Solutions to exam : 1FA352 Quantum Mechanics 10 hp 1
|
|
- Lawrence Russell
- 5 years ago
- Views:
Transcription
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) Show that e ipa/ x is an eigenstate to the position operator What is the eigenvalue? Hint: Consider [x, e ipa/ ] (a) An operator U is unitary if it satisfies U U = UU = With U = e ipa/ where p is Hermitian, it follows that U = e ipa/ and, therefore, U U = e ipa/ e ipa/ = = e ipa/ e ipa/ = UU (b) With the position operator x x = x x, we should demonstrate that x e ipa/ x = C e ipa/ x where C is the eigenvalue: x e ipa/ x = {[x, e ipa/ ] + e ipa/ x} x = ( using [x i, F ( p)] = i F p i from the collection of formulae ) = i eipa/ x + e ipa/ x x = i ia p eipa/ x + e ipa/ x x = ( a + x )e ipa/ x, ie eigenstate of x with eigenvalue x a This means a translation by a, which is in accordance with e ipa/ being the translation operator Problem (4 p): Consider the one-dimensional harmonic oscillator: states n, energies E n = ω(n + ) (a) Show that the expectation value n x n = for all n (b) Construct the state α = c + c that makes α x α as large as possible (c, c can be chosen real) (c) With the oscillator in state α at time t =, derive the state vector (in Schrödinger picture) for t > and calculate the expectation value α x α as a function of time (a) Using x = mω (a+a ) and a n = n n, a n = n + n+ (from collection of formulae) one obtains n x n = mω n a+a ( ) n = mω n n n + n + n n + = due to orthonormality m n = δ mn of the energy eigenstates of the harmonic oscillator In words, operator x is given by creation and annihilation operators that shift the state n to n ± which is orthogonal to n, giving zero expectation value (b) x = α x α = (c + c )x(c + c ) = c c x + c c x + c c x + c c x Using x, a, a as in (a) one obtains x = mω (c c + c c ) = c mω c, where in the last step the normalisation condition c + c = has been used together with c, c real since a complex phase does not affect the following Maximum of x is then obtained by d x /dc = giving c = / (maximum since d x /dc < for c = / ), and then c = / Hence x max = /mω for α = ( + )/ (up to an overall phase factor) (c): Applying the time evolution operator e iht/, and using e iht/ n = e iω(n+/)t n, one gets α, t = e iht/ α, t = = e iht/ ( + )/ = (e iωt/ + e 3iωt/ )/ as the state vector for t > in the Schrödinger picture The expectation value is then α, t x α, t = / (e iωt/ +e 3iωt/ )x(e iωt/ +e 3iωt/ )/ = ( x +e iωt x + e iωt x + x )/ = (again using x a + a as in (a)) = /mω cos ωt
2 Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem 3 (4 p): (a) Derive the relation m m j j ; m m j j ; jm = for Clebsch-Gordan coefficients and comment on its physical interpretation (b) An electron is in a p-orbital (l = ) Give, with motivations/explanations, all possible states jm of total angular momentum j of the electron, expressed in terms of its spin and orbital angular momentum (a) When combining two angular momenta j, j, the new states with total angular momentum j can be expressed (using the completeness relation) as j j ; j, m = m m j j ; m m j j ; m m j j ; j, m in the old direct-product basis Multiplying with j j ; jm gives the left-hand side j j ; jm j j ; jm = and the right-hand side m m j j ; jm j j ; m m j j ; m m j j ; jm = m m j j ; m m j j ; jm, ie the desired relation is obtained Since the Clebsch-Gordan coefficients are the expansion coefficients of the new j j ; jm basis states in the old direct-product basis, the given relation expresses the conservation of probability, ie the sum of squared amplitude coefficients should be unity (b) Add orbital angular momentum l = (m l =,,,) and spin angular momentum s = / (m s = /, /) to total angular momentum j = l s,, l + s = /, 3/ (m j = j, j +, j, j for j = / and j = 3/, resp) and m j = m l + m s Express new states (eigenstates to L, S, J, J z ) as linear combinations in old direct product basis (eigenstates to L, S, L z, S z ) using change of basis via completeness relation: l, s; j, m j = m l m s l, s; m l, m s l, s; m l, m s l, s; j, m j, where the scalar products are the expansion coefficients called Clebsch-Gordan coefficients Reading them from the table gives (suppressing l, s): j = 3, m j = + 3 = m l = +, m s = + j = 3, m j = + = +, +, + j =, m 3 3 j = + = +,, j = 3, m j = =, +, + j =, m 3 3 j = =,, j = 3, m j = 3 = m l =, m s = Here one should note the orthogonality between all states, in particular those on the same line due to the relative + and signs To derive all these states, one starts with the extreme state j = 3, m j = + 3 = m l = +, m s = + with coefficient (chosen real by convention) since only possibility is maximum m l and m s Then one operates with ladder operators J = L +S on respective sides giving j = 3, m j = + = above Iterating with the ladder operators gives the following states with j = 3/ The state j =, m j = + is then constructed to be orthogonal to j = 3, m j = + and have sum of squared coefficients equal unity The ladder operators are then iteratively applied to get the remaining j = / states Problem 4 (5 p): A hydrogen atom in its ground state nlm = is at time t = placed in a time-dependent electric field E(t) = E ẑ exp( t/τ) along ẑ direction, with E the field strength (a) In first-order time-dependent perturbation theory, which ones of the excited states lm are accessible in a direct transition? Motivate your answer (b) Calculate the transition probability from the ground state to these excited states You do not need to perform the radial integral, but perform all other integrals Hint: Energy levels: E n = e a ; Bohr radius: a n = ; Y me (θ, φ) =
3 Solutions to exam 6--6: FA35 Quantum Mechanics hp 3 (a) The perturbation is an electric field E(t) along the ẑ direction, which gives a perturbation potential dependent on z, V (t) = ee z exp( t/τ), using the potential in electrodynamics The initial state is and the final states are lm, with l =, and m =,, + The transition amplitudes in the zero th and first order are c () (t) = δ, c () lm (t) = i t with the difference in the energy levels is ω = E E elements of the perturbation dt e iω t V lm, (t ), = e ( 3e a ) = 4 8a V lm, (t ) = lm ( ee ) ze t /τ = ee e t /τ lm z and the matrix The possible transitions from to lm are the ones where the matrix elements of the operator z are non-zero The parity of the z operator is odd and lm states (or Yl m ) have parity ( ) l, which excludes transition from to states with even l Moreover, z is the zero-component of a rank- spherical tensor (z Y ), so the Wigner-Ekhardt theorem applies and gives the selection rules l = l ± and m = m Therefore, the only non-zero matrix element is z (which can also be obtained by detailed inspection of the polar angle integrals) Thus, the only direct transition in first-order perturbation theory is (b) We calculate the matrix element z using the position representation and write the wave function in spherical coordinates Ψ nlm (x) = x nlm = R nl (r) Yl m (θ, φ), z = d 3 x Ψ (x) z Ψ (x) = d 3 x R (r)y (θ, φ) z R (r)y (θ, φ) = r drdω R (r)y (θ, φ) r cos θ R (r)y (θ, φ) = dr r R (r)r (r) dω Y (θ, φ) cos θy (θ, φ) where we have separated the radial part and the angular part The radial integral does not need to be estimated and we can denote it I r = dr r R (r)r (r) = = 4 6 a ( 3) 5 The angular integral can be estimated using the spherical harmonics Y and Y π 3 I θ,φ = dω Y (θ, φ) cos θy (θ, φ) = dφ d(cos θ) cos θ cos θ 3 = π d(cos θ) cos θ = 3 The transition amplitude to first order is c () (t) = i = i ( ee ) t t dt e iω t V, (t ) = i t dt e iω t ( ee ) e t /τ z dt e iω t e t /τ I r I θ,φ = iee exp(iω t t ) τ iω I r I θ,φ τ The transition probability for is finally given by ( ) P (t) = c () (t) ee [ ] = + e t τ e t τ cos(ω t) ω + Ir 3 τ
