Modern Physics 2005 Test 2 due Friday, Oct. 14, at 9 am

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 you’ve 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 book’s 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 earth’s 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 u_n(x). Let’s consider another set of functions, w_n(x), where . These functions are of course normalized, and their squared magnitudes do not differ at all from the squared magnitudes of the u_n’s. 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 particle’s 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?