QM2 Exercises, Fall 2008

(Please refer to Merzbacher, 3rd edition )

Set-8. (due 11/3/08)

8.1. Use eq. (17.37) to derive the D-matrix for j=1.

8.2. Go through the derivation of eq. (17.101) to (17.105) of Merzbacher, then complete the calculation of all the matrix elements that you need to do in problem 7.2.

8.3. Exercise 17.30.

Set-9. (due 11/10/08)

9.1.  A one-particle system is in the angular momentum state . Measurement is made of  along the z' axis which makes an angle  with respect to the original z-axis. What is the expectation value of   that you will get?

9.2. (a) Along the z'-axis, an eigenstate  was prepared. Write down this eigenstate with respect to the original coordinate system (xyz). The angle between the z-axis and the z'-axis is  (Note that you start with on the xyz-frame and then rotate the state to the new x'y'z' system using the D-matrix. Note that we have not defined the x- and y-axis in this problem).

The calculation of the D-matrix elements can be found from  eq. (37) at this link

link1:  http://www1.elsevier.com/homepage/saa/eccc3/paper48/eccc3.html#seccion2

(b) Confirm that this state indeed is an eigenstate of Lz' with eigenvalue m=0. The calculation should be done in the xyz frame.

9.3. For two spin 1/2 particles, let . Use the product spin functions as basis functions, construct the 4x4 matrix of . Diagonalize the matrix to obtain the eigenvalues. For the degenerate states, construct states that are also eigenstates of Sz.  This is an alternative method of finding eigenstates of J2 and Jz without using the C-G coefficients.

……………………………………………………………….done …………………………………..

Set-1 (due 9/2)  Chapter 11

1.      Exercises 11.6;

2.      Exercise 11.7;

3.      Exercise 11.16;

4.      Exercise 11. 21.

5.       Prove eq. (11.115).

6.      Problem 3 for j=3/2. Page 255.

Set-2. (due 9/9) Chapter 12.

1.     Using (12.1), show that (12.2) is true, then show [H, L2]=0.

2.     Use Mathematica or some software, plot  spherical Bessel function for order 0 and 1, and show that for large argument, it is indeed as given by eq. (12.20).

3.     Exercise 12.4—why there is no bound state if V(r)=0 everywhere?

4.     On page 261, the Hilbert space can be generated by plane waves or by spherical waves. What are the eigenvalues and their range used to label the base functions  in each set?

5.     Exercise 12.5.

6.     Exercise 12.7.

Set -3.  (due 9/16 but collected 9/25)  Chapter 13

1.     Exercise 13.1

2.     Exercise 13.7.

3.     Go thru the derivation from eq. (13.52) to the Rutherford result for the differential cross section below eq. (13.53).

4.     Exercise 13.9. The answer what happens if it is a potential step instead of a potential well.

Set-4. (due 9/23 but collected 9/25)

1.     Problem 2, page 275.

2.     Problem 16, page 277.

3.      If two protons are fixed in space and are separated by a distance “a”, use the Born approximation to calculate the differential cross section by an electron with incident momentum k. Assume that the axis between the two protons is perpendicular to the direction of the incident electron. Note that this is similar to a double-slit experiment. You should be able to use the result from problem 3 of set-3.

4.     Exercise 13.14

5.     Follow steps from eq. (13.65) to reproduce eq. (13.70).

Set-5. (due 9/30)

From the handout.

Set -6.   (due 10/7/08)

Derive the Fano shape function for a general resonance. Use the handout page. Do the graph carefully and label them accordingly.

See page 495 of Fano and Cooper, Rev. Mod. Phys. 40 (1968)   for examples of such resonances.

Set 7.   (due 10/20/08) homework is in the pdf file link below

See the e-mail sent to you. Or the .pdf file  assignment.pdf