Lecture 45: The Eigenvalue Problem of L z and L 2 in Three Dimensions, ct d: Operator Method Date Revised: 2009/02/17 Date Given: 2009/02/11

Size: px
Start display at page:

Download "Lecture 45: The Eigenvalue Problem of L z and L 2 in Three Dimensions, ct d: Operator Method Date Revised: 2009/02/17 Date Given: 2009/02/11"

Transcription

1 Page 757 Lecture 45: The Eigenvalue Problem of L z and L 2 in Three Dimensions, ct d: Operator Method Date Revised: 2009/02/17 Date Given: 2009/02/11

2 The Eigenvector-Eigenvalue Problem of L z and L 2 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 758 Operator Method We ve found the eigenfunctions and eigenvalues in the standard pedestrian way. Let s now use some clever operator methods that recall how we used raising and lowering operators to determine the eigenvalues of the SHO without having to explicitly find the eigenfunctions. We shall see that this method leads to a simpler way to find the eigenfunctions too, just as we were able to obtain all the eigenfunctions of the SHO by applying the raising operator in the position basis to the simple Gaussian ground-state wavefunction. Let s assume we know nothing about the eigenvalue spectrum of L 2 and L z except that the operators commute so they have simultaneous eigenvectors. Denote an eigenstate of L 2 and L z with eigenvalues α and β by α, β. That is L 2 α, β = α α, β L z α, β = β α, β

3 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 759 We define angular momentum raising and lowering operators: L ± = L x ± i L y They are named this way because they satisfy so that [L z, L ± ] = ± L ± L z (L ± α, β ) = (± L ± + L ± L z ) α, β = (± + β) (L ± α, β ) That is, when α, β has L z eigenvalue β, the state obtained by applying a raising or lowering operator in the state, L ± α, β, is an eigenvector of L z with eigenvalue β ±. The raising and lowering operators commute with L 2, [L 2, L ± ] = 0 so we are assured that α, β and L ± α, β have the same eigenvalue α of L 2.

4 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 760 So, our space will break down into subspaces that are eigenspaces of L 2, which will be further decomposed into subspaces that are eigenspaces of L z. L ± moves between these subspaces of a particular L 2 eigenspace. Explicitly, we have L ± α, β = C ± (α, β) α, β ± We run into the same problem we had with the SHO raising and lowering operators, which is that we so far have no condition that puts a lower or upper limit on the L z eigenvalue β. Heuristically, it would be unphysical to have β 2 > α. This can be seen rigorously as follows: α, β `L 2 L 2 z α, β = α, β `L2 x + L 2 y α, β The latter expression is nonnegative because the eigenvalues of L 2 x and L 2 y are all nonnegative because the eigenvalues of L x and L y are real because they are Hermitian. So we see α β 2 0, or α β 2 as desired.

5 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 761 So, we require there to be states α, β max and α, β min that satisfy L + α, β max = 0 L α, β min = 0 where by 0 we mean the null vector, usually referred to as 0, which may be confusing in this situation. We need to rewrite these expressions in terms of L 2 and L z to further reduce them; let s apply L and L + to do this: L L + α, β max = 0 L +L α, β min = 0 `L2 L 2 z L z α, βmax = 0 `L2 L 2 z + L z α, βmin = 0 `α β 2 max β max α, βmax = 0 β max (β max + ) = α `α β 2 min + β min α, βmin = 0 β min (β min ) = α which implies β min = β max

6 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 762 In order for the raising chain begun at β min and the lowering chain begun at β max to terminate, it is necessary that there be a k + and k such that Therefore (L +) k++1 α, β min α, β max (L ) k +1 α, β max α, β min β min + k + = β max β max k = β min So we have k + = k k β max β min = k Since β min = β max, we then have β max = k 2 α = β max (β max + ) = 2 k 2 «k k = 0, 1, 2,... For k even, we recover the allowed eigenvalues we obtained via the differential equation method. The k odd eigenvalues are a different beast, though, and are associated with spin, a degree of freedom that behaves like angular momentum in many ways but is not associated with orbital motion of a particle.

