PY354 Modern Physics Final Exam December, 16, 2003
Name:______BU ID#______
Section time:______
This is a closed book exam. Any formulas you are likely to need, and would have trouble remembering are provided on the back page. Please do not use formulas or expressions stored in your calculators. Please write all your work in the space provided, including calculations and answers. Please circle answers wherever you can. If you need more space, write on the back of these exam pages, and make a note so the grader can follow. This is a long exam, and in all likelihood, you will not finish, so don't be upset by leaving things incomplete. Please note the point totals for each problem and section as they will guide you in how most effectively to apportion your time. There is a total of 120 points. Good luck!!
Problem 1. (20 pts)
Consider a square well with infinite sides of width L:
(a) (8 pts) Calculate, symbolically, the energy and wavelength of a photon emitted when a transition between the n=5 and the ground state is made.
(b) (4 pts) Write down the expression for the probability that a particle in the nth state will be found in the first 1/3 of a well of width L.
(c) (4 pts) Compare, with respect to unity, and as a function of the quantum number n, the ratio of the probability that the particle will be found in the first third of the well to the probability that it will be found in the middle third. Write down the expression and argue mathematically, but do not solve the integrals. Feel free to make a graph.
(d) (4 pts) In quantum mechanics, we often talk about the correspondence principle. What is the correspondence principle, and what does it mean?
Problem 2. (16 pts)
The energy levels of a 3D quantum box, of unequal lengths Lx, Ly, Lz on each side are given by:
.
(a.) (10 pts) If write down the energy levels and degeneracy for the first 7 states. Hint: First simplify the above equation, then count the energy of the states in units of
(b) (6 pts) Explain why the second x-direction state is not reached until the ?? energy level. A conceptual explanation here is fine.
Problem 3. (26 pts) Schrödinger Equation: quantum mechanical tunneling
In this problem, you will derive the approximate transmission probability of a particle with energy E incident on a potential barrier of energy U, U>E. In the figure below the wavefunction in region I is shown split into two parts, one for the incoming wave and one for the reflected wave. The number of particles per unit time which impinge on the barrier is given by
where kI+ is the wavevector (related to the group velocity). The transmission probability is given by the ratio of the outgoing flux to the incoming flux, or
(a) (8 pts) Identify the three regions in the figure and then write down the Schrödinger equation and the trial wavefunctions for each of the three regions. Be sure to write down the value of the wavevector in each region.
(b) (10 pts) Write down the boundary conditions for this problem, and then apply each of the boundary conditions appropriately to yield the set of coupled equations. Do not solve the coupled equations.
is the equation that would result from solving the equations above.
(c) (8 pts) Make the following approximations to find the transmission probability in simpler form: Assume that the height of the barrier is high relative to the incident energy E of the particles, and that the barrier is relatively wide such that kII L >1, where kII is the wavevector (real) in the barrier region. Show, finally, that the transmission is proportional to
Problem 4. (8 pts) Conceptual elements of quantum mechanical tunneling:
(a) (4 pts) Imagine that a barrier looks like the figure to the right. Order the regions in terms of the wavelength in each of the three regions, from shortest to longest wavelength. For this case E>UII.
(b) (4 pts) Calculate the ratio of the kinetic energy difference between region I and region III if the former has a wavelength a factor of two different than the latter.
Problem 5. (12 pts) Addition of angular momentum
(a) (6 pts) Consider an atom which has a single electron in the n=3 state. Determine the possible values of the total angular momentum.
(b) (6 pts) Now consider an atom which has two electrons in the d-shell. Determine the possible values of the total angular momentum. Does the total symmetry of the wavefunction change with j?
Problem 6. (18 pts) Hydrogen atom:
(a) (4 pts) In the hydrogen atom, for fixed orbital quantum number l, what happens to the number of radial anti-nodes as the principle quantum number n goes up? Explain.
(b) (4 pts) In the hydrogen atom, for fixed principle quantum number n, what happens to the number of radial anti-nodes as the orbital angular quantum number l goes up? Explain. Feel free to draw.
(c) (10 pts) The radial probability density in the Hydrogen atom is given by where the radial wavefunction takes the form for principle quantum number n. Find the most probable location of the radial component of the wavefunction for any n.
Problem 7 (10 pts) Wave packets:
(a) (6 pts) The equation for the dispersion of a massive particle is: , and for a massless photon is . For each case determine the phase and group velocities.
(b) (4 pts) Show how the group velocity is related to the particle’s classical velocity for massive particles.
Problem 8 (10 pts): Molecules and Symmetrization
(a) (2 pts) What is the difference between Fermions and Bosons in terms of the symmetry of the total wavefunction?
(b) (8 pts) For two electrons in a simple square well we argued that the ground state is generally found to be a spin triplet. On the other hand, for the two electrons in a Hydrogen molecule, the ground state is generally found to be a spin singlet. Explain this difference, and show what the total wavefunction is in both cases.
Formulas for PY354 Final Exam, Dec 16th, 2002
Harmonic oscillator wavefunctions:
Gaussian integrals:
;;;
R+T=1 for free particle
Symmetric and antisymmetric spatial wavefunction:
Useful constants:
9