1. Define the annihilation and creation operators for the
simple harmonic oscillator in one dimension, using either notation
A and A^{+} (G3, p112, G2 p131) or a and a^{+} (L4 p193, L3 p201): a) Do G3 problem 11, p118 (not in G2). This problem can usefully be preceded by G3, problems 5-6, and the definition of the unit operator in problem 7. b) Use these problem(s) as a means of clarifying your notes on the SHO, and submit a short summary of these notes for assessment. Include in your discussion the derivation of how you find the ground state wave function of the SHO, including normalization, by an operator method. |
2. Familiarize yourself with the properties of angular
momentum operators and the first few spherical harmonics, by reading one of the
following references (G3 chap 7, p120-128, G2 chap 11, p188-197, Liboff, chap 9
(L4 p349-365, L3 p362-378), in conjunction with your lecture notes.
Then do: a) G3 problem 2, p129; b) G3 problem 6, p129; c) G3 problem 7, p129. |
3. Do G3 problem 5, p129 (G2 prob 5, p201). This is an example
of a problem which can be classified compactly using operator methods. Look up
a suitable quantum chemistry textbook (ask for a reference if you need to), and
list some molecules which have such rotational spectra, giving reasons why they do. A suitable extension to this problem are G3, problem 1, p 129, and problems on HCl and Carbon2 molecules on previous Comprehensive Exams. |
4. At time t = 0, a 1D oscillator is equally likely to
be in its ground or first excited states, and it has zero probability of
being in any other state. Moreover the expectation value of p = 0 at t
= 0. The system is in a pure quantum state.
a) compute the expectation value of p(t) as an explicit function of time, evaluating all necessary matrix elements. b) Compute as a function of time the probability to find the oscillator in each of its energy eigenstates. |
This problem was set in the Track 1 Quantum Mechanics exam in January 1995, and a rather similar t-dependent oscillator problem was set in January 2004. |
5. Do problem 5 from G2 chapter 14, p250 (not in G3), for angular momentum l = 1 and compare your answer with question 3 above. Derive the matrix that you need to diagonalize for l = 2, but don't attempt to solve this equation unless you have access to a good algebraic matrix solver. For this matrix, make the simplifications which occur if I_{2} = I_{1}, and use this to work out the explicit answers, comparing them with question 3. |
6.The conceptual parts of this problem have often been set on
the examinations (mid-term or final) in the past.
A particle with mass m moves in a double-well potential, where there are two
one-dimensional wells of depth V(x) = -V_{0} and width
(b-a), where the inner edges of the two wells are at x = b and -b.
a) Write down the form of bound state wave functions in the different regions of the potential, taking full account of symmetry. b) Sketch (i.e. Draw) the wave function of the ground state and the first excited state in this potential. Describe what happens to the energies of these two states as the wells are placed further apart (i.e. as b is increased (slowly)). c) Picture the second excited state in this potential. What can you say about this state in the d-function limit, where V(x) = -V_{0} is increased, and (b-a) is decreased, such that the product V_{0} (b-a) is kept constant. (d)- an optional part for 50% extra credit: Complete the quantitative aspect of the above questions, by proving the relationships given in G3 problem 11, p92-93 (G2 prob 16, p112), and then taking the d-function limit. |
7. This problem comes from a book by N. Zettili, Quantum Mechanics:
Concepts and Applications, which has many problems and worked examples. This one is
exercise 5.13 on page 312. Consider a system which is described by the state Y = (3/8)^{1/2}Y_{11} + (1/8)^{1/2}Y_{10} + AY_{1,-1}, where A is a real constant.
(a) Calculate A so that |Y> is normalized; |
8. A particle is in an eigenstate of L^{2} and L_{x} with eigenvalues 2h^{2} and 0 respectively. What are the possible results and corresponding probabilities of measurements of L_{z}? |
This problem was set in the Quantum Mechanics exam in August 2004. |
9. If the above range of problems do not give you enough choice, or you feel you need more or less challenge..., arrange to substitute another problem, or conceivably more than one problem, by personal agreement with me well before the problem set is due. This can include previous Comprehensive Exam problems. |
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