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

### These problems may be done in your own time, or substituted for questions on sheet 3, 4 or 5. Note that a full set of recent Comps questions (password protected) can be downloaded as .pdf files.

 1. A photon of wavelength l is incident upon a stationary electron, collides with the electron and scatters inelastically through an angle q, achieving the final wavelength l '. To lowest order in |l '-l |/l, and assuming that the electron is non-relativistic, obtain an expression for  l'( q) in terms of l and physical constants. This problem on Compton scattering is discussed  by Gasiorowicz (G2 and G3) in chapter 1, and was set in the Track 2 Basic Quantum Mechanics exam in January 1996.

 2. A spinless, nonrelativistic particle of mass m moves in three dimensions in a potential which is non-zero only in a thin shell of radius r0. The corresponding potential is V(r) = - Dd(r - r0), where D and r0 are positive constants, and d is the Dirac d-function. Assume that the angular momentum of the system is l = 0. Show that there is a bound state of the system only if D is larger than some positive constant D0, and determine the value of D0. Find the bound state wavefunction for the system, and sketch it. You need not normalize the wavefunction. This problem was set in the Track 2 Basic QM exam in August 1995.

 3. a) Two neutrons are trapped in a spherical potential V(r) = 0 if r < R and V(r) = + infinity if r > R. Ignoring the interactions between the neutrons, find the ground state wave function of system (including the spin wave function). b) Assume now that the neutrons interact with each other through the potential Vi = As1. s2, where si is the Pauli spin matrix for the i-th neutron, and A is a constant. Treating Vi as a perturbation, find the approximate ground state energy. This problem was set in the Track 2 Basic QM exam in August 1995.

 4. A spherically symmetric attractive potential has the form V(r) = - V0r-a, where V0 > 0 and 2 > a > 1. By using a trial wavefunction R(r) = exp(-br) show that the lowest energy bound state for a particle of mass m should be found below This problem was set in the Track 2 Basic QM exam in August 1998.

 5. Denote the energy eigenstates of a one-dimensional simple harmonic oscillator by |n>. You are told that, at t = 0, there is zero probability that a measurement of the energy will give a value greater than 3hw/2. Moreover, at t = 0, the expectation value of p is as large as possible, consistent with the above information. Determine the oscillator's state vector |t> and the expectation value of p for all subsequent times. Hint: Do not forget about the relative phases of your basis states. It might be convenient to use an operator like... (the explicit form of the operator A for the SHO was given). This problem was set in the QM exam in January 2004.

 6. Consider a two level system which obeys the Schrödinger equation: where W1,W2 and V are real numbers. 1) Obtain expressions for the energy levels and normalized eigenstates of this Hamiltonian. 2) Assume now that W1 = W2 and that at t = 0 the system is intially in the state a = 1, b = 0. Find the time t it takes for the system to reach a (relative) maximum probability of being in the state a = 0, b = 1. (Note: the above Schroedinger equation is an approximation to the following physical situation: two atoms can bind an electron, one with energy W1 and the other with energy W2. The electron initally on the left atom is represented by a = 1, b = 0; on the right atom by a = 0, b = 1. If the atoms are close together, the electron can hop between the atoms.) This problem was set in the Track 2 Basic QM exam in August 1996.

 7. Each of two identical, non-relativisitic, spin-1 particles of mass M are independently bound to the same center of force by a common harmonic-oscillator potential (with spring constant k). The two particles do not interact with each other. 1) (60%) Write down the two-particle wave-function of the ground state. What is the ground state energy? What is its degeneracy? 2) (40%) A magnetic field is then applied along the z-axis, which adds a term to the Hamiltonian H' = A(Sz,1 + Sz,2), where Sz,i is the spin operator (in the z-direction) of particle i, and A is a constant. Neglect the interaction of this magnetic field with the orbital angular momentum of either particle. To first order in H', determine how the ground state is shifted and/or split. This problem was set in the Track 2 Basic QM exam in January 1996.

