Your name:

Please read these directions BEFORE beginning.

There are 6 problems, plus the essay assigned two weeks ago...

Please use separate sheets for each numbered problem, and write on one side only. Write your name on the top of each page, and staple your solutions together, with this page on top, thanks.

You may have two 8.5 x11 sheet of paper with your own notes.

Feel free to look over my crib sheet before the exam begins.

Don't do integrals if you don't have to! Use symmetry and operator arguments when possible. However, you should always show your work, and explain your reasoning if you feel you can skip algebra by making use of symmetry, etc.

This exam will focus on new material since Exam II, especially quantum mechanics in 3D, but naturally any material from the earlier parts of this course may be included. Primary new topics we've covered are:

Ch. 7 Operator Methods in Quantum Mechanics

* Raising and Lowering operators, and the application to the Harmonic Oscillator.

* Time dependence of operators.

Ch. 9-10 Quantum Mechanics in 3 Dimensions

* The method of separation of variables. (examples in this section includes separation in both Cartesian and Spherical coordinates)

* Consequences of rotational invariance, Noether's theorem. (Symmetry <-> Conservation Law)

* Angular momentum as an operator, commutation relations.

* Radial equation - interpretation, boundary conditions.

* Solutions of radial equation for infinite spherical box, spherical Bessel fns.

* Finite spherical box (positive energy solutions (phase shift) and negative energy solutions)

Ch. 11 Angular momentum

* Eigenfunctions of L_z alone, example of rotating solid body.

* Operator method revisited for angular momentum, raising and lowering operators.

* Spherical Harmonic functions. (Derivation, properties, use)

* Expansion theorem in 3-D.

Ch. 12 Hydrogen atom

* Method of solution via power series.

* Energies, and degeneracies.

* Notation of radial wavefunctions and Ylm's, including sketches + qualititive aspects.

* Relations between quantum numbers N,l, and n_r.

Ch. 8 N-Particle Systems

* Consequence of translational symmetry. (conservation of total momentum)

* 2 particle systems, and reduction to center of mass and relative coordinates.

* Systems of identical particles, general Pauli principle and its consequences.

* Symmetric and Antisymmetric wavefunctions, and the exchange operator P_12.