Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Solutions to Problem Set 4
Spring 2010
1. (a) Consider the Born approximation as the first term of the Born series. Show that: (i) the Born approximation for the forward scattering amplitude [i.e. at = 0] is purely real, and therefore (ii) the
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Problem Set 4 DUE: MAY 25, 2010
Spring 2010
1. (a) Consider the Born approximation as the first term of the Born series. Show that: (i) the Born approximation for the forward scattering amplitude [i.e. at = 0] is purely real, and therefore (ii
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Final, 2011 Instructions: The are three questions worth 100 points each. Feel free to consult 2 sides each of two sheets of notes. 1. Calculate the firstorder relativistic correction to all of the energy levels (enumerated by n) of the simple har
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Solutions to the Final Exam
Spring 2010
1. An electron is placed in a potential V (r) = e2 + (r 2  3z 2 ) , r
where is a small parameter. Neglect the spin of the electron. (a) Compute the shifts of the n = 2 energy levels (you may neglect fi
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Final Exam Extras
Spring 2010
This sheet contains additional information to help you solve the problems of the final exam more efficiently. 1. For = 0, the unperturbed n = 2 energy eigenfunctions are given by: 1 200 (r) = 1 210 (r) = 2 1 211 (
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Final Exam
Spring 2010
FINAL EXAM INSTRUCTIONS: This is an open book exam. You are permitted to consult the textbooks of Shankar and Baym, your handwritten notes, and any class handouts that are posted to the course website. One mathematical r
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Problem Set 5
Spring 2010
DUE: JUNE 8, 2010 FINAL EXAM ALERT: The final exam will be take place from 11 am2 pm on Wednesday June 9, 2010 in ISB 235. Please note the change of date, time and location. The exam will cover the entire course. Duri
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Problem Set 3
Spring 2010
DUE: TUESDAY, MAY 11, 2010 MIDTERM ALERT: There is a change of plans for the midterm exam. The midterm exam will be take place from 1011:45 am on Thursday May 13, 2010 in ISB 231 (our usual classroom). The exam will c
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Problem Set 2
Spring 2010
DUE: TUESDAY, APRIL 27, 2010
1 1. We wish to determine the correct form of the Schrodinger equation for a spin 2 1 particle in an external electromagnetic field. The wavefunction for a spin 2 particle is a twocompo
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Solutions to the Midterm Exam
Spring 2010
1. Suppose we define G(t)
dxK(x, t, ; x, 0)

(1)
where K(x, t; x , t ) is the propagator. Assume that the system has a timeindependent Hamiltonian and a discrete energy level spectrum. (a) Prove tha
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Midterm Exam
Spring 2010
MIDTERM EXAM INSTRUCTIONS: This is an open book exam. You are permitted to consult the textbooks of Shankar and Baym, your handwritten notes, and any class handouts that are posted to the course website. One mathematic
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Solutions to Problem Set 1
Spring 2010
1. Prove that x , t  T [X (t1)X (t2 ) X (tn )] x , t = D [x(t)] x(t1 )x(t2 ) x(tn ) eiS [x(t)]/ ,
where T is the timeordered product symbol, S [x(t)] is the action [which depends on the path x(t)], X (
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Solutions to Problem Set 2
Spring 2010
1. We wish to determine the correct form of the Schrodinger equation for a spin 1 2 1 particle in an external electromagnetic eld. The wavefunction for a spin 2 particle is a twocomponent spinor wave f
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Solutions to Problem Set 3
Spring 2010
where E2 > E1 . The quantities a and b are to be regarded as perturbations that are of the same order but small compared with E2  E1 . We shall write write the Hamiltonian matrix as H = H (0) + H (1) , w
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Solutions to Problem Set 5
Spring 2010
1. Tritium (the isotope H3 ), which is initially in its ground state, undergoes spontaneous beta decay, emitting an electron of maximum energy of about 17 keV. The nucleus remaining is He3 . NOTE: In this
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Supplement
8A
A Useful Theorem
The following useful result appears in Paulis 1930 Handbuch Article on Quantum Theory: Consider eigenvalues and eigenfunctions of a Hamiltonian that depends on some parameterfor example, the mass of the electron, or the cha
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 221A Fall 2005 Notes 16 Time Reversal
16.1. Introduction We have now considered the spacetime symmetries of translations, proper rotations, and spatial inversions (that is, improper rotations) and the operators that implement these symmetries on
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
Physics 216
Problem Set 1
Spring 2010
DUE: TUESDAY, APRIL 13, 2010
1. Prove that x , t  T [X (t1)X (t2 ) X (tn )] x , t = D [x(t)] x(t1 )x(t2 ) x(tn ) eiS [x(t)]/ ,
where T is the timeordered product symbol, S [x(t)] is the action [which depends on the
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
1
ERRATA for 13th Printing May 26, 2006
Page viii:Third line from below must read as follows: of friendly and warm cooperation. I thank Ron Johnson, Editor at of Springer for his tireless eorts on behalf of this book, and Chris Bostock, Daniel Keren and J
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Midterm, 2011 Instructions: Your solutions are due Saturday at 9 AM, via email or in my office (I should be present on Saturday morning, but if you want to drop it friday night, or miss me, you can slip it under the door.) Late midterms will not b
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Midterm, 2011 Instructions: Your solutions are due Saturday at 9 AM, via email or in my office (I should be present on Saturday morning, but if you want to drop it friday night, or miss me, you can slip it under the door.) Late midterms will not b
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Problem set 1, Due 4/6
1. In class, we estimated the groundstate energy of the 3D SHO (see Shankar 12.6.42) using a `guess' of 0 (a; ) = R(a; r)Y00 (, ), r Suppose we try to get the first energy E1 using the trial function 1 = R(a; r)Y10 , Where
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Problem set 2, Due 4/13
1. A mass m is attached by a massless rod of length l to a pivot p and swings in a vertical plane under the influence of gravity, with deflection angle from the vertical. (a) In the small angle approximation find the energy
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Problem set 3, Due 4/20
1. Consider the hydrogen atom in an excited n = 2 state, which is subjected to an external uniform electric field E. Do not neglect the spin of the electron. Assume that the field E is sufficiently weak so that eEa0 is smal
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Problem set 4, Due 4/27
1. A particle of mass M is in a onedimensional harmonic oscillator potential V1 = 1 kx2 . 2 (a) It is initially in its ground state. The spring constant is suddenly doubled (k 2k) so that the new potential is V2 = kx2 . Th
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Problem set 5, Due 5/4
MIDTERM REMINDER: The course midterm starts after class wednesday 5/4, and concludes at 9 am Saturday 5/7. 1. In class we derived for a plane EM wave of frequency with linear polarization direction incident upon ^ a bound el
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Problem set 6, Due 5/19
1. A diatomic molecule, such as H2 or O2 , consisting of two identical atoms, is modeled by two identical spherically symmetric potentials centered on points A and B, which are separated by a displacement R that points from
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Problem set 7, Due 5/26
1. Consider the elastic scattering of photons off electrons in atoms, assuming that the incident photon energies are large compared to the atomic binding energies. However, you should assume that the photon wavelength is st
Advanced Topics in NonRelativistic Quantum Mechanics
PHYS 216

Spring 2011
PH 216, Problem set 1, Due 4/6
1. In class, we estimated the groundstate energy of the 3D SHO (see Shankar 12.6.42) using a `guess' of 0 (a; ) = R(a; r)Y00 (, ), r Suppose we try to get the first energy E1 using the trial function 1 = R(a; r)Y10 , Where