 8. A C2 molecule can be regarded as a rigid rotator at low temperatures. a) Write down its rotator Hamiltonian in terms of (i) its total angular momentum L, the distance d in between the two C atoms, and the mass M of one C atom. b) Find the energies and orbital degeneracies if (i) its ground state E0, and (ii) its first excited states E0. c) Suppose a weak perturbation V = acosq is turned on at time t = 0. What transitions from the ground to the excited states (if any) will the perturbation induce? This problem was set in the Track 2 Basic Quantum Mechanics exam in August 2001.

 9. The absorption spectrum of HCl gas is the subject of this problem. 1) (30%) Explain the meaning of the term absorption spectrum of a gas, and explain how such a spectrum can be measured in the laboratory. 2) (70%) In HCl gas, an number of absorption lines have been observed with the following wavenumbers (in cm-1): 83.03, 103.73, 124.30, 145.03, 165.51, and 185.86. Are these vibrational or rotational transitions? (You may assume that transitions involve quantum numbers that change by only one unit.) Explain your reasoning briefly. 2a) If the transitions are vibrational, estimate the spring constant (in dyne/cm). 2b) If the transitions are rotational, estimate the separation between the H and Cl nuclei. What J values do they correspond to, and what is the moment of inertia of HCl (in gm-cm2)? This problem was set in the Track 2 Basic QM exam in January 1996.

 10. A particle of charge q and mass m moves in one dimension under the action of a uniform electric field E. The corresponding potential is V(x) = - qEx. Initially (i.e. at t = 0) the average (i.e. expectation value) of position and momentum are < x > = x0 and < p > = 0. Proceeding quantum mechanically.... a) Determine d< p >/dt, and solve it to find < p > as a function of time. b) Determine d< x >/dt, and solve it to find < x > as a function of time. c) Compare your results with classical physics. We haven't spent much time on this topic: time evolution of operators. But it did come up sometimes on the (Basic) QM Comprehensive exam; this example was set in August 2001, and a Simple Harmonic Oscillator was set in January 2004. The theory is given by Gasiorowicz (G3 p103-105, 116-117, G2 p 124-128, 139-141). Maybe someone should produce a web page on the topic!

 11. Consider angular momentum L, coordinate R, and linear momentum P. a) Let (Operator) L± = Lx ± iLy and R± = X ± iY. Find the commutators [L±,R±] and [L±,Rm]. b) Find [L±,Z], [Lz,R±], and [Lz,Z]. This example was on the QM exam in January 2005. Try going back to square-one and writing L = R x P. Liboff, problem 9.7, has many of the worked out. Other angular momentum questions were set in January and August 2004, downloadable from the department web site.

 12. A spin-1/2 particle has a spin angular momentum s and a magnetic moment m = gs. At t=0 the particle is placed in a magnetic field of magnitude H0 along the z axis, with the expectation value of the spin in an arbitrary direction. The Hamiltonian of the spin in the field is the ordinary Zeeman Hamiltonian H = -m.B = gH0sz. a) The time development of the system may be described by the time development operator T(t) in spin space, such that at time t the spin wave function is y(t) = T(t)y(0), where y(0) is the wave function at t=0. Write the 1st-order ordinary differential equation obeyed by the operator T(t). b) Solve this differential equation, expressing T(t) as an exponential operator function of time. c) Write the matrix of T(t) in a representation with sz diagonal. (The Pauli spin matrices were given). This question was set on the Quantum Mechanics examination in January 2005.

 13. Show that for the ground state of the hydrogen atom: a) the most probable value of r is la, where we have a = h2/me2, and give a value for l. b) the mean value of r-1 is aa-1, and give a value for a. c) the mean value of r-2 is ba-2, and give a value for b. This problem on expectation vales for the hydrogen atom was set in the Track II Basic Quantum Mechanics exam in January 2002. If you do this problem, remember to normalize the ground state wavefunction.

 14. A two-dimensional oscillator has the Hamiltonian H = 1/2(px2 + py2) + 1/2(1 + axy)(x2 + y2), where a << 1. (a) Give the eigenstates for the three lowest levels when a = 0. (b) Evaluate the first-order perturbation of these levels for a ¹ 0. This question was set on the QM exam in August 2004.