Graduate Course: Quantum Physics

John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona 

Quantum Physics: Problem Set #3, initial version

This problem set is due Monday 31st March at the beginning of class, but you are strongly encouraged to submit half the problems a week earlier. Five problems are needed for a complete score, but you can buy 'credit' for future problem sets by doing more. Everyone should do problem 2 and either problem 6 or 8. You can also submit previous Comprehensive Exam problems, and if you are doing a Project, you should be choosing it soon.

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 I2 = I1, 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) = -V0 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) = -V0 is increased, and (b-a) is decreased, such that the product V0 (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)1/2Y11 + (1/8)1/2Y10 + AY1,-1, where A is a real constant.

(a) Calculate A so that |Y> is normalized;
(b) Find L+Y;
(c) Calculate the expectation values of Lx and L2 in the state |Y>;
(d) Find the probability associated with a measurement that gives zero for the z-component of the angular momentum;
(e) Calculate <F|Lz|Y> and <F|L_|Y>, where F = (8/15)1/2Y21 + (4/15)1/2Y10 + (3/15)1/2Y2,-1.  

8. A particle is in an eigenstate of L2 and Lx with eigenvalues 2h2 and 0 respectively. What are the possible results and corresponding probabilities of measurements of Lz?
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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