Physics 237, Final Exam
Wednesday May 5, 2010
7.15 pm – 10.15 pm
Do not turn the pages of the exam until you are instructed to do so.
Exam rules: You may use only a writing instrument and your equation sheet while taking this test. You may not consult any calculators, computers, books, or each other.
1. Problems 1, 2, and 3 must be answered in booklet # 1.
2. Problems 4, 5, 6, and 7 must be answered in booklet # 2.
3. The answers need to be well motivated and expressed in terms of the variables used in the problem. You will receive partial credit where appropriate, but only when we can read your solution. Answers that are not motivated will not receive any credit, even if correct.
At the end of the exam, you need to hand in your exam, your equation sheet, and the three blue exam booklets. All items must be clearly labeled with your name, your student ID number, and the day/time of your workshop.
ID number: ______________________________________________
Workshop Day/Time: ______________________________________
INTENTIONALLY LEFT BLANK
Problem 1 (35 points) ANSWER IN BOOKLET 1
Consider the motion of an electron of mass m under the influence of the following potential V:
a) What is the general solution of the time-independent Schrdinger equation for this system?
b) What is the energy of the solution obtained in part a)?
c) What is the zero-point energy of the system?
d) If at time t = 0, the electron is in a state corresponding to the first excited state, what is the probability that the electron will be in that same state at time t = lC/c where lC is the Compton wavelength of the electron?
Problem 2 (30 points) ANSWER IN BOOKLET 1
Consider one electron in the n = 2 shell of a one-electron atom. If we ignore the spin-orbit coupling, then the wavefunctions in the n = 2 shell are degenerate. Assume that the electron can be found with equal probability in each of the degenerate n = 2 states.
a) Write down the wavefunction describing this electron in terms of the eigenfunctions of the one-electron atom. Make sure your wavefunction is properly normalized.
b) What is the shape of the probability density distribution of this electron? You will need to specify the r dependence, the q dependence, and the f dependence of the probability density distribution.
Problem 3 (35 points) ANSWER IN BOOKLET 1
Consider a He atom. When He is in its ground state, both electrons are in the 1s state.
a) What is the proper spectroscopic notation of the ground state of He?
If we shine ultraviolet light on the He atoms, we can excite the atom into its first few excited states. These excited states have one electron in the 1s state and one electron in the 2s or in the 2p state.
b) What is the proper spectroscopic notation of the excited states in He when the electrons are in a (1s)(2s) configuration?
c) What is the proper spectroscopic notation of the excited states in He when the electrons are in a (1s)(2p) configuration?
d) Draw an energy-level diagram, showing all the states of He discussed in parts a), b), and c). Label each state with the proper spectroscopic notation. Do not ignore the spin-orbit coupling.
e) In the diagram obtained in part d), indicate which transitions can occur between the excited states discussed in parts b) and c) and the ground state discussed in part a).
f) If we put the atom in an external magnetic field of strength B, we see an increase in the number of transitions we can observe. How many transitions will we be able to observe now?
Problem 4 (30 points) ANSWER IN BOOKLET 2
Consider a photon, moving in one dimension in a region where V = 0.
a) Starting with the relativistic expression for total energy in terms of linear momentum and mass, formulate the Schrdinger equation for the photon.
b) Use separation of variables to solve the Schrdinger equation. What is the wavefunction that describes the photon?
Problem 5 (30 points) ANSWER IN BOOKLET 2
An energy diagram for Hydrogen is shown in the Figure below.
Two techniques can be used to confirm the energies of the levels of Hydrogen: studies of emission spectra and studies of absorption spectra.
At low temperatures, the absorption spectrum is dominated by transitions that are part of the Lyman series. Estimate the temperature at which Balmer lines will be observed in the absorption spectrum. Note: you do not need to evaluate the numerical expression you obtain for the temperature T.
Problem 6 (35 points) ANSWER IN BOOKLET 2
a) Consider the following energy-level diagram for atoms with 14 nucleons. Energy levels with similar properties are connected with dashed lines.
Identify the isospin and the z component of the isospin for all states shown in the diagram.
b) Consider the following reactions:
For each of these reactions, state the fastest interaction through which it can proceed. If the reaction is forbidden, indicate why.
c) Consider the 4 lowest energy levels observed for the two-nucleon system, shown in the Figure below.
For each of the four states shown in the Figure, determine the total spin. Which of these states are stable? What is the cause of the energy difference between the state shown for 2He and the state shown for 0n?
Problem 7 (5 points) ANSWER IN BOOKLET 2
Match the following players
a) Y Berra
b) B Williams
c) M Mantle
d) D Jeter
e) L Gehrig
to the following At Bat (AB) numbers of these players for the New York Yankees:
Note: these number were correct on May 1, 2010 but may have changed on the day of the exam.