7 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 763 The last point is a very important one: the k odd values arose only from the assumption of the angular momentum operator commutation relations. They did not come from the differential equations, which is what ties all of this to the behavior of spatial wavefunctions; the differential equations method does not permit k odd. This is the source of our statement that the k odd values are not associated with orbital angular momentum. In detail, the restriction to k even comes from the requirement that the wavefunction be single-valued in φ, which is required by Hermiticity of L z. Such a requirement would not hold for a particle spin s z-component operator because there will be no spatial wavefunction to consider. Thus, the above proof tells us which values of k are allowed, and then other restrictions can further reduce the set. Unlike Shankar, who gives a bit more detailed of a hint at what is meant by spin, we will delay discussion until we have time to do it thoroughly. For now it is not important to have a physical picture of the states that result in half-integral values of L z.

8 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 764 Given that the spectrum of eigenvalues we have derived is more general than just orbital angular momentum L, we will follow standard notation and use J instead of L. We will denote the eigenvalues as follows: We will denote by j the value of k/2. j may take on any nonnegative integral or half-integral value. The J 2 eigenvalue is α = 2 j(j + 1). However, we will replace α in α, β by j for brevity. The J z eigenvalue β can take on values from j to j in steps of size. We define m = β/. We will replace β in α, β by m for consistency with the notation we developed via the differential equation method. Therefore, simultaneous eigenstates of J 2 and J z will be denoted by j, m and will have J 2 eigenvalue α = 2 j (j + 1) and J z eigenvalue β = m.

9 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 765 Summary Let us take a step back and see what we have done and where we should go. What we have done: We are considering problems in two or three spatial dimensions in cylindrical and spherical coordinates with an eye toward working with Hamiltonians that are invariant under rotations and hence depend on the cylindrical coordinate ρ or the radial coordinate r. Since a continuous symmetry transformation of a Hamiltonian derives from a generator operator that commutes with the Hamiltonian, we knew it would be useful to find the generator and its eigenvalues and eigenvectors to help us reduce or solve the eigenvector-eigenvalue problem of the full Hamiltonian. This led us to write explicit forms for L and L 2 and to obtain their eigenvectors and eigenfunctions, both in the position basis and in the more natural basis of their eigenstates. We have thus been able to organize the Hilbert space into subspaces of specific values of the angular momentum magnitude.

10 Section 14.5 Rotations and Orbital Angular Momentum: The Eigenvector-Eigenvalue Problem of L z and L 2 Page 766 We have two important tasks left: To understand the full structure of the Hilbert space in terms of the eigenstates of J 2 and J z ; i.e., let s write down explicit forms for all the operators we have considered: J x, J y, J +, J and rotation operators. To understand the connection between the { j, m } basis and the position basis eigenstates essentially, to show that we can obtain the position basis eigenstates from the structure of the Hilbert space in terms of the { j, m } basis. We consider these tasks next.

Page 404. Lecture 22: Simple Harmonic Oscillator: Energy Basis Date Given: 2008/11/19 Date Revised: 2008/11/19

Page 404. Lecture 22: Simple Harmonic Oscillator: Energy Basis Date Given: 2008/11/19 Date Revised: 2008/11/19 Page 404 Lecture : Simple Harmonic Oscillator: Energy Basis Date Given: 008/11/19 Date Revised: 008/11/19 Coordinate Basis Section 6. The One-Dimensional Simple Harmonic Oscillator: Coordinate Basis Page

More information

Lecture #1. Review. Postulates of quantum mechanics (1-3) Postulate 1

Lecture #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 information

Section 9 Variational Method. Page 492

Section 9 Variational Method. Page 492 Section 9 Variational Method Page 492 Page 493 Lecture 27: The Variational Method Date Given: 2008/12/03 Date Revised: 2008/12/03 Derivation Section 9.1 Variational Method: Derivation Page 494 Motivation

More information

Rotations in Quantum Mechanics

Rotations in Quantum Mechanics Rotations in Quantum Mechanics We have seen that physical transformations are represented in quantum mechanics by unitary operators acting on the Hilbert space. In this section, we ll think about the specific

More information

Page 712. Lecture 42: Rotations and Orbital Angular Momentum in Two Dimensions Date Revised: 2009/02/04 Date Given: 2009/02/04

Page 712. Lecture 42: Rotations and Orbital Angular Momentum in Two Dimensions Date Revised: 2009/02/04 Date Given: 2009/02/04 Page 71 Lecture 4: Rotations and Orbital Angular Momentum in Two Dimensions Date Revised: 009/0/04 Date Given: 009/0/04 Plan of Attack Section 14.1 Rotations and Orbital Angular Momentum: Plan of Attack

More information

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition

More information

2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.

2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x. Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and

More information

Time part of the equation can be separated by substituting independent equation

Time part of the equation can be separated by substituting independent equation Lecture 9 Schrödinger Equation in 3D and Angular Momentum Operator In this section we will construct 3D Schrödinger equation and we give some simple examples. In this course we will consider problems where

More information

Physics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I

Physics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I Physics 342 Lecture 26 Angular Momentum Lecture 26 Physics 342 Quantum Mechanics I Friday, April 2nd, 2010 We know how to obtain the energy of Hydrogen using the Hamiltonian operator but given a particular

More information

Total Angular Momentum for Hydrogen

Total 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 information

Addition of Angular Momenta

Addition 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 information

which implies that we can take solutions which are simultaneous eigen functions of

which implies that we can take solutions which are simultaneous eigen functions of Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3-D Quantum mechanics in 3-D For most physical systems, the dynamics is in 3-D. The solutions to the general 3-d problem are quite complicated,

More information

3. Quantum Mechanics in 3D

3. 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 information

Lecture 19 (Nov. 15, 2017)

Lecture 19 (Nov. 15, 2017) Lecture 19 8.31 Quantum Theory I, Fall 017 8 Lecture 19 Nov. 15, 017) 19.1 Rotations Recall that rotations are transformations of the form x i R ij x j using Einstein summation notation), where R is an

More information

Generators for Continuous Coordinate Transformations

Generators for Continuous Coordinate Transformations Page 636 Lecture 37: Coordinate Transformations: Continuous Passive Coordinate Transformations Active Coordinate Transformations Date Revised: 2009/01/28 Date Given: 2009/01/26 Generators for Continuous

More information

Page 684. Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02

Page 684. Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02 Page 684 Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02 Time Transformations Section 12.5 Symmetries: Time Transformations Page 685 Time Translation

More information

Physics 70007, Fall 2009 Answers to Final Exam

Physics 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 information

The 3 dimensional Schrödinger Equation

The 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 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

Angular Momentum. Andreas Wacker Mathematical Physics Lund University

Angular Momentum. Andreas Wacker Mathematical Physics Lund University Angular Momentum Andreas Wacker Mathematical Physics Lund University Commutation relations of (orbital) angular momentum Angular momentum in analogy with classical case L= r p satisfies commutation relations

More information

Atomic Systems (PART I)

Atomic Systems (PART I) Atomic Systems (PART I) Lecturer: Location: Recommended Text: Dr. D.J. Miller Room 535, Kelvin Building d.miller@physics.gla.ac.uk Joseph Black C407 (except 15/1/10 which is in Kelvin 312) Physics of Atoms

More information

d 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5)

d 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5) LECTURE 12 Maxwell Velocity Distribution Suppose we have a dilute gas of molecules, each with mass m. If the gas is dilute enough, we can ignore the interactions between the molecules and the energy will

More information

Physics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I

Physics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I Physics 342 Lecture 27 Spin Lecture 27 Physics 342 Quantum Mechanics I Monday, April 5th, 2010 There is an intrinsic characteristic of point particles that has an analogue in but no direct derivation from

More information

Implications of Time-Reversal Symmetry in Quantum Mechanics

Implications 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

Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor

Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor It turns out that the boundary condition of the wavefunction going to zero at infinity is sufficient to quantize the value of energy that

More information

Angular momentum. Quantum mechanics. Orbital angular momentum

Angular 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 information

Lecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II

Lecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II Lecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II We continue our discussion of symmetries and their role in matrix representation in this lecture. An example

More information

PHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep

PHY 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 information

Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 20, March 8, 2006

Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 20, March 8, 2006 Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer Lecture 20, March 8, 2006 Solved Homework We determined that the two coefficients in our two-gaussian

More information

Physics 606, Quantum Mechanics, Final Exam NAME ( ) ( ) + V ( x). ( ) and p( t) be the corresponding operators in ( ) and x( t) : ( ) / dt =...

Physics 606, Quantum Mechanics, Final Exam NAME ( ) ( ) + V ( x). ( ) and p( t) be the corresponding operators in ( ) and x( t) : ( ) / dt =... Physics 606, Quantum Mechanics, Final Exam NAME Please show all your work. (You are graded on your work, with partial credit where it is deserved.) All problems are, of course, nonrelativistic. 1. Consider

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

MAT265 Mathematical Quantum Mechanics Brief Review of the Representations of SU(2)

MAT265 Mathematical Quantum Mechanics Brief Review of the Representations of SU(2) MAT65 Mathematical Quantum Mechanics Brief Review of the Representations of SU() (Notes for MAT80 taken by Shannon Starr, October 000) There are many references for representation theory in general, and

More information

26 Group Theory Basics

26 Group Theory Basics 26 Group Theory Basics 1. Reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian.

More information

Angular momentum & spin

Angular momentum & spin Angular momentum & spin January 8, 2002 1 Angular momentum Angular momentum appears as a very important aspect of almost any quantum mechanical system, so we need to briefly review some basic properties

More information

Problem 1: A 3-D Spherical Well(10 Points)

Problem 1: A 3-D Spherical Well(10 Points) Problem : A 3-D Spherical Well( Points) For this problem, consider a particle of mass m in a three-dimensional spherical potential well, V (r), given as, V = r a/2 V = W r > a/2. with W >. All of the following

More information

QM and Angular Momentum

QM and Angular Momentum Chapter 5 QM and Angular Momentum 5. Angular Momentum Operators In your Introductory Quantum Mechanics (QM) course you learned about the basic properties of low spin systems. Here we want to review that

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

Quantum Theory of Angular Momentum and Atomic Structure

Quantum Theory of Angular Momentum and Atomic Structure Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the

More information

General Exam Part II, Fall 1998 Quantum Mechanics Solutions

General 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 information

CHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM. (From Cohen-Tannoudji, Chapter XII)

CHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM. (From Cohen-Tannoudji, Chapter XII) CHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM (From Cohen-Tannoudji, Chapter XII) We will now incorporate a weak relativistic effects as perturbation

More information

Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction

Coupling 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 information

Physics 221A Fall 2017 Notes 20 Parity

Physics 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 information

Isotropic harmonic oscillator

Isotropic harmonic oscillator Isotropic harmonic oscillator 1 Isotropic harmonic oscillator The hamiltonian of the isotropic harmonic oscillator is H = h m + 1 mω r (1) = [ h d m dρ + 1 ] m ω ρ, () ρ=x,y,z a sum of three one-dimensional

More information

Harmonic Oscillator I

Harmonic Oscillator I Physics 34 Lecture 7 Harmonic Oscillator I Lecture 7 Physics 34 Quantum Mechanics I Monday, February th, 008 We can manipulate operators, to a certain extent, as we would algebraic expressions. By considering

More information

9 Electron orbits in atoms

9 Electron orbits in atoms Physics 129b Lecture 15 Caltech, 02/22/18 Reference: Wu-Ki-Tung, Group Theory in physics, Chapter 7. 9 Electron orbits in atoms Now let s see how our understanding of the irreps of SO(3) (SU(2)) can help

More information

Physics 221A Fall 1996 Notes 14 Coupling of Angular Momenta

Physics 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 information

Lecture Notes 2: Review of Quantum Mechanics

Lecture Notes 2: Review of Quantum Mechanics Quantum Field Theory for Leg Spinners 18/10/10 Lecture Notes 2: Review of Quantum Mechanics Lecturer: Prakash Panangaden Scribe: Jakub Závodný This lecture will briefly review some of the basic concepts

More information

The Particle in a Box

The Particle in a Box Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:

More information

P3317 HW from Lecture and Recitation 10

P3317 HW from Lecture and Recitation 10 P3317 HW from Lecture 18+19 and Recitation 10 Due Nov 6, 2018 Problem 1. Equipartition Note: This is a problem from classical statistical mechanics. We will need the answer for the next few problems, and

More information

Physics 221A Fall 2017 Notes 27 The Variational Method

Physics 221A Fall 2017 Notes 27 The Variational Method Copyright c 2018 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 27 The Variational Method 1. Introduction Very few realistic problems in quantum mechanics are exactly solvable, so approximation methods

More information

In this lecture we will go through the method of coupling of angular momentum.

In this lecture we will go through the method of coupling of angular momentum. Lecture 3 : Title : Coupling of angular momentum Page-0 In this lecture we will go through the method of coupling of angular momentum. We will start with the need for this coupling and then develop the

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

1 Revision to Section 17.5: Spin

1 Revision to Section 17.5: Spin 1 Revision to Section 17.5: Spin We classified irreducible finite-dimensional representations of the Lie algebra so(3) by their spin l, where l is the largest eigenvalue for the operator L 3 = iπ(f 3 ).

More information

Two and Three-Dimensional Systems

Two and Three-Dimensional Systems 0 Two and Three-Dimensional Systems Separation of variables; degeneracy theorem; group of invariance of the two-dimensional isotropic oscillator. 0. Consider the Hamiltonian of a two-dimensional anisotropic

More information

Particle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision)

Particle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision) Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e - e + - + e - e + - + e - e + - + e - e + - Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 Non-Relativistic QM (Revision)

More information

Sample Quantum Chemistry Exam 2 Solutions

Sample Quantum Chemistry Exam 2 Solutions Chemistry 46 Fall 7 Dr. Jean M. Standard Name SAMPE EXAM Sample Quantum Chemistry Exam Solutions.) ( points) Answer the following questions by selecting the correct answer from the choices provided. a.)

More information

Introduction to Quantum Mechanics PVK - Solutions. Nicolas Lanzetti

Introduction to Quantum Mechanics PVK - Solutions. Nicolas Lanzetti Introduction to Quantum Mechanics PVK - Solutions Nicolas Lanzetti lnicolas@student.ethz.ch 1 Contents 1 The Wave Function and the Schrödinger Equation 3 1.1 Quick Checks......................................

More information

Angular Momentum Algebra

Angular Momentum Algebra Angular Momentum Algebra Chris Clark August 1, 2006 1 Input We will be going through the derivation of the angular momentum operator algebra. The only inputs to this mathematical formalism are the basic

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

Lecture 4 Quantum mechanics in more than one-dimension

Lecture 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 information

Are these states normalized? A) Yes

Are these states normalized? A) Yes QMII-. Consider two kets and their corresponding column vectors: Ψ = φ = Are these two state orthogonal? Is ψ φ = 0? A) Yes ) No Answer: A Are these states normalized? A) Yes ) No Answer: (each state has

More information

Physics 221A Fall 2018 Notes 22 Bound-State Perturbation Theory

Physics 221A Fall 2018 Notes 22 Bound-State Perturbation Theory Copyright c 2018 by Robert G. Littlejohn Physics 221A Fall 2018 Notes 22 Bound-State Perturbation Theory 1. Introduction Bound state perturbation theory applies to the bound states of perturbed systems,

More information

C/CS/Phys C191 Particle-in-a-box, Spin 10/02/08 Fall 2008 Lecture 11

C/CS/Phys C191 Particle-in-a-box, Spin 10/02/08 Fall 2008 Lecture 11 C/CS/Phys C191 Particle-in-a-box, Spin 10/0/08 Fall 008 Lecture 11 Last time we saw that the time dependent Schr. eqn. can be decomposed into two equations, one in time (t) and one in space (x): space

More information

Particle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation

Particle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 45 Particle Physics Dr. Alexander Mitov µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 46 Non-Relativistic

