The graduate students in PHYS5050 are expected to complete all assignments and tests as the PHYS4011 students, except for a final assignment in which they will complete some literature review work on a topic of choice.

__Final Exam:__ On Thursday, April 14, 10 am, come to PSE230 to pick
up the question sheet. Return at 1 pm (out of 100%), for every hour after
that: minus 10% until 5 pm. To be returned in PSE230.

__ Assignment 1:__ The Maple assignment is for hand-in,
Monday, Jan. 17, the pencil-paper assignment is for self-study.

Pencil-paper: Problem 4.40 in Griffiths: Prove the 3-dimensional
virial theorem for stationary states: 2__Solution: __ page1;
page2

Maple: do Exercises 1 and 2 in the worksheet HatomParabolic.mws (Theme 1). Write up a summary of what you found in Exercise 1 and hand in. From Exercise 2 comment on the graphs produced on the computer screen, but there is no need to hand them in. Do not hand in a Maple printout, just your findings.

__ Assignment 2:__ The Maple assignment is for hand-in,
Monday, Jan. 24, the pencil-paper assignment is for self-study.

Pencil-paper: Problem 6.1/6.4 in Griffiths: Consider a particle moving in a onedimensional infinite square well potential (0 < x < a). Add a Dirac-delta function potential at the centre of the well: W(x)=A*delta(x-a/2), where A=const.

a) Find the first-order energy corrections for all states. Why are the energies of the even-n states not perturbed?

b) Find the first three non-zero terms in the infinite series for the first-order correction to the ground-state wavefunction.

c) Find the second-order energy correction for the ground state. Can you do it for all states?

Maple: do Exercises 1 to 3 in the worksheet PerturbedOscillator.mws (Theme 1). Do not hand in a Maple printout, just your findings.

__ Assignment 3:__ The Maple assignment is for hand-in,
Monday, Feb. 7 (part A), the pencil-paper assignment is for self-study.

Pencil-paper for self study: Problem 6.40 a in Griffiths:

Find the first-order correction to the ground state of hydrogen in the presence of a uniform external electric field of strength F by using a wavefunction of the form (A+B*r+C*r^2)*exp(-r/a)*cos(theta) in the equation (H0-E0)|psi1> = - (W-E1)|psi0>.

Then use the equation E2 =

Maple assignment for hand-in: Maple worksheet which contains 2 Exercises which comprise part A: Coulomb1dNumerical.mws . Hand-in: Wednesday, Feb.9 for part A. In Part B you will solve for the same states in a matrix diagonalization. The worksheet for this partB: Coulomb1dim.mws . Hand-in date for part B: February 21.

**
Lecture notes and reference to Maple worksheets:** Notes .

__Assignment 4:__Pencil-paper part for self-study. Problem 4.41 in Griffiths:
Define the probability current in quantum mechanics as the vector
J = I*hbar/(2*m*)[psi Del(psi^*) - (psi^*) Del(psi)], where Del is the
gradient operator, and psi^* is the complex conjugate of the wavefunction.
(a) Show that a continuity equation can be derived from the Schroedinger
equation and its complex conjugate with this current J and rho=(psi^*)(psi).
(b) Calculate the current for a hydrogenic state with quantum numbers
(n,l,m)=(2,1,1). (c) Interpret (m)J as the flow of mass, and therefore
express angular momentum as a volume integral of the cross product
expression (m)(r cross J). Calculate this for the (2,1,1) (i.e. 2p_1)
state of hydrogen and make your observation.

__Maple part of Assignment4: (hand-in: Friday, March 11).__
Carry out Exercise 3 in the Maple worksheet TDPT.mws. Compare the transition
probabilities from TDPT with the close-coupling results for excited states
k=1,2,3 in a time interval (pulse duration) when differences become apparent.
Summarize your results.

__Assignment 5 (hand-in:, Monday, March 21):__ This is a pencil/paper assignment
for which you can use some Maple for illustration. Problem 5.20 in Griffiths
(requires you to study Section 5.3.2 Band structure, pp. 224-228 on your own).
Replace the periodic sequence of
delta-spikes used as a potential by delta-wells.
Construct the graph shown in figure 5.6 with a choice of beta=-1.5.
Not much new work for
the E>0 states, but the E<0 states need to be calculated
(define the wavenumber kappa=sqrt(-2*m*E)/hbar in this case).
How many states are there in the first allowed band?

Lecture Notes as a Word File. Can be printed in one go, but try out one page first, may need to adjust Page Setup. Notes

__Textbook(s).__
The course can be supported by any decent quantum mechanics book.
The bookstore erroneously lists Shankar's book as required, it
is a recommended text for those who don't own a book on QM yet.
If you have a copy of Liboff, or of Griffiths' book, you can use
it. A book I really like for showing explicit details on many
topics to be covered is the 2nd volume of Cohen-Tannoudji et al.
on Quantum Mechanics. I have ordered some copies for both the
York bookstore, and for Uni-Text. You can also try the UoT bookstore
or Amazon,Chapters, etc.

One book referenced in this course is HA Bethe + EE Salpeter: Quantum Mechanics of One- and Two-Electron Atoms. QC174.17 P7 B47. It is also permanently in the library as part of the physics encyclopedia (Handbuch der Physik), QC21 H327, vol.35

We will use Maple for some assignments and classroom demonstrations. If you don't know what that is, then ask around. Maple9.5,9,8, and even 7 are quite acceptable versions. Get yourself an ACADLABS account and look at my Computational Physics using Maple website.

__Marking scheme:__

6 Maple-based assignments: 30%

Midterm exam: 30%

Final Exam: 40%

Pencil-paper assignments for test preparation will be announced, these will be not for hand-in, but for self-study, and solutions will be posted.

__Topics.__
The blue-book (minicalendar) description will be followed more or
less with the exception of topic 1 (H atom) covered in 4010 extensively.
We will start with stationary perturbation theory, relativistic corrections
to the Schroedinger-Coulomb problem, fine structure (magnetic
interactions), and somwhere we may include an introduction
to the Dirac equation (relativistic quantum mechanics), or lectures on
scattering theory in non-rel. quantum mechanics.