PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10: Solutions

Size: px
Start display at page:

Download "PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10: Solutions"

Transcription

1 PHYS851 Quantum Mechanics I, Fall 009 HOMEWORK ASSIGNMENT 10: Solutions Topics Covered: Tensor product spaces, change of coordinate system, general theory of angular momentum Some Key Concepts: Angular momentum: commutation relations, raising and lowering operators, eigenstates and eigenvalues. 1. [10 pts] Consider the position eigenstate r. In spherical coordinates, this state is written as rθφ, where R rθφ = r e r (θ,φ) rθφ. In cartesian coordinates, the same state is written xyz, where R xyz = (x e x + y e y + z e z ) xyz. Evaluate the following: Most of these can be evaluated in many different ways, I am just giving one possibility for each: (a) rθφ r θ φ = 1 r sinθ δ(r r )δ(θ θ )δ(φ φ ) (b) rθφ xyz = δ(r sin θ cos φ x)δ(r sin θ sin φ y)δ(r cos θ z) (c) rθφ p x p y p z = [π ] 3/ exp [ i (p xr sin θ cos φ + p y r sin θ sinφ + p z r cos θ) ] (d) rθφ R r θ φ = 1 r sin θ e r(θ,φ)δ(r r )δ(θ θ )δ(φ φ ) (e) rθφ Z r θ φ = cot θ r e r(θ,φ)δ(r r )δ(θ θ )δ(φ φ ) (f) rθφ P z r θ φ = i z rθφ r θ φ now z = r z r + θ z θ + φ z φ = secθ r 1 r sin θ θ so that rθφ P z r θ φ = i sec θ cscθ r 3 δ(r r )δ(θ θ )δ(φ φ sec θ csc θ ) i r δ (r r )δ(θ θ )δ(φ φ ) i cos θ csc3 θ r 3 δ(r r )δ(θ θ )δ(φ φ ) + i csc θ r 3 δ(r r )δ (θ θ )δ(φ φ ) 1

2 . [10 pts] Consider a system consisting of two spin-less particles with masses m 1 and m, and charges q 1 and q. (a) Write the quantum mechanical Hamiltonian that describes this system. H = P 1 + P q 1 q + m 1 m 4πǫ 0 R 1 R (b) Define a suitable tensor-product-state basis to describe the system. Basis: { r 1, r (1) } where r 1, r (1) = r 1 (1) r) () (c) Evaluate the expression b H ψ, where ψ is an arbitrary state of the system, and b should be replaced by one of your basis states. r 1, r H ψ = [ m 1 1 m + (d) Do the same for N identical particles of mass m and charge q. Basis: { r 1,..., r N }, where r 1,..., r N = r 1 (1)... r N (N) r 1,..., r N H ψ = N j + m j j=1 N k=j+1 ] q 1 q ψ( r 1, r ) 4πǫ 0 r 1 r q j q k ψ( r 1,..., r N ) 4πǫ 0 r j r k

3 3. [0 pts] Let L = R P, where L, R, and P are the three-dimensional vector operators for angular momentum, position, and linear momentum, respectively. For µ, ν {x, y, z}, evaluate the following expressions: (a) [R µ,r ν ]= 0 (b) [P µ,p ν ]= 0 (c) [R µ,p ν ]= i δ µ,ν Use these results to prove explicitly that [L x,l y ] = i L z, then use a symmetry argument to obtain similar expressions for the commutators [L y,l z ] and [L x,l z ]. We start from L x = Y P z ZP y, L y = ZP x XP z, and L z = XP y Y P x, so that [L x,l y ] = [Y P z ZP y,zp x XP z ] = [Y P z,zp x ] [Y P z,xp z ] [ZP y,zp x ] + [ZP y,xp z ] = Y P x [P z,z] P y X[Z,P z ] = i Y P x + i XP y = i L z For any right-handed triplet of unit-vectors, the choice which to call e x, is arbitrary, but once the choice is made, the labels of the other two are fixed due to handedness. Thus cyclic permutations of x, y, z are all equivalent to different labeling choices. This means that with different choices, our result would become [L y,l z ] = i L x or [L z,l x ] = i L y. 3

4 4. [10 pts] Show explicitly that J = J x + J y + J z commutes with J z, then use a symmetry argument to show that J must also commute with J x and J y. Then, answer the following (be sure to explain your reasoning): [Jx,J z] = Jx J z J z Jx = Jx J z J x J z J x + J x J z J x J z Jx = J x [J x,j z ] + [J x,j z ]J x = i J x J y i J y J x [Jy,J z ] = JyJ z J z Jy = JyJ z J y J z J y + J y J z J y J z Jy = J y [J y,j z ] + [J y,j z ]J y = i J y J x i J x J y [Jz,J z] = 0 Adding these results together gives [J,J z ] = 0 (a) Do simultaneous eigenstates of J x and J z exist? No, because J x and J z do not commute. (b) Do simultaneous eigