1. Divide the interval into N segments and using the central difference approximation, set up the numerical eigenvalue problem.
2. Using Maple solve the eigenvalue problem for the first 3 eigenvalues using two different choices of N.
3. (Extra credit) Work the same problem using the Numerov technique.

PHY 741 -- Assignment #11

Continue reading Chap. 8 in Shankar.

1. Work problem 2.8.7 in Shankar, finding the action of the harmonic oscillator.

PHY 741 -- Assignment #12

Continue reading Chap. 12 in Shankar.

1. Determine the first 6 eigenvalues of the Schrödinger equation (in convenient units) for a particle of mass m in a two-dimensional spherical well of radius a. For this purpose you will need to know the zeros of cylindrical Bessel functions J_m(x) which you can obtain from Maple or in the list below.

   m	First zero	Second zero
   0	2.40483	5.52008
   1	3.83171	7.01559
   2	5.13562	8.41724
   3	6.38016	9.76102
   4	7.58834	11.06471

PHY 741 -- Assignment #13

Continue reading Chap. 12 in Shankar.

1. Work problem #12.5.3 on page 329 in Shankar.

PHY 741 -- Assignment #14

PHY 741 -- Assignment #15

Finish reading Chap. 12 in Shankar.

1. Determine the first 6 eigenvalues of the Schrödinger equation (in convenient units) for a particle of mass m in a three-dimensional spherical well of radius a. For this purpose you will need to know the zeros of spherical Bessel functions j_l(x) which you can obtain from Maple or in the list below.

   l	First zero	Second zero
   0	3.14159	6.28319
   1	4.49341	7.72525
   2	5.76346	9.09501
   3	6.98793	10.41712
   4	8.18256	11.70491

PHY 741 -- Assignment #16

PHY 741 -- Assignment #17

PHY 741 -- Assignment #18

Finish reading Chap. 15 in Shankar.

1. Derive Eq. 15.2.20 on page 415 in Shankar and evaluate it for a few simple cases.

PHY 741 -- Assignment #19

Finish reading Chap. 15 in Shankar.

1. Consider a many electron atom with two electrons in an incomplete shell -- 3d².
   1. How many degenerate states are in this configuration?
   2. Enumerate the corresponding states in the term notation ^(2S+1)L_J and check that there are the same number of states.
2. Consider a many electron atom with two electrons in two incomplete shells with the designation 3d¹4p¹. Answer the same questions a. and b. as above.

PHY 741 -- Assignment #20

Continue reading Chap. 16 in Shankar.

1. Work problem 16.1.3 in Shankar.

PHY 741 -- Assignment #21

Finish reading Chap. 17 in Shankar.

Consider an electron in the |n=2,l=1,m,m_s> state of a hydrogen atom. Taking H⁰ to be the unperturbed hydrogen atom, determine the lowest order correction in the presence of a perturbation of the form
H¹= A L · S + B ( L_z + g S_z),
where A and B are parameters, and g is the electron g-factor.

1. First consider A = 0.
2. Now consider B = 0. You may either use degenerate perturbation theory or evaluate the problem in terms of the total angular momentum J=L+S.
3. Now consider both A and B to be non-zero. You will need to set up a 6x6 matrix. Plot your results as a function of B/A.

PHY 741 -- Assignment #22

PHY 741 -- Assignment #23

Continue reading Chap. 18 in Shankar.

Consider a low energy electron being scattered by a weak spherical potential well:
V(r) = -V₀ for r ≤ a
V(r) = 0 for r > a

1. Find the differential cross section to lowest order in the potential strenth using the Born approximation.
2. Evaluate the lowest order partial wave phase shift and the corresponding differential cross section. Compare your result with the Born approximation.