More information

Chapter 2 The Group U(1) and its Representations

Chapter 2 The Group U(1) and its Representations Chapter 2 The Group U(1) and its Representations The simplest example of a Lie group is the group of rotations of the plane, with elements parametrized by a single number, the angle of rotation θ. It is

More information

Massachusetts Institute of Technology Physics Department

Massachusetts 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 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

Lecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators

Lecture 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 information

Physics 115C Homework 2

Physics 115C Homework 2 Physics 5C Homework Problem Our full Hamiltonian is H = p m + mω x +βx 4 = H +H where the unperturbed Hamiltonian is our usual and the perturbation is H = p m + mω x H = βx 4 Assuming β is small, the perturbation

More information

Lecture 4 Quantum mechanics in more than one-dimension

Lecture 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 information

ECEN 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 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 information

PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 8: Solutions. Topics covered: hydrogen fine structure

PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 8: Solutions. Topics covered: hydrogen fine structure PHYS85 Quantum Mechanics II, Spring HOMEWORK ASSIGNMENT 8: Solutions Topics covered: hydrogen fine structure. [ pts] Let the Hamiltonian H depend on the parameter λ, so that H = H(λ). The eigenstates and

More information

Physics 401: Quantum Mechanics I Chapter 4

Physics 401: Quantum Mechanics I Chapter 4 Physics 401: Quantum Mechanics I Chapter 4 Are you here today? A. Yes B. No C. After than midterm? 3-D Schroedinger Equation The ground state energy of the particle in a 3D box is ( 1 2 +1 2 +1 2 ) π2

More information

Lecture 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7

Lecture 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7 Lecture 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7 In Lecture 1 and 2, we have discussed how to represent the state of a quantum mechanical system based the superposition

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

Harmonic Oscillator. Robert B. Griffiths Version of 5 December Notation 1. 3 Position and Momentum Representations of Number Eigenstates 2

Harmonic Oscillator. Robert B. Griffiths Version of 5 December Notation 1. 3 Position and Momentum Representations of Number Eigenstates 2 qmd5 Harmonic Oscillator Robert B. Griffiths Version of 5 December 0 Contents Notation Eigenstates of the Number Operator N 3 Position and Momentum Representations of Number Eigenstates 4 Coherent States

More information

Spin-Orbit Interactions in Semiconductor Nanostructures

Spin-Orbit Interactions in Semiconductor Nanostructures Spin-Orbit Interactions in Semiconductor Nanostructures Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A. http://www.physics.udel.edu/~bnikolic Spin-Orbit Hamiltonians

More information

1 Mathematical preliminaries

1 Mathematical preliminaries 1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical

More information

129 Lecture Notes More on Dirac Equation

129 Lecture Notes More on Dirac Equation 19 Lecture Notes More on Dirac Equation 1 Ultra-relativistic Limit We have solved the Diraction in the Lecture Notes on Relativistic Quantum Mechanics, and saw that the upper lower two components are large

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

The Hamiltonian and the Schrödinger equation Consider time evolution from t to t + ɛ. As before, we expand in powers of ɛ; we have. H(t) + O(ɛ 2 ).

The Hamiltonian and the Schrödinger equation Consider time evolution from t to t + ɛ. As before, we expand in powers of ɛ; we have. H(t) + O(ɛ 2 ). Lecture 12 Relevant sections in text: 2.1 The Hamiltonian and the Schrödinger equation Consider time evolution from t to t + ɛ. As before, we expand in powers of ɛ; we have U(t + ɛ, t) = I + ɛ ( īh ) H(t)

More information

Chemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron):

Chemistry 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 information

1 Commutators (10 pts)

1 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 information

Quantum mechanics in one hour

Quantum mechanics in one hour Chapter 2 Quantum mechanics in one hour 2.1 Introduction The purpose of this chapter is to refresh your knowledge of quantum mechanics and to establish notation. Depending on your background you might

More information

Problem 1: Step Potential (10 points)

Problem 1: Step Potential (10 points) Problem 1: Step Potential (10 points) 1 Consider the potential V (x). V (x) = { 0, x 0 V, x > 0 A particle of mass m and kinetic energy E approaches the step from x < 0. a) Write the solution to Schrodinger

