetting that this may depend
on time. (d) Determine <p>, not
forgetting that this may depend on time. (e) Are these two expectation values related to each other as
they ought to be?