Physics 742 Graduate Quantum Mechanics 2 Solutions to Second Exam, Spring 2017

Size: px
Start display at page:

Download "Physics 742 Graduate Quantum Mechanics 2 Solutions to Second Exam, Spring 2017"

Transcription

1 Physics 74 Graduate Quantum Mechanics Solutions to Second Exam Spring 17 The points for each question are marked. Each question is worth points. Some possibly useful formulas appear at the end of the test. 1. A particle lies in a 1d moving harmonic oscillator 1 with potential V x t m xa t where a(t) is a smoothly increasing function from a t A. It is initially in at to the ground state with wave function 1/4 m x x m e. What is the probability that it is still in the ground state at t if the increase is (a) adiabatic (b) sudden? In the adiabatic approximation the lowest energy state of the initial Hamiltonian with probability 1. Hence there is no calculation to do. In the sudden approximation the probability is given by P. The ground state for the harmonic oscillator is always the same but we must replace x by the position relative to the minimum a(t) of the potentail at each moment so the ground state at time t is given by 1/4 xtm m xat exp We simply replace a(t) by A at the final time so we have * m m m xxdx exp xa exp x dx m m exp x Ax A x dx m ma m ma exp exp x x dx m ma 1 ma exp exp m 4 m ma ma ma exp exp exp. 4 4 We then square this to get the probability so we have adiabatic: P 1 sudden: P exp m A.

2 . A particle has time-dependent Hamiltonian H H W t where in some basis t H 1 and W t e where the perturbation W(t) applies only for t >. At t = the system is in the ground state of H. Find the probability as t that it goes to each of the excited states to leading order in the perturbation W. We use the probability for the non-initial state 1 i t leading order scattering matrix S i dtw te have energy n We therefore have for n = 1 so the frequency differences are E E F F F I P I F S together with the. The three eigenstates of H n 1 1 S1 dt W t e dt e e i i i i 1 1 S dt W t e dt e e i i i i It is then straightforward to get the probabilities it 1 t i t it t i t. 1 P 1 S i i P S i i 4

3 3. Show that the following wave function is an eigenstate of the free Dirac Hamiltonian: E cp ip 1 e where mc with energy E and find a relationship for E in terms of p and m. To check that it is an eigenstate of the Hamiltonian with energy E we have to simply let the Hamiltonian act on it. We find Ecp ip Ecp ip H cα P mc e c pmc e mc mc 4 cp mc E cp ip cp E cp m c ip e e 3 mc cp mc mc E cp mc p 4 cp E cp m c ip e mc E where we used the fact that is an eigenstate of with eigenvalue +1 to simplify. Now if this is an eigenstate with eigenvalue E we must have H E or in other words 4 cp E cp m c ip EE cp ip e e. mc E Emc The lower of the pair of these equation is obviously true and the upper will be true provided 4 cp Ecp m c E Ecp This is the correct relativistic formula 4 cpe c p m c E Ecp 4 c p m c E 4 E c p m c..

4 4. A single photon is detected by absorbing photons so that the probability of finding a photon is proportional to Er. Calculate this quantity first for a single photon in the state 1 q 1 with wave number q q x ˆ ˆ x q y and polariation y ε ˆ 1. Then compute it again for a single photon in the superposition state 1 1 q 1 1 q 1. We note that in every case we are going from one photon to ero and hence we must annihilate the photon; indeed we must annihilate only the photon that is present. We therefore have k ikr * ikr Er 1 q1 i akεke ak εke 1 q1 k V cq iqxxiqyy iε e a 1 q 1 iˆ e. V q iq r q1 q1 V Therefore for a single photon we would have cq cq Er 1 q1 ˆ ˆ V V where q q q. For the superposition state we have x y cq 1 iqxxiqyy iqxxiqyy ˆ i e e Er Er 1 q1 Er 1 q1 V iqyy iqyy cq iqxx y 1 cq iqxx ie ˆ e e ie ˆ cos qy V V We therefore have in this case cq Er cos qy y. V We note in this case that we have the periodic changes in intensity that one would expect from two waves interfering whereas a single wave yields just a smooth probability distribution.

5 5. An electron of mass m is in the state 111 of the 3D asymmetric harmonic oscillator 1 with potential V R m X 4Y 9Z. Find the decay rate to each of the possible final states with lower energy due to emission of a single photon in the dipole approximation. The unperturbed Hamiltonian is just three harmonic oscillators with angular frequencies and 3 in the x y and directions respectively. The eigenstates will be of the form n p q with energies Enpq n p 3q np3q3. The three operators R can only raise or lower one of the three labels in n p q and only by one unit so since we want to decrease the energy the only possible final states will be and 11 and the only non-ero matrix elements of R will be the x y and - components of these respectively. We then use our formula for the operator X in one dimension in each case but modify the frequency to the appropriate multiple of in each case. We therefore have: r R xˆ xˆ ax ax 111 m m r R yˆ yˆ ay ay 111 m 4m r R ˆ ˆ a a 111. m3 6m The frequency differences in each case are simply given by IF and 3 respectively. We then simply substitute these into the decay rate formulas to yield c m 3mc c 4m 3mc c 6m mc

6 1D H.O.: 1 V x m x X aa m an nn1 a n n n 1 1 Electric field operator k ikr * Er i ak εke ak εke V ikr k Possibly Helpful Formulas: Free Dirac Hamiltonian Spontaneous H cα P mc icα mc Decay: 4 1 σ 3 IF r α 3c 1 σ 1 i 1 x 1 y i 1 Time-dependent Perturbation Theory 1 T S i dtw t e i t Possibly Helpful Integrals: Ax Bx B 4 A n x n 1 e dx e x e dxn!. A

Quantum Physics III (8.06) Spring 2008 Final Exam Solutions

Quantum Physics III (8.06) Spring 2008 Final Exam Solutions Quantum Physics III (8.6) Spring 8 Final Exam Solutions May 19, 8 1. Short answer questions (35 points) (a) ( points) α 4 mc (b) ( points) µ B B, where µ B = e m (c) (3 points) In the variational ansatz,

More information

Physics 742 Graduate Quantum Mechanics 2 Midterm Exam, Spring [15 points] A spinless particle in three dimensions has potential 2 2 2

Physics 742 Graduate Quantum Mechanics 2 Midterm Exam, Spring [15 points] A spinless particle in three dimensions has potential 2 2 2 Physics 74 Graduate Quantum Mechanics Midterm Exam, Spring 4 [5 points] A spinless particle in three dimensions has potential V A X Y Z BXYZ Consider the unitary rotation operator R which cyclically permutes

More information

Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.

Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.

More information

( ) in the interaction picture arises only

( ) in the interaction picture arises only Physics 606, Quantum Mechanics, Final Exam NAME 1 Atomic transitions due to time-dependent electric field Consider a hydrogen atom which is in its ground state for t < 0 For t > 0 it is subjected to a

More information

PH 451/551 Quantum Mechanics Capstone Winter 201x

PH 451/551 Quantum Mechanics Capstone Winter 201x These are the questions from the W7 exam presented as practice problems. The equation sheet is PH 45/55 Quantum Mechanics Capstone Winter x TOTAL POINTS: xx Weniger 6, time There are xx questions, for

More information

ψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions.

ψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions. 1. Quantum Mechanics (Fall 2004) Two spin-half particles are in a state with total spin zero. Let ˆn a and ˆn b be unit vectors in two arbitrary directions. Calculate the expectation value of the product

More information

( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II -- Exam 1

( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II -- Exam 1 Physics 624, Quantum II -- Exam 1 Please show all your work on the separate sheets provided (and be sure to include your name) You are graded on your work on those pages, with partial credit where it is

More information

Simple Harmonic Oscillator

Simple Harmonic Oscillator Classical harmonic oscillator Linear force acting on a particle (Hooke s law): F =!kx From Newton s law: F = ma = m d x dt =!kx " d x dt + # x = 0, # = k / m Position and momentum solutions oscillate in

More information

Lecture 12. The harmonic oscillator

Lecture 12. The harmonic oscillator Lecture 12 The harmonic oscillator 107 108 LECTURE 12. THE HARMONIC OSCILLATOR 12.1 Introduction In this chapter, we are going to find explicitly the eigenfunctions and eigenvalues for the time-independent

More information

a = ( a σ )( b σ ) = a b + iσ ( a b) mω 2! x + i 1 2! x i 1 2m!ω p, a = mω 2m!ω p Physics 624, Quantum II -- Final Exam

a = ( a σ )( b σ ) = a b + iσ ( a b) mω 2! x + i 1 2! x i 1 2m!ω p, a = mω 2m!ω p Physics 624, Quantum II -- Final Exam Physics 624, Quantum II -- Final Exam Please show all your work on the separate sheets provided (and be sure to include your name). You are graded on your work on those pages, with partial credit where

More information

Quantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie

Quantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie Quantum Mechanics II Lecture (www.sp.phy.cam.ac.u/~dar/pdf) David Ritchie Michaelmas. So far we have found solutions to Section 4:Transitions Ĥ ψ Eψ Solutions stationary states time dependence with time

More information

( ) /, so that we can ignore all

( ) /, so that we can ignore all Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The ac-stark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)

More information

1. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants.

1. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants. Sample final questions.. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants. 2. A one-dimensional harmonic oscillator, originally in the ground state,

More information

Phys 622 Problems Chapter 5

Phys 622 Problems Chapter 5 1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit

More information

Quantum Physics 2006/07

Quantum Physics 2006/07 Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 12, 2011 1:00PM to 3:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of this

More information

University of Michigan Physics Department Graduate Qualifying Examination

University of Michigan Physics Department Graduate Qualifying Examination Name: University of Michigan Physics Department Graduate Qualifying Examination Part II: Modern Physics Saturday 17 May 2014 9:30 am 2:30 pm Exam Number: This is a closed book exam, but a number of useful

More information

Physics 741 Graduate Quantum Mechanics 1 Solutions to Midterm Exam, Fall x i x dx i x i x x i x dx

Physics 741 Graduate Quantum Mechanics 1 Solutions to Midterm Exam, Fall x i x dx i x i x x i x dx Physics 74 Graduate Quantum Mechanics Solutions to Midterm Exam, Fall 4. [ points] Consider the wave function x Nexp x ix (a) [6] What is the correct normaliation N? The normaliation condition is. exp,

More information

2 Quantization of the Electromagnetic Field

2 Quantization of the Electromagnetic Field 2 Quantization of the Electromagnetic Field 2.1 Basics Starting point of the quantization of the electromagnetic field are Maxwell s equations in the vacuum (source free): where B = µ 0 H, D = ε 0 E, µ

More information

Atomic Transitions and Selection Rules

Atomic Transitions and Selection Rules Atomic Transitions and Selection Rules The time-dependent Schrodinger equation for an electron in an atom with a time-independent potential energy is ~ 2 2m r2 (~r, t )+V(~r ) (~r, t )=i~ @ @t We found

More information

C. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.

C. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0. Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is

More information

Time Independent Perturbation Theory Contd.

Time Independent Perturbation Theory Contd. Time Independent Perturbation Theory Contd. A summary of the machinery for the Perturbation theory: H = H o + H p ; H 0 n >= E n n >; H Ψ n >= E n Ψ n > E n = E n + E n ; E n = < n H p n > + < m H p n

More information

Graduate Written Exam Part I (Fall 2011)

Graduate Written Exam Part I (Fall 2011) Graduate Written Exam Part I (Fall 2011) 1. An elevator operator in a skyscraper, being a very meticulous person, put a pendulum clock on the wall of the elevator to make sure that he spends exactly 8

More information

AA 242B / ME 242B: Mechanical Vibrations (Spring 2016)

AA 242B / ME 242B: Mechanical Vibrations (Spring 2016) AA 242B / ME 242B: Mechanical Vibrations (Spring 206) Solution of Homework #3 Control Tab Figure : Schematic for the control tab. Inadequacy of a static-test A static-test for measuring θ would ideally

More information

Quantum Physics III (8.06) Spring 2016 Assignment 3

Quantum Physics III (8.06) Spring 2016 Assignment 3 Quantum Physics III (8.6) Spring 6 Assignment 3 Readings Griffiths Chapter 9 on time-dependent perturbation theory Shankar Chapter 8 Cohen-Tannoudji, Chapter XIII. Problem Set 3. Semi-classical approximation

More information

(a) [15] Use perturbation theory to compute the eigenvalues and eigenvectors of H. Compute all terms up to fourth-order in Ω. The bare eigenstates are

(a) [15] Use perturbation theory to compute the eigenvalues and eigenvectors of H. Compute all terms up to fourth-order in Ω. The bare eigenstates are PHYS85 Quantum Mechanics II, Spring 00 HOMEWORK ASSIGNMENT 6: Solutions Topics covered: Time-independent perturbation theory.. [0 Two-Level System: Consider the system described by H = δs z + ΩS x, with

More information

Perturbation Theory. Andreas Wacker Mathematical Physics Lund University

Perturbation Theory. Andreas Wacker Mathematical Physics Lund University Perturbation Theory Andreas Wacker Mathematical Physics Lund University General starting point Hamiltonian ^H (t) has typically noanalytic solution of Ψ(t) Decompose Ĥ (t )=Ĥ 0 + V (t) known eigenstates