4 Solutions to exam 6--6: FA35 Quantum Mechanics hp 4 Problem 5 (4 p): Consider the elastic scattering of particles, with mass m and wave vector k, on an atom represented by a 3D δ-potential V (x) = V δ(x), with V > (a) Calculate the differential scattering cross-section and the total scattering cross-section using the Born approximation (b) Perform the same calculations for a molecule represented by a double δ-potential V (x) = V (δ(x + R) + δ(x R)), where R k (a) The scattering amplitude in Born approximation is given by f () (k, k) = m (π) 3 k V k = m d 3 x e i(k k) x V (x ) Using the single δ-potential V (x) = V δ(x), the expression becomes f () (k, k) = m d 3 x e i(k k) x ( V )δ(x ) = m V k) ei(k = mv π The differential scattering cross-section and the total scattering cross-section become ( ) dσ dω = f () (k, k) mv =, σ π tot = dω dσ ( ) dω = mv σ π atom (b) Using the double δ-potential V (x) = V (δ(x + R) + δ(x R)), the amplitude is f () (k, k) = m d 3 x e i(k k) x ( V )(δ(x +R)+δ(x R)) = m V ( ) e i(k k) R + e i(k k) R With the notation that θ = (k, k) and considering that R k, the scattering amplitude becomes f () (k, k) = mv π ( e ikr( cos θ) + e ikr( cos θ)) The differential scattering cross-section becomes ( ) dσ dω = f () (k, k) mv ( = + e ikr( cos θ) + e ikr( cos θ)), π and the total scattering cross-section after performing the angular integrals is σ tot = dω dσ dω = π d(cos θ) dσ ( ) ( dω = mv + sin(4kr) ) π 4kR Finally, we note that ( σ molecule = σ atom + sin(4kr) ) 4kR
5 Solutions to exam 6--6: FA35 Quantum Mechanics hp 5 Problem 6 (3 p): To determine possible violation of the Bell inequality, one needs to evaluate expectation values of operators on the form a σ b σ, where a = (a x, a y, a z ) and b = (b x, b y, b z ) are unit vectors and σ= (σ x, σ y, σ z ) are Pauli matrices Suppose Alice and Bob share the Bell state Φ + = ( + ) (a) Calculate the expectation value Φ + a σ b σ Φ + (b) Suppose these are spin / states and Alice decides to measure S z, ie she chooses her vector a = (,, ) Which measurement must Bob choose to get a maximum expectation value? (a) With the notation a = (a x, a y, a z ) and b = (b x, b y, b z ) we can calculate the expectation value using the properties of the Pauli matrices σ x, σ y, σ z Φ + a σ b σ Φ + = ( + )(a σ b σ)( + ) = ( a σ b σ + a σ b σ ) + a σ b σ + a σ b σ = [ a z b z + (a x ia y ) (b x ib y ) ] + (a x + ia y ) (b x + ib y ) + ( a z ) ( b z ) = a x b x a y b y + a z b z Here we made use of a σ = a x σ x + a y σ y + a z σ z = a z, and so on (b) If Alice measures S z, then a x =, a y =, a z =, in which case the expectation value becomes Φ + a σ b σ Φ + = b z The maximum value is, when Bob also measures S z, ie b = (,, ) Simply put, this corresponds to the case where Alice and Bob measure S z on entangled spins and discover their measurements are fully correlated
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 information8.05 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 informationApproximation 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 informationQuantum 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 informationQuantum 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 informationPhysics 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 informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationQuantum Computing Lecture 3. Principles of Quantum Mechanics. Anuj Dawar
Quantum Computing Lecture 3 Principles of Quantum Mechanics Anuj Dawar What is Quantum Mechanics? Quantum Mechanics is a framework for the development of physical theories. It is not itself a physical
More informationTime 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 informationPhys 622 Problems Chapter 6
1 Problem 1 Elastic scattering Phys 622 Problems Chapter 6 A heavy scatterer interacts with a fast electron with a potential V (r) = V e r/r. (a) Find the differential cross section dσ dω = f(θ) 2 in the
More informationUNIVERSITY 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 22, 2016 9:00 a.m. 1:00 p.m. Do any four problems. Each problem is worth 25 points.
More informationQuantum 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 informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler April, 20 Lecture 2. Scattering Theory Reviewed Remember what we have covered so far for scattering theory. We separated the Hamiltonian similar to the way in
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationPhysics 580: Quantum Mechanics I Department of Physics, UIUC Fall Semester 2017 Professor Eduardo Fradkin
Physics 58: Quantum Mechanics I Department of Physics, UIUC Fall Semester 7 Professor Eduardo Fradkin Problem Set No. 5 Bound States and Scattering Theory Due Date: November 7, 7 Square Well in Three Dimensions
More informationGeneral Exam Part II, Fall 1998 Quantum Mechanics Solutions
General Exam Part II, Fall 1998 Quantum Mechanics Solutions Leo C. Stein Problem 1 Consider a particle of charge q and mass m confined to the x-y plane and subject to a harmonic oscillator potential V
More informationPHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep
Problem set 1 Due Sep 15 2005 1. Let V be the set of all complex valued functions of a real variable θ, that are periodic with period 2π. That is u(θ + 2π) = u(θ), for all u V. (1) (i) Show that this V
More informationAddition of Angular Momenta
Addition of Angular Momenta What we have so far considered to be an exact solution for the many electron problem, should really be called exact non-relativistic solution. A relativistic treatment is needed
More informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
More informationTime 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 informationQuantum 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 informationPHYSICS DEPARTMENT, PRINCETON UNIVERSITY PHYSICS 505 FINAL EXAMINATION. January 18, 2013, 1:30 4:30pm, A06 Jadwin Hall SOLUTIONS
PHYSICS DEPARTMENT, PRINCETON UNIVERSITY PHYSICS 55 FINAL EXAMINATION January 18, 13, 1:3 4:3pm, A6 Jadwin Hall SOLUTIONS This exam contains five problems Work any three of the five problems All problems
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
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 informationPhys 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 informationUNIVERSITY 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 August 23, 208 9:00 a.m. :00 p.m. Do any four problems. Each problem is worth 25 points.
More informationIn the following, we investigate the time-dependent two-component wave function ψ(t) = ( )
Ph.D. Qualifier, Quantum mechanics DO ONLY 3 OF THE 4 QUESTIONS Note the additional material for questions 1 and 3 at the end. PROBLEM 1. In the presence of a magnetic field B = (B x, B y, B z ), the dynamics
More informationPhysics 139B Solutions to Homework Set 5 Fall 2009
Physics 39B Solutions to Homework Set 5 Fall 9 Liboff, problem 35 on pages 749 75 A one-dimensional harmonic oscillator of charge-to-mass ratio e/m, and spring constant K oscillates parallel to the x-axis
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals Laporte Selection Rule Polarization Dependence Spin Selection Rule 1 Laporte Selection Rule We first apply this
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationChemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron):
April 6th, 24 Chemistry 2A 2nd Midterm. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (-electron): E n = m e Z 2 e 4 /2 2 n 2 = E Z 2 /n 2, n =, 2, 3,... where Ze is
More informationNon-relativistic scattering
Non-relativistic scattering Contents Scattering theory 2. Scattering amplitudes......................... 3.2 The Born approximation........................ 5 2 Virtual Particles 5 3 The Yukawa Potential
More informationQuantum 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 informationKet space as a vector space over the complex numbers
Ket space as a vector space over the complex numbers kets ϕ> and complex numbers α with two operations Addition of two kets ϕ 1 >+ ϕ 2 > is also a ket ϕ 3 > Multiplication with complex numbers α ϕ 1 >
More informationLecture #1. Review. Postulates of quantum mechanics (1-3) Postulate 1
L1.P1 Lecture #1 Review Postulates of quantum mechanics (1-3) Postulate 1 The state of a system at any instant of time may be represented by a wave function which is continuous and differentiable. Specifically,
More informationPhysics 221A Fall 1996 Notes 14 Coupling of Angular Momenta
Physics 1A Fall 1996 Notes 14 Coupling of Angular Momenta In these notes we will discuss the problem of the coupling or addition of angular momenta. It is assumed that you have all had experience with
More informationChem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.
Chem 44 Review for Exam Hydrogenic atoms: The Coulomb energy between two point charges Ze and e: V r Ze r Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative
More informationImplications of Time-Reversal Symmetry in Quantum Mechanics
Physics 215 Winter 2018 Implications of Time-Reversal Symmetry in Quantum Mechanics 1. The time reversal operator is antiunitary In quantum mechanics, the time reversal operator Θ acting on a state produces
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More informationAngular momentum. Quantum mechanics. Orbital angular momentum
Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular
More informationGraduate Quantum Mechanics I: Prelims and Solutions (Fall 2015)
Graduate Quantum Mechanics I: Prelims and Solutions (Fall 015 Problem 1 (0 points Suppose A and B are two two-level systems represented by the Pauli-matrices σx A,B σ x = ( 0 1 ;σ 1 0 y = ( ( 0 i 1 0 ;σ
More informationdoes not change the dynamics of the system, i.e. that it leaves the Schrödinger equation invariant,
FYST5 Quantum Mechanics II 9..212 1. intermediate eam (1. välikoe): 4 problems, 4 hours 1. As you remember, the Hamilton operator for a charged particle interacting with an electromagentic field can be
More informationSingle Electron Atoms
Single Electron Atoms In this section we study the spectrum and wave functions of single electron atoms. These are hydrogen, singly ionized He, doubly ionized Li, etc. We will write the formulae for hydrogen
More informationLECTURES 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 informationPreliminary Examination - Day 1 Thursday, August 9, 2018
UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, August 9, 8 This test covers the topics of Thermodynamics and Statistical Mechanics (Topic ) and Quantum Mechanics (Topic
More informationCollection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators
Basic Formulas 17-1-1 Division of Material Science Hans Weer The de Broglie wave length λ = h p The Schrödinger equation Hψr,t = i h t ψr,t Stationary states Hψr,t = Eψr,t Collection of formulae Quantum
More informationChm 331 Fall 2015, Exercise Set 4 NMR Review Problems
Chm 331 Fall 015, Exercise Set 4 NMR Review Problems Mr. Linck Version.0. Compiled December 1, 015 at 11:04:44 4.1 Diagonal Matrix Elements for the nmr H 0 Find the diagonal matrix elements for H 0 (the
More informationQualifying 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 informationGoal: find Lorentz-violating corrections to the spectrum of hydrogen including nonminimal effects
Goal: find Lorentz-violating corrections to the spectrum of hydrogen including nonminimal effects Method: Rayleigh-Schrödinger Perturbation Theory Step 1: Find the eigenvectors ψ n and eigenvalues ε n
More informationPHY413 Quantum Mechanics B Duration: 2 hours 30 minutes
BSc/MSci Examination by Course Unit Thursday nd May 4 : - :3 PHY43 Quantum Mechanics B Duration: hours 3 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More informationPhysics 221A Fall 2017 Notes 20 Parity
Copyright c 2017 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 20 Parity 1. Introduction We have now completed our study of proper rotations in quantum mechanics, one of the important space-time
More informationClassical field theory 2012 (NS-364B) Feynman propagator
Classical field theory 212 (NS-364B Feynman propagator 1. Introduction States in quantum mechanics in Schrödinger picture evolve as ( Ψt = Û(t,t Ψt, Û(t,t = T exp ı t dt Ĥ(t, (1 t where Û(t,t denotes the
More informationQuantum Mechanics FKA081/FIM400 Final Exam 28 October 2015
Quantum Mechanics FKA081/FIM400 Final Exam 28 October 2015 Next review time for the exam: December 2nd between 14:00-16:00 in my room. (This info is also available on the course homepage.) Examinator:
More informationQUANTUM 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 informationIntro to Nuclear and Particle Physics (5110)
Intro to Nuclear and Particle Physics (5110) March 13, 009 Nuclear Shell Model continued 3/13/009 1 Atomic Physics Nuclear Physics V = V r f r L r S r Tot Spin-Orbit Interaction ( ) ( ) Spin of e magnetic
More informationPhysics 828 Problem Set 7 Due Wednesday 02/24/2010
Physics 88 Problem Set 7 Due Wednesday /4/ 7)a)Consider the proton to be a uniformly charged sphere of radius f m Determine the correction to the s ground state energy 4 points) This is a standard problem
More informationThe Postulates of Quantum Mechanics
p. 1/23 The Postulates of Quantum Mechanics We have reviewed the mathematics (complex linear algebra) necessary to understand quantum mechanics. We will now see how the physics of quantum mechanics fits
More informationQuantization of the E-M field
April 6, 20 Lecture XXVI Quantization of the E-M field 2.0. Electric quadrupole transition If E transitions are forbidden by selection rules, then we consider the next term in the expansion of the spatial
More information1 Commutators (10 pts)
Final Exam Solutions 37A Fall 0 I. Siddiqi / E. Dodds Commutators 0 pts) ) Consider the operator  = Ĵx Ĵ y + ĴyĴx where J i represents the total angular momentum in the ith direction. a) Express both
More informationCh 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 informationSommerfeld (1920) noted energy levels of Li deduced from spectroscopy looked like H, with slight adjustment of principal quantum number:
Spin. Historical Spectroscopy of Alkali atoms First expt. to suggest need for electron spin: observation of splitting of expected spectral lines for alkali atoms: i.e. expect one line based on analogy
More informationIV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance
IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance The foundation of electronic spectroscopy is the exact solution of the time-independent Schrodinger equation for the hydrogen atom.
