Physics 237, Final Exam
Wednesday May 9, 2012
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, and 6 must be answered in booklet # 2.
3. Problems 7, 8, and 9 must be answered in booklet # 3.
4. 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: ______________________________________
One-Electron Atoms – Details
The following table lists the n = 1, n = 2, and n = 3 wavefunctions of the one-electron atom.
In these wavefunctions, the parameter a0 is defined as
The energy of each wavefunction is equal to
The radial probability density for the electron in a one-electron atom for n = 1, 2, 3 and various values of l.
Ground-state Properties of the Deuteron
á Binding energy: DE = 2.22 MeV
á Nuclear spin: 1
á Nuclear parity: even
á Magnetic dipole moment: µ = +0.857µn
á Electric quadrupole moment: q = +2.7 × 10-31 m2
á Charge distribution half-value radius: a = 2.1 F
Problem 1 (20 points) ANSWER IN BOOKLET 1
The energy of a linear harmonic oscillator is equal to . The angular frequency of this oscillator is . Use the uncertainty principle to determine the minimum energy of the oscillator, expressed in terms of the frequency v where
Problem 2 (20 points) ANSWER IN BOOKLET 1
Consider ¹ - capture by a deuteron. A slow ¹ – in liquid deuterium looses energy by a variety of mechanisms until it finally ends up in the lowest Bohr orbit around the (pn) nucleus. It is then captured through the action of the strong force. The result of this capture is the following reaction:
¹ - + d --> n + n.
a) What it the angular momentum of the initial state?
b) What are the possible states of the exit channel? List all possible states with L up to 3 using spectroscopic notation.
c) Based on the conservation properties of the strong force, which of the states in part b) do you expect to be able to observe in this reaction?
d) How can you use this information to determine the parity of the pion?
Problem 3 (30 points) ANSWER IN BOOKLET 1
Consider the following wavefunction describing a single particle in a one-dimensional world:
The constant a is known.
a) What is the value of N?
b) What is the expectation value of x?
c) What is the uncertainty in x?
d) What is the expectation value of p?
e) What it the uncertainty in p?
f) Do your answers to part c) and e) agree with the Heisenberg uncertainly principle?
Problem 4 (30 points) ANSWER IN BOOKLET 2
Consider a particle of mass m and energy E, approaching a step potential of height V0 at x = 0. You do not know in which region (x < 0 or x > 0) the potential is non-zero and you do not know from which direction the particle is approaching the step potential. You also do not know if E is larger or smaller than V0. But you are able to measure the probability density distribution in the x < 0 region. Three different cases are shown in the Figure below. For each case, sketch the step potential and indicate the energy of the particle (E > V0 or E < V0) and its initial direction. You need to motivate your answer.
Problem 5 (35 points) ANSWER IN BOOKLET 2
Consider the first few low-lying states of the Helium atom. When the Helium atom is in its ground state, both electrons are in n = 1 states. When the Helium atom is in one of the first excited states, one electron is in an n = 1 state and the other electron is in an n = 2 state. The spatial wavefunction of electron i can be written as
where a contains information about the spatial quantum numbers.
The spin wavefunction of the two-electron system, , can be written as combinations of the following four functions
a) Write down all possible total wavefunctions for the ground state of Helium (n1 = 1 and n2 = 1).
b) Write down all possible total wavefunctions for the first low-lying excited states of Helium (n1 = 1 and n2 = 2).
c) Make an energy diagram showing the location of the low-lying energy levels of Helium (n1 = 1 and n2 = 1, 2), assuming there is no Coulomb interaction between the electrons. Label each level with the n and l values of each of the two electrons and their total spin. Note: the actual location of these energy levels is not important, but their relative position is.
d) Now include the effect of the Coulomb interaction between the electrons, but ignore the exchange force. What is the effect of the Coulomb interaction on the energy of the states shown in the diagram constructed in c)? In the energy diagram, indicate the shifts of the energy levels of the low-lying states of Helium due to the Coulomb interaction. Label each level with the n and l values of each of the two electrons and their total spin. Note: the relative shifts of the levels are important and need to be correctly motivated and drawn.
e) Finally, include the effect of the exchange force. What is the effect of the exchange force on the energy of the states shown in the diagram constructed in d)? In the energy diagram, indicate the shifts of the energy levels of the low-lying states of Helium due to the exchange force. Label each level with the n and l values of each of the two electrons and their total spin. Note: the relative shifts of the levels are important and need to be correctly motivated and drawn.
Problem 6 (20 points) ANSWER IN BOOKLET 2
Measurements made on the line spectrum emitted by a certain atom of intermediate Z show that the ratio of the separation energies between three adjacent levels of increasing energy in a particular multiplet is approximately 3 to 5 (that is the energy difference between the 2nd and the 3rd member of the multiplet is 5/3 of the energy difference between the 1st and the 2nd member of the multiplet.)
a) What are the jÕ quantum number that can be assigned to these states?
b) What is the lÕ quantum number that can be assigned to these states?
c) What is the sÕ quantum number that can be assigned to these states?
Problem 7 (20 points) ANSWER IN BOOKLET 3
Consider multi-electron atoms. To study the properties of the levels occupied by the electrons, ionization energies and x-ray energies of these atoms are measured. The following two figures show the results of the measurements of these energies as function of the nuclear charge of the atom.
What do we learn about the energy levels of the electrons in these atoms on the basis of these measurements?
Problem 8 (20 points) ANSWER IN BOOKLET 3
Consider the following potential energy distribution:
Three acceptable eigenfunctions for this potential are shown in the figure below.
The three wavefunctions have the same value at x = x0. Use this information to rank the three wavefunctions in order of their energy (lowest energy, middle energy, highest energy). Your answer must be well motivated (e.g. node counting is not sufficient justification).
Problem 9 (5 points) ANSWER IN BOOKLET 3
Who is the greatest closer in baseball who has saved 608 games, the most in Major League history, and finished a record 892 games. He is shown in the picture below, taken by your instructor, when he broke the Major League record.
a) Brian Wilson
b) Mariano Rivera
c) Jonathan Papelbon
d) Scott Downs
e) Santiago Casilla
f) Craig Kimbrel
g) John Axford
h) Chris Perez
i) Joel Hanrahan
j) JJ Putz