More information

The Dirac Equation. Topic 3 Spinors, Fermion Fields, Dirac Fields Lecture 13

The Dirac Equation. Topic 3 Spinors, Fermion Fields, Dirac Fields Lecture 13 The Dirac Equation Dirac s discovery of a relativistic wave equation for the electron was published in 1928 soon after the concept of intrisic spin angular momentum was proposed by Goudsmit and Uhlenbeck

More information

Further Quantum Mechanics Problem Set

Further Quantum Mechanics Problem Set CWPP 212 Further Quantum Mechanics Problem Set 1 Further Quantum Mechanics Christopher Palmer 212 Problem Set There are three problem sets, suitable for use at the end of Hilary Term, beginning of Trinity

More information

Quantization of the E-M field. 2.1 Lamb Shift revisited 2.1. LAMB SHIFT REVISITED. April 10, 2015 Lecture XXVIII

Quantization of the E-M field. 2.1 Lamb Shift revisited 2.1. LAMB SHIFT REVISITED. April 10, 2015 Lecture XXVIII .. LAMB SHFT REVSTED April, 5 Lecture XXV Quantization of the E-M field. Lamb Shift revisited We discussed the shift in the energy levels of bound states due to the vacuum fluctuations of the E-M fields.

More information

The energy of the emitted light (photons) is given by the difference in energy between the initial and final states of hydrogen atom.

The energy of the emitted light (photons) is given by the difference in energy between the initial and final states of hydrogen atom. Lecture 20-21 Page 1 Lectures 20-21 Transitions between hydrogen stationary states The energy of the emitted light (photons) is given by the difference in energy between the initial and final states of

More information

LECTURES ON QUANTUM MECHANICS

LECTURES ON QUANTUM MECHANICS LECTURES ON QUANTUM MECHANICS GORDON BAYM Unitsersity of Illinois A II I' Advanced Bock Progrant A Member of the Perseus Books Group CONTENTS Preface v Chapter 1 Photon Polarization 1 Transformation of

More information

Classical Mechanics Comprehensive Exam

Classical Mechanics Comprehensive Exam Name: Student ID: Classical Mechanics Comprehensive Exam Spring 2018 You may use any intermediate results in the textbook. No electronic devices (calculator, computer, cell phone etc) are allowed. For

More information

Transition probabilities and couplings

Transition probabilities and couplings L6 Transition probabilities and couplings Mario Barbatti A*Midex Chair Professor mario.barbatti@univ amu.fr Aix Marseille Université, Institut de Chimie Radicalaire LIGT AND Fermi s golden rule LIGT AND

More information

J10M.1 - Rod on a Rail (M93M.2)

J10M.1 - Rod on a Rail (M93M.2) Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight

More information

Creation and Destruction Operators and Coherent States

Creation and Destruction Operators and Coherent States Creation and Destruction Operators and Coherent States WKB Method for Ground State Wave Function state harmonic oscillator wave function, We first rewrite the ground < x 0 >= ( π h )1/4 exp( x2 a 2 h )

More information

The Two Level Atom. E e. E g. { } + r. H A { e e # g g. cos"t{ e g + g e } " = q e r g

The Two Level Atom. E e. E g. { } + r. H A { e e # g g. cost{ e g + g e }  = q e r g E e = h" 0 The Two Level Atom h" e h" h" 0 E g = " h# 0 g H A = h" 0 { e e # g g } r " = q e r g { } + r $ E r cos"t{ e g + g e } The Two Level Atom E e = µ bb 0 h" h" " r B = B 0ˆ z r B = B " cos#t x

More information

Time dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012

Time dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012 Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system

More information

Lecture 6 Photons, electrons and other quanta. EECS Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku

Lecture 6 Photons, electrons and other quanta. EECS Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku Lecture 6 Photons, electrons and other quanta EECS 598-002 Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku From classical to quantum theory In the beginning of the 20 th century, experiments

More information

QUANTUM THEORY OF LIGHT EECS 638/PHYS 542/AP609 FINAL EXAMINATION

QUANTUM THEORY OF LIGHT EECS 638/PHYS 542/AP609 FINAL EXAMINATION Instructor: Professor S.C. Rand Date: April 5 001 Duration:.5 hours QUANTUM THEORY OF LIGHT EECS 638/PHYS 54/AP609 FINAL EXAMINATION PLEASE read over the entire examination before you start. DO ALL QUESTIONS

More information

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum Mechanics for Scientists and Engineers. David Miller Quantum Mechanics for Scientists and Engineers David Miller Time evolution of superpositions Time evolution of superpositions Superposition for a particle in a box Superposition for a particle in a box

More information

Physical Chemistry II Exam 2 Solutions

Physical Chemistry II Exam 2 Solutions Chemistry 362 Spring 208 Dr Jean M Standard March 9, 208 Name KEY Physical Chemistry II Exam 2 Solutions ) (4 points) The harmonic vibrational frequency (in wavenumbers) of LiH is 4057 cm Based upon this

More information

EE 223 Applied Quantum Mechanics 2 Winter 2016

EE 223 Applied Quantum Mechanics 2 Winter 2016 EE 223 Applied Quantum Mechanics 2 Winter 2016 Syllabus and Textbook references Version as of 12/29/15 subject to revisions and changes All the in-class sessions, paper problem sets and assignments, and

More information

Physics PhD Qualifying Examination Part I Wednesday, January 21, 2015

Physics PhD Qualifying Examination Part I Wednesday, January 21, 2015 Physics PhD Qualifying Examination Part I Wednesday, January 21, 2015 Name: (please print) Identification Number: STUDENT: Designate the problem numbers that you are handing in for grading in the appropriate

More information

Quantum Mechanics Solutions. λ i λ j v j v j v i v i.

Quantum Mechanics Solutions. λ i λ j v j v j v i v i. Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =

More information

16. GAUGE THEORY AND THE CREATION OF PHOTONS

16. GAUGE THEORY AND THE CREATION OF PHOTONS 6. GAUGE THEORY AD THE CREATIO OF PHOTOS In the previous chapter the existence of a gauge theory allowed the electromagnetic field to be described in an invariant manner. Although the existence of this

More information

Physics PhD Qualifying Examination Part I Wednesday, August 26, 2015

Physics PhD Qualifying Examination Part I Wednesday, August 26, 2015 Physics PhD Qualifying Examination Part I Wednesday, August 26, 2015 Name: (please print) Identification Number: STUDENT: Designate the problem numbers that you are handing in for grading in the appropriate

More information

Matrix Representation

Matrix Representation Matrix Representation Matrix Rep. Same basics as introduced already. Convenient method of working with vectors. Superposition Complete set of vectors can be used to express any other vector. Complete set

More information

8.04 Spring 2013 March 12, 2013 Problem 1. (10 points) The Probability Current

8.04 Spring 2013 March 12, 2013 Problem 1. (10 points) The Probability Current Prolem Set 5 Solutions 8.04 Spring 03 March, 03 Prolem. (0 points) The Proaility Current We wish to prove that dp a = J(a, t) J(, t). () dt Since P a (t) is the proaility of finding the particle in the

More information

4.1 Time evolution of superpositions. Slides: Video Introduction to time evolution of superpositions

4.1 Time evolution of superpositions. Slides: Video Introduction to time evolution of superpositions 4.1 Time evolution of superpositions Slides: Video 4.1.1 Introduction to time evolution of superpositions Time evolution of superpositions Quantum mechanics for scientists and engineers David Miller 4.1

More information

Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14

Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14 Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14 Because the wavelength of light (400-700nm) is much greater than the diameter of an atom (0.07-0.35

More information

(relativistic effects kinetic energy & spin-orbit coupling) 3. Hyperfine structure: ) (spin-spin coupling of e & p + magnetic moments) 4.

(relativistic effects kinetic energy & spin-orbit coupling) 3. Hyperfine structure: ) (spin-spin coupling of e & p + magnetic moments) 4. 4 Time-ind. Perturbation Theory II We said we solved the Hydrogen atom exactly, but we lied. There are a number of physical effects our solution of the Hamiltonian H = p /m e /r left out. We already said

More information

Lecture 6 Quantum Mechanical Systems and Measurements

Lecture 6 Quantum Mechanical Systems and Measurements Lecture 6 Quantum Mechanical Systems and Measurements Today s Program: 1. Simple Harmonic Oscillator (SHO). Principle of spectral decomposition. 3. Predicting the results of measurements, fourth postulate

More information

Lecture-XXVI. Time-Independent Schrodinger Equation

Lecture-XXVI. Time-Independent Schrodinger Equation Lecture-XXVI Time-Independent Schrodinger Equation Time Independent Schrodinger Equation: The time-dependent Schrodinger equation: Assume that V is independent of time t. In that case the Schrodinger equation

More information

Suppression of Radiation Excitation in Focusing Environment * Abstract

Suppression of Radiation Excitation in Focusing Environment * Abstract SLAC PUB 7369 December 996 Suppression of Radiation Excitation in Focusing Environment * Zhirong Huang and Ronald D. Ruth Stanford Linear Accelerator Center Stanford University Stanford, CA 94309 Abstract

More information

11 Perturbation Theory

11 Perturbation Theory S.K. Saikin Oct. 8, 009 11 Perturbation Theory Content: Variational Principle. Time-Dependent Perturbation Theory. 11.1 Variational Principle Lecture 11 If we need to compute the ground state energy of

More information

Fundamentals of Spectroscopy for Optical Remote Sensing. Course Outline 2009

Fundamentals of Spectroscopy for Optical Remote Sensing. Course Outline 2009 Fundamentals of Spectroscopy for Optical Remote Sensing Course Outline 2009 Part I. Fundamentals of Quantum Mechanics Chapter 1. Concepts of Quantum and Experimental Facts 1.1. Blackbody Radiation and

More information

SCE3337 Quantum Mechanics III (Quantum Mechanics and Quantum Optics) Teaching Team

SCE3337 Quantum Mechanics III (Quantum Mechanics and Quantum Optics) Teaching Team SCE3337 Quantum Mechanics III 1 SCE3337 Quantum Mechanics III (Quantum Mechanics and Quantum Optics) Teaching Team Dr Robert Sang Robert Sang can be found in Science II Level Room.15, non-personal contact

More information

Qualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!

Qualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets! Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics

More information

The general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation

The general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation Lecture 17 Page 1 Lecture 17 L17.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent

More information

Accelerator Physics NMI and Synchrotron Radiation. G. A. Krafft Old Dominion University Jefferson Lab Lecture 16

Accelerator Physics NMI and Synchrotron Radiation. G. A. Krafft Old Dominion University Jefferson Lab Lecture 16 Accelerator Physics NMI and Synchrotron Radiation G. A. Krafft Old Dominion University Jefferson Lab Lecture 16 Graduate Accelerator Physics Fall 17 Oscillation Frequency nq I n i Z c E Re Z 1 mode has

More information

CONTENTS. vii. CHAPTER 2 Operators 15

CONTENTS. vii. CHAPTER 2 Operators 15 CHAPTER 1 Why Quantum Mechanics? 1 1.1 Newtonian Mechanics and Classical Electromagnetism 1 (a) Newtonian Mechanics 1 (b) Electromagnetism 2 1.2 Black Body Radiation 3 1.3 The Heat Capacity of Solids and

More information

eigenvalues eigenfunctions

eigenvalues eigenfunctions Born-Oppenheimer Approximation Atoms and molecules consist of heavy nuclei and light electrons. Consider (for simplicity) a diatomic molecule (e.g. HCl). Clamp/freeze the nuclei in space, a distance r

More information

UGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM PH 05 PHYSICAL SCIENCE TEST SERIES # 1. Quantum, Statistical & Thermal Physics

UGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM PH 05 PHYSICAL SCIENCE TEST SERIES # 1. Quantum, Statistical & Thermal Physics UGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM BOOKLET CODE SUBJECT CODE PH 05 PHYSICAL SCIENCE TEST SERIES # Quantum, Statistical & Thermal Physics Timing: 3: H M.M: 00 Instructions. This test

More information

CHM Physical Chemistry II Chapter 12 - Supplementary Material. 1. Einstein A and B coefficients

CHM Physical Chemistry II Chapter 12 - Supplementary Material. 1. Einstein A and B coefficients CHM 3411 - Physical Chemistry II Chapter 12 - Supplementary Material 1. Einstein A and B coefficients Consider two singly degenerate states in an atom, molecule, or ion, with wavefunctions 1 (for the lower

More information

A. F. J. Levi 1 EE539: Engineering Quantum Mechanics. Fall 2017.

A. F. J. Levi 1 EE539: Engineering Quantum Mechanics. Fall 2017. A. F. J. Levi 1 Engineering Quantum Mechanics. Fall 2017. TTh 9.00 a.m. 10.50 a.m., VHE 210. Web site: http://alevi.usc.edu Web site: http://classes.usc.edu/term-20173/classes/ee EE539: Abstract and Prerequisites

More information

Solutions to exam : 1FA352 Quantum Mechanics 10 hp 1

Solutions to exam : 1FA352 Quantum Mechanics 10 hp 1 Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem (4 p): (a) Define the concept of unitary operator and show that the operator e ipa/ is unitary (p is the momentum operator in one dimension) (b)

More information

Lecture 0. NC State University

Lecture 0. NC State University Chemistry 736 Lecture 0 Overview NC State University Overview of Spectroscopy Electronic states and energies Transitions between states Absorption and emission Electronic spectroscopy Instrumentation Concepts

More information

Physics 139B Solutions to Homework Set 4 Fall 2009

Physics 139B Solutions to Homework Set 4 Fall 2009 Physics 139B Solutions to Homework Set 4 Fall 9 1. Liboff, problem 1.16 on page 594 595. Consider an atom whose electrons are L S coupled so that the good quantum numbers are j l s m j and eigenstates

More information

Non-relativistic Quantum Electrodynamics

Non-relativistic Quantum Electrodynamics Rigorous Aspects of Relaxation to the Ground State Institut für Analysis, Dynamik und Modellierung October 25, 2010 Overview 1 Definition of the model Second quantization Non-relativistic QED 2 Existence

More information

Second quantization: where quantization and particles come from?

Second quantization: where quantization and particles come from? 110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 10, 2018 10:00AM to 12:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of

More information

Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015)

Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Interaction of x-ray with matter: - Photoelectric absorption - Elastic (coherent) scattering (Thomson Scattering) - Inelastic (incoherent) scattering

More information

+E v(t) H(t) = v(t) E where v(t) is real and where v 0 for t ±.

+E v(t) H(t) = v(t) E where v(t) is real and where v 0 for t ±. . Brick in a Square Well REMEMBER: THIS PROBLEM AND THOSE BELOW SHOULD NOT BE HANDED IN. THEY WILL NOT BE GRADED. THEY ARE INTENDED AS A STUDY GUIDE TO HELP YOU UNDERSTAND TIME DEPENDENT PERTURBATION THEORY

More information

Units and dimensions

Units and dimensions Particles and Fields Particles and Antiparticles Bosons and Fermions Interactions and cross sections The Standard Model Beyond the Standard Model Neutrinos and their oscillations Particle Hierarchy Everyday

More information

Quantum Mechanics Solutions

Quantum Mechanics Solutions Quantum Mechanics Solutions (a (i f A and B are Hermitian, since (AB = B A = BA, operator AB is Hermitian if and only if A and B commute So, we know that [A,B] = 0, which means that the Hilbert space H

More information

International Physics Course Entrance Examination Questions

International Physics Course Entrance Examination Questions International Physics Course Entrance Examination Questions (May 2010) Please answer the four questions from Problem 1 to Problem 4. You can use as many answer sheets you need. Your name, question numbers

More information

Open quantum systems

Open quantum systems Open quantum systems Wikipedia: An open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment. The open quantum system can be viewed as

More information

van Quantum tot Molecuul

van Quantum tot Molecuul 10 HC10: Molecular and vibrational spectroscopy van Quantum tot Molecuul Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ j.rojo@vu.nl Molecular and Vibrational Spectroscopy Based

More information

in terms of the classical frequency, ω = , puts the classical Hamiltonian in the form H = p2 2m + mω2 x 2

in terms of the classical frequency, ω = , puts the classical Hamiltonian in the form H = p2 2m + mω2 x 2 One of the most important problems in quantum mechanics is the simple harmonic oscillator, in part because its properties are directly applicable to field theory. The treatment in Dirac notation is particularly

More information

0 + E (1) and the first correction to the ground state energy is given by

0 + E (1) and the first correction to the ground state energy is given by 1 Problem set 9 Handout: 1/24 Due date: 1/31 Problem 1 Prove that the energy to first order for the lowest-energy state of a perturbed system is an upper bound for the exact energy of the lowest-energy

More information

22.02 Intro to Applied Nuclear Physics

22.02 Intro to Applied Nuclear Physics 22.02 Intro to Applied Nuclear Physics Mid-Term Exam Solution Problem 1: Short Questions 24 points These short questions require only short answers (but even for yes/no questions give a brief explanation)

More information

Ch 125a Problem Set 1

Ch 125a Problem Set 1 Ch 5a Problem Set Due Monday, Oct 5, 05, am Problem : Bra-ket notation (Dirac notation) Bra-ket notation is a standard and convenient way to describe quantum state vectors For example, φ is an abstract

More information

B2.III Revision notes: quantum physics

B2.III Revision notes: quantum physics B.III Revision notes: quantum physics Dr D.M.Lucas, TT 0 These notes give a summary of most of the Quantum part of this course, to complement Prof. Ewart s notes on Atomic Structure, and Prof. Hooker s

More information

Quantum ElectroDynamics III

Quantum ElectroDynamics III Quantum ElectroDynamics III Feynman diagram Dr.Farida Tahir Physics department CIIT, Islamabad Human Instinct What? Why? Feynman diagrams Feynman diagrams Feynman diagrams How? What? Graphic way to represent

More information

Quantum Mechanics: Fundamentals

Quantum Mechanics: Fundamentals Kurt Gottfried Tung-Mow Yan Quantum Mechanics: Fundamentals Second Edition With 75 Figures Springer Preface vii Fundamental Concepts 1 1.1 Complementarity and Uncertainty 1 (a) Complementarity 2 (b) The

More information

QUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer

QUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer Franz Schwabl QUANTUM MECHANICS Translated by Ronald Kates Second Revised Edition With 122Figures, 16Tables, Numerous Worked Examples, and 126 Problems ff Springer Contents 1. Historical and Experimental

More information

Advanced Quantum Mechanics

Advanced Quantum Mechanics Advanced Quantum Mechanics Rajdeep Sensarma sensarma@theory.tifr.res.in Quantum Dynamics Lecture #3 Recap of Last lass Time Dependent Perturbation Theory Linear Response Function and Spectral Decomposition

More information

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II January 20, 2017 9:00 a.m. 1:00 p.m. Do any four problems. Each problem is worth 25 points.

More information

Math Questions for the 2011 PhD Qualifier Exam 1. Evaluate the following definite integral 3" 4 where! ( x) is the Dirac! - function. # " 4 [ ( )] dx x 2! cos x 2. Consider the differential equation dx

More information

Problem Set 5 Solutions

Problem Set 5 Solutions Chemistry 362 Dr Jean M Standard Problem Set 5 Solutions ow many vibrational modes do the following molecules or ions possess? [int: Drawing Lewis structures may be useful in some cases] In all of the

More information

8 Quantized Interaction of Light and Matter

8 Quantized Interaction of Light and Matter 8 Quantized Interaction of Light and Matter 8.1 Dressed States Before we start with a fully quantized description of matter and light we would like to discuss the evolution of a two-level atom interacting

More information

8.05 Quantum Physics II, Fall 2011 FINAL EXAM Thursday December 22, 9:00 am -12:00 You have 3 hours.

8.05 Quantum Physics II, Fall 2011 FINAL EXAM Thursday December 22, 9:00 am -12:00 You have 3 hours. 8.05 Quantum Physics II, Fall 0 FINAL EXAM Thursday December, 9:00 am -:00 You have 3 hours. Answer all problems in the white books provided. Write YOUR NAME and YOUR SECTION on your white books. There

More information

Chapter 3: Relativistic Wave Equation

Chapter 3: Relativistic Wave Equation Chapter 3: Relativistic Wave Equation Klein-Gordon Equation Dirac s Equation Free-electron Solutions of the Timeindependent Dirac Equation Hydrogen Solutions of the Timeindependent Dirac Equation (Angular

More information

Physics 43 Exam 2 Spring 2018

Physics 43 Exam 2 Spring 2018 Physics 43 Exam 2 Spring 2018 Print Name: Conceptual Circle the best answer. (2 points each) 1. Quantum physics agrees with the classical physics limit when a. the total angular momentum is a small multiple

More information

Approximation Methods in QM

Approximation Methods in QM Chapter 3 Approximation Methods in QM Contents 3.1 Time independent PT (nondegenerate)............... 5 3. Degenerate perturbation theory (PT)................. 59 3.3 Time dependent PT and Fermi s golden

More information

Light - Atom Interaction

Light - Atom Interaction Light - Atom Interaction PHYS261 fall 2006 Go to list of topics - Overview of the topics - Time dependent QM- two-well problem - Time-Dependent Schrödinger Equation - Perturbation theory for TDSE - Dirac

More information

Quantum Physics III (8.06) Spring 2005 Assignment 10

Quantum Physics III (8.06) Spring 2005 Assignment 10 Quantum Physics III (8.06) Spring 2005 Assignment 10 April 29, 2005 Due FRIDAY May 6, 2005 Please remember to put your name and section time at the top of your paper. Prof. Rajagopal will give a review

More information

Lecture notes for QFT I (662)

Lecture notes for QFT I (662) Preprint typeset in JHEP style - PAPER VERSION Lecture notes for QFT I (66) Martin Kruczenski Department of Physics, Purdue University, 55 Northwestern Avenue, W. Lafayette, IN 47907-036. E-mail: markru@purdue.edu

More information

4.3 Lecture 18: Quantum Mechanics

4.3 Lecture 18: Quantum Mechanics CHAPTER 4. QUANTUM SYSTEMS 73 4.3 Lecture 18: Quantum Mechanics 4.3.1 Basics Now that we have mathematical tools of linear algebra we are ready to develop a framework of quantum mechanics. The framework

More information