More information

Statistical Interpretation

Statistical Interpretation Physics 342 Lecture 15 Statistical Interpretation Lecture 15 Physics 342 Quantum Mechanics I Friday, February 29th, 2008 Quantum mechanics is a theory of probability densities given that we now have an

More information

221B Lecture Notes Spontaneous Symmetry Breaking

221B Lecture Notes Spontaneous Symmetry Breaking B Lecture Notes Spontaneous Symmetry Breaking Spontaneous Symmetry Breaking Spontaneous Symmetry Breaking is an ubiquitous concept in modern physics, especially in condensed matter and particle physics.

More information

St Hugh s 2 nd Year: Quantum Mechanics II. Reading. Topics. The following sources are recommended for this tutorial:

St Hugh s 2 nd Year: Quantum Mechanics II. Reading. Topics. The following sources are recommended for this tutorial: St Hugh s 2 nd Year: Quantum Mechanics II Reading The following sources are recommended for this tutorial: The key text (especially here in Oxford) is Molecular Quantum Mechanics, P. W. Atkins and R. S.

More information

Lecture 10. Central potential

Lecture 10. Central potential Lecture 10 Central potential 89 90 LECTURE 10. CENTRAL POTENTIAL 10.1 Introduction We are now ready to study a generic class of three-dimensional physical systems. They are the systems that have a central

More information

Structure of diatomic molecules

Structure of diatomic molecules Structure of diatomic molecules January 8, 00 1 Nature of molecules; energies of molecular motions Molecules are of course atoms that are held together by shared valence electrons. That is, most of each

More information

3 Symmetry Protected Topological Phase

3 Symmetry Protected Topological Phase Physics 3b Lecture 16 Caltech, 05/30/18 3 Symmetry Protected Topological Phase 3.1 Breakdown of noninteracting SPT phases with interaction Building on our previous discussion of the Majorana chain and

More information

Problem 1: Spin 1 2. particles (10 points)

Problem 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 information

11.D.2. Collision Operators

11.D.2. Collision Operators 11.D.. Collision Operators (11.94) can be written as + p t m h+ r +p h+ p = C + h + (11.96) where C + is the Boltzmann collision operator defined by [see (11.86a) for convention of notations] C + g(p)

More information

4 Matrix product states

4 Matrix product states Physics 3b Lecture 5 Caltech, 05//7 4 Matrix product states Matrix product state (MPS) is a highly useful tool in the study of interacting quantum systems in one dimension, both analytically and numerically.

More information

Quantum Mechanics for Mathematicians: The Heisenberg group and the Schrödinger Representation

Quantum Mechanics for Mathematicians: The Heisenberg group and the Schrödinger Representation Quantum Mechanics for Mathematicians: The Heisenberg group and the Schrödinger Representation Peter Woit Department of Mathematics, Columbia University woit@math.columbia.edu November 30, 2012 In our discussion

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

ΑΜ Α and ΒΜ Β angular momentum basis states to form coupled ΑΒCΜ C basis states RWF Lecture #4. The Wigner-Eckart Theorem

ΑΜ Α and ΒΜ Β angular momentum basis states to form coupled ΑΒCΜ C basis states RWF Lecture #4. The Wigner-Eckart Theorem MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 5.74 RWF Lecture #4 4 The Wigner-Ecart Theorem It is always possible to evaluate the angular

More information

( ) dσ 1 dσ 2 + α * 2

( ) dσ 1 dσ 2 + α * 2 Chemistry 36 Dr. Jean M. Standard Problem Set Solutions. The spin up and spin down eigenfunctions for each electron in a many-electron system are normalized and orthogonal as given by the relations, α

More information

Quantum Physics II (8.05) Fall 2002 Outline

Quantum Physics II (8.05) Fall 2002 Outline Quantum Physics II (8.05) Fall 2002 Outline 1. General structure of quantum mechanics. 8.04 was based primarily on wave mechanics. We review that foundation with the intent to build a more formal basis

More information

Lecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics

Lecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of

More information