More informationPhysics Qual - Statistical Mechanics ( Fall 2016) I. Describe what is meant by: (a) A quasi-static process (b) The second law of thermodynamics (c) A throttling process and the function that is conserved
More informationAngular Momentum. Classical. J r p. radius vector from origin. linear momentum. determinant form of cross product iˆ xˆ J J J J J J
Angular Momentum Classical r p p radius vector from origin linear momentum r iˆ ˆj kˆ x y p p p x y determinant form of cross product iˆ xˆ ˆj yˆ kˆ ˆ y p p x y p x p y x x p y p y x x y Copyright Michael
More informationOptical Lattices. Chapter Polarization
Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose- Hubbard model used to describe ultracold atoms in optical lattices. We show how the AC-Stark
More information5.111 Lecture Summary #6
5.111 Lecture Summary #6 Readings for today: Section 1.9 (1.8 in 3 rd ed) Atomic Orbitals. Read for Lecture #7: Section 1.10 (1.9 in 3 rd ed) Electron Spin, Section 1.11 (1.10 in 3 rd ed) The Electronic
More informationJ10M.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 informationSolution Exercise 12
Solution Exercise 12 Problem 1: The Stark effect in the hydrogen atom a) Since n = 2, the quantum numbers l can take the values, 1 and m = -1,, 1.We obtain the following basis: n, l, m = 2,,, 2, 1, 1,
More information5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:
5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ 1 0 0 0 ξ 1 0 0 0 ξ 3, ξ
More informationChapter 6: SYMMETRY IN QUANTUM MECHANICS
Chapter 6: SYMMETRY IN QUANTUM MECHANICS Since the beginning of physics, symmetry considerations have provided us with an extremely powerful and useful tool in o ur effort in understanding nature. Gradually
More informationPhysics 70007, Fall 2009 Answers to Final Exam
Physics 70007, Fall 009 Answers to Final Exam December 17, 009 1. Quantum mechanical pictures a Demonstrate that if the commutation relation [A, B] ic is valid in any of the three Schrodinger, Heisenberg,
More informationLectures on Quantum Mechanics
Lectures on Quantum Mechanics Steven Weinberg The University of Texas at Austin CAMBRIDGE UNIVERSITY PRESS Contents PREFACE page xv NOTATION xviii 1 HISTORICAL INTRODUCTION 1 1.1 Photons 1 Black-body radiation
More informationψ 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 informationONE AND MANY ELECTRON ATOMS Chapter 15
See Week 8 lecture notes. This is exactly the same as the Hamiltonian for nonrigid rotation. In Week 8 lecture notes it was shown that this is the operator for Lˆ 2, the square of the angular momentum.
More informationNotes on Quantum Mechanics
Notes on Quantum Mechanics K. Schulten Department of Physics and Beckman Institute University of Illinois at Urbana Champaign 405 N. Mathews Street, Urbana, IL 61801 USA (April 18, 2000) Preface i Preface
More informationUniversity of Illinois at Chicago Department of Physics. Quantum Mechanics Qualifying Examination. January 7, 2013 (Tuesday) 9:00 am - 12:00 noon
University of Illinois at Chicago Department of Physics Quantum Mechanics Qualifying Examination January 7, 13 Tuesday 9: am - 1: noon Full credit can be achieved from completely correct answers to 4 questions
More informationCold molecules: Theory. Viatcheslav Kokoouline Olivier Dulieu
Cold molecules: Theory Viatcheslav Kokoouline Olivier Dulieu Summary (1) Born-Oppenheimer approximation; diatomic electronic states, rovibrational wave functions of the diatomic cold molecule molecule
More informationLecture Notes. Quantum Theory. Prof. Maximilian Kreuzer. Institute for Theoretical Physics Vienna University of Technology. covering the contents of
Lecture Notes Quantum Theory by Prof. Maximilian Kreuzer Institute for Theoretical Physics Vienna University of Technology covering the contents of 136.019 Quantentheorie I and 136.027 Quantentheorie II
More information16.1. PROBLEM SET I 197
6.. PROBLEM SET I 97 Answers: Problem set I. a In one dimension, the current operator is specified by ĵ = m ψ ˆpψ + ψˆpψ. Applied to the left hand side of the system outside the region of the potential,
More informationTotal Angular Momentum for Hydrogen
Physics 4 Lecture 7 Total Angular Momentum for Hydrogen Lecture 7 Physics 4 Quantum Mechanics I Friday, April th, 008 We have the Hydrogen Hamiltonian for central potential φ(r), we can write: H r = p
More informationPhysics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom
Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228 Happy April Fools Day Example / Worked Problems What is the ratio of the
More informationQuantum Physics II (8.05) Fall 2004 Assignment 3
Quantum Physics II (8.5) Fall 24 Assignment 3 Massachusetts Institute of Technology Physics Department Due September 3, 24 September 23, 24 7:pm This week we continue to study the basic principles of quantum
More informationFinal Examination. Tuesday December 15, :30 am 12:30 pm. particles that are in the same spin state 1 2, + 1 2
Department of Physics Quantum Mechanics I, Physics 57 Temple University Instructor: Z.-E. Meziani Final Examination Tuesday December 5, 5 :3 am :3 pm Problem. pts) Consider a system of three non interacting,
More informationGROUP THEORY IN PHYSICS
GROUP THEORY IN PHYSICS Wu-Ki Tung World Scientific Philadelphia Singapore CONTENTS CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 PREFACE INTRODUCTION 1.1 Particle on a One-Dimensional Lattice 1.2 Representations
More information量子力学 Quantum mechanics. School of Physics and Information Technology
量子力学 Quantum mechanics School of Physics and Information Technology Shaanxi Normal University Chapter 9 Time-dependent perturation theory Chapter 9 Time-dependent perturation theory 9.1 Two-level systems
More informationThe rotation group and quantum mechanics 1 D. E. Soper 2 University of Oregon 30 January 2012
The rotation group and quantum mechanics 1 D. E. Soper 2 University of Oregon 30 January 2012 I offer here some background for Chapter 3 of J. J. Sakurai, Modern Quantum Mechanics. 1 The rotation group
More information20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R
20 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 2. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian
More informationUGC 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 informationDecays, resonances and scattering
Structure of matter and energy scales Subatomic physics deals with objects of the size of the atomic nucleus and smaller. We cannot see subatomic particles directly, but we may obtain knowledge of their
More informationCoupling of Angular Momenta Isospin Nucleon-Nucleon Interaction
Lecture 5 Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction WS0/3: Introduction to Nuclear and Particle Physics,, Part I I. Angular Momentum Operator Rotation R(θ): in polar coordinates the
More informationINTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM
INTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM I. The interaction of electromagnetic fields with matter. The Lagrangian for the charge q in electromagnetic potentials V
More information3. Quantum Mechanics in 3D
3. Quantum Mechanics in 3D 3.1 Introduction Last time, we derived the time dependent Schrödinger equation, starting from three basic postulates: 1) The time evolution of a state can be expressed as a unitary
More informationLecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators
Lecture Spin, orbital, and total angular momentum 70.00 Mechanics Very brief background MATH-GA In 9, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform
More informationLecture: Scattering theory
Lecture: Scattering theory 30.05.2012 SS2012: Introduction to Nuclear and Particle Physics, Part 2 2 1 Part I: Scattering theory: Classical trajectoriest and cross-sections Quantum Scattering 2 I. Scattering
More information( ) 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 informationLSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (07/2017) =!2 π 2 a cos π x
LSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (7/17) 1. For a particle trapped in the potential V(x) = for a x a and V(x) = otherwise, the ground state energy and eigenfunction
More informationSemi-Classical Theory of Radiative Transitions
Semi-Classical Theory of Radiative Transitions Massimo Ricotti ricotti@astro.umd.edu University of Maryland Semi-Classical Theory of Radiative Transitions p.1/13 Atomic Structure (recap) Time-dependent
More informationPhysics 215 Quantum Mechanics I Assignment 8
Physics 15 Quantum Mechanics I Assignment 8 Logan A. Morrison March, 016 Problem 1 Let J be an angular momentum operator. Part (a) Using the usual angular momentum commutation relations, prove that J =
More informationSchrödinger equation for the nuclear potential
Schrödinger equation for the nuclear potential Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 January 24, 2011 NUCS 342 (Lecture 4) January 24, 2011 1 / 32 Outline 1 One-dimensional
More information