PY354 Modern PhysicsMidterm ExamOctober, 29 2002
Name:______BU ID#______
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 write all your work in the space provided, including calculations and answers. Please circle answers where appropriate. If you need more space, write on the back. This is a long exam and in all likelihood, you will not finish, so don't be upset by leaving a few things incomplete. There are two parts, one section of short answers, one of long. Please note the point totals for each section and questions within each section, as they will guide you in how most effectively to apportion your time. Good luck!!
1. (6 pts): Anna is driving past Bob in a very fast car. As she zips by, she turns on her lightsin greeting, both front and rear simultaneously. If her car is 4m long and she is traveling at v/c=0.36, then, according to a stationary Bob on the sidewalk, which light comes on first, what is the time difference he perceives between the two events, and why?
2. (8 pts): Prove that if information cannot travel faster than the speed of light c, then it is impossible to have two events, causally related in one frame, violate causality in another inertial frame (that is, their order is reversed). (Hint: Start with an equation for, and write it in terms that shows the ordering of the relationship between the two events in different frames, that is, a direct relationship betweenand ).
3. (7 pts): What is the speed of a particle whose kinetic energy EK is twice its internal (rest mass) energy Einternal= mc2?
4. (8 pts): Explain in three or four sentences at most, why the Davidson-Germer electron diffraction experiment (your third lab) displayed the wave nature of particles, and why the diffraction maxima appeared as concentric rings on the phosphor screen in the laboratory you did.
5. (20 pts): Consider an electron double slit experiment where the slits have a width and separation of twice the electron de Broglie wavelength of the impinging electrons. Imagine you have a special device, very small, which can be accurately placed in front of one of the slits, easily blocking one or the other slit and allowing you to observe the results on the screen beyond.
a.) Both slits are open and you observe an interference pattern. What property of electrons are you measuring?
b.) Now you close one slit. What is the change in the pattern on the screen, and what property of the electron have you measured (and how)?
c.) The slits are changed such that they are no longer exactly the same width. With slit #1 open 360 electrons per second are detected in the peak and with slit #2 open 640 electrons per second are detected in the peak. How many electrons are detected in an interference maximum AND and interference minimum when both slits are opened? Explain your reasoning.
d.) Now, imagine that one slit is again closed and the open slit increases in width by a factor of 10. The screen shows a single-slit diffraction pattern. Using the language of the Uncertainty Principle, explain why an interference pattern is now observed.
6. (5 pts): Determine the probability density for the following wavefunction:
7. (15 pts): Well-defined Observables:
a.) (4 pts) Explain the relationship between a stationary state and one with a well defined energy.
b.) (4 pts) Show that the function does not have a well defined energy.
c.) (7 pts) Write down any state which has a non-trivial (non-zero) well-defined value of the momentum and determine that value.
8. (30 pts): Consider a particle confined to a harmonic oscillator potential, shown schematically on the right.
a.) (3 pts) Write down the energy for the harmonic oscillator energy eigenvalues in terms of the principle quantum number n,, the angular frequency, and .
b.) (3 pts) Draw on the figure (approximately) the first three levels, paying close attention to form of the wavefunction both inside and outside the potential.
c.) (3 pts) Draw in the probability density of these first three energy levels. (You can put them in as dashed, or on a new plot, if you would like.)
d.) (3 pts) The ground state, n=0 level of the harmonic oscillator is a Gaussian. What do you expect the product of the uncertainty in position and uncertainty in momentum to be? Explain.
e.) (5 pts) Write down, but do not solve, the full integrals that make up the product of the uncertainty in position times the uncertainty in momentum. Reduce and simplify as much as possible, without doing the integrals (What elements are zero?).
e.) (5 pts) The Schrödinger Equation for the harmonic oscillator is and the first excited state wavefunction has the form:
. Determine the energy of this state by direct substitution into the Schrödinger equation and solving for the coefficients.
f.) (8 pts) Determine the probability that the particle is further from the origin than the classical turning point. You do not have to solve the integrals, just express them in simplest form.
Formulas for PY354 Midterm Exam, October 24, 2002
Harmonic oscillator wavefunctions:
Gaussian integrals: