QM2 Exercises, Fall 2008

(Ple= ase refer to Merzbacher, 3^rd edition )

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

= 8.1. Use eq. (17.37) to derive the D-matrix for j=3D1.

8.2. Go through the derivati= on 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 <= /span>. Measurement is made of <= /span> along the z= ' axis which makes an angle <= /span> with respec= t to the original z-axis. What is the expectation value of <= /span> that you wi= ll get?

9.2. (a) Along the z'-axis, an eigenstate <= /span> was prepare= d. Write down this eigenstate with respect to the original coordinate system (xyz). The angle between the z-axis and the z'-axis is <= /span> (Note that = you start with <= /span>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 el= ements can be found from eq. (37) at= this link

link1: http://www1.elsevier.com/homepage/saa/eccc3/paper48/eccc3.html#seccio= n2

(b) Confirm that this state indeed is an eigenstate of L_z' with eigenvalue m=3D0. The calculation should be done in the xyz frame.

9.3. For two spin 1/2 particles, let <= /span>. Use the product spin functions<= /span> as basis functions, construct the 4x4 matrix of <= /span>. Diagonalize the matrix to obtain the eigenvalues. For t= he degenerate states, construct states that are also eigenstates of S_{z. This is an alternative method of f=
inding
eigenstates of J² and J_z without using the C-G
coefficients.}

…………………R= 30;…………………………&#= 8230;…………….done …………………………̷= 0;……..

= Set-1= (due 9/2) Chapter 11

1.&n= bsp; Exercises 11.6; <= o:p>

2.&n= bsp; Exercise 11.7;

3.&n= bsp; Exercise 11.16; <= o:p>

4.&n= bsp; Exercise 11. 21.<= o:p>

5.&n= bsp; Prove eq. (11.115).

6.&n= bsp; Problem 3 for j= =3D3/2. Page 255.

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

<= span style=3D'mso-list:Ignore'>1.&n= bsp; Using (12.1), show that (12.2) is true, then show [H, L²]=3D0.

<= span style=3D'mso-list:Ignore'>2.&n= bsp; 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).

<= span style=3D'mso-list:Ignore'>3.&n= bsp; Exercise 12.4—why there is no bound state if V(r)=3D0 everywhere?

<= span style=3D'mso-list:Ignore'>4.&n= bsp;    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?

<= span style=3D'mso-list:Ignore'>5.&n= bsp;    Exercise 12.5.

6.     Exercise 12.7.

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

1.&n= bsp;    Exercise 13.1

2.&n= bsp;    Exercise 13.7.

3.&n= bsp;    Go thru the derivation from eq. (13.52) to the Rutherford result for the differential c= ross section below eq. (13.53).

4.&n= bsp;    Exercise 13.9. The answer what happens if it is a potential step instead of a potential well.<= o:p>

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

1.&n= bsp;    Problem 2, page 2= 75.

2.&n= bsp;    Problem 16, page = 277.

3.&n= bsp;    If two protons are fixed in space a= nd are separated by a distance “a”, use the Born approximation to calculate the differential cross section by an electron with incident momen= tum 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-s= lit experiment. You should be able to use the result from problem 3 of set-3.

4.&n= bsp;    Exercise 13.14

5.&n= bsp;    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)

    Deriv= e the Fano shape function for a general resonance. Use the handout page. Do the g= raph carefully and label them accordingly.<= /span>

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

        =

        = Set 7.   (d= ue 10/20/08) homework is in the pdf file link below
        =             &nb= sp; See the e-mail sent to you. Or the .pdf file assignment.pdf

=