Homework on the Wavefunction (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material and be in your own words. Students who turn in identical (or close to identical) homework assignments will be asked to explain their answers orally to the TA or prof. A student who cannot explain how he or she arrived at a given answer will be charged with academic dishonesty.

You should justify all of your answers for full credit.

Match each wavefunction expression or graph on the left with the situation on the right which it could describe. If a wavefunction does not have a match, say so. Explain how you arrive at each of your answers.

1. c(x) = A sin (2px/L)
D. This is the ground state of an infinite square well of width L. A. An electron in a finite well of width L

2.
None. This is similar to C, but the probability of transmission is 1/4 and the probability of reflection is 3/4. B. A photon that has passed through (or been reflected by) a symmetric beam splitter giving a 50% chance of being reflected and a 50% chance of being transmitted

3.
None. This is close to F, an electron tunneling through a barrier, but the wavefunction inside the barrier is incorrect. Since the electron is classically forbidden in that region, the wavefunction should exponentially decay (like it does outside of a finite well), not oscillate. C. A photon that has passed through (or been reflected by) an unsymmetric beam splitter giving a 25% chance of being reflected and a 75% chance of being transmitted

4.
C. The first term represents a transmitted ray. The probability of transmission is the (magnitude) square of the coefficient, or 3/4. The second term is the reflected ray, with a probability of 1/4. D. An electron in an infinitely-deep well of width L

5.
A. This is the first excited state (n = 2, since two "humps") of a finite square well of width L. E. A photon that has passed through two symmetric beamsplitters, with a 50% chance of being reflected at each.

6.
B. The first term represents a transmitted ray. The probability of transmission is the (magnitude) square of the coefficient, or 1/2. The second term is the reflected ray, also with a probability of 1/2. F. An electron tunneling through a barrier of width L

1.	c(x) = A sin (2px/L) D. This is the ground state of an infinite square well of width L.	A.	An electron in a finite well of width L
2.	None. This is similar to C, but the probability of transmission is 1/4 and the probability of reflection is 3/4.	B.	A photon that has passed through (or been reflected by) a symmetric beam splitter giving a 50% chance of being reflected and a 50% chance of being transmitted
3.	None. This is close to F, an electron tunneling through a barrier, but the wavefunction inside the barrier is incorrect. Since the electron is classically forbidden in that region, the wavefunction should exponentially decay (like it does outside of a finite well), not oscillate.	C.	A photon that has passed through (or been reflected by) an unsymmetric beam splitter giving a 25% chance of being reflected and a 75% chance of being transmitted
4.	C. The first term represents a transmitted ray. The probability of transmission is the (magnitude) square of the coefficient, or 3/4. The second term is the reflected ray, with a probability of 1/4.	D.	An electron in an infinitely-deep well of width L
5.	A. This is the first excited state (n = 2, since two "humps") of a finite square well of width L.	E.	A photon that has passed through two symmetric beamsplitters, with a 50% chance of being reflected at each.
6.	B. The first term represents a transmitted ray. The probability of transmission is the (magnitude) square of the coefficient, or 1/2. The second term is the reflected ray, also with a probability of 1/2.	F.	An electron tunneling through a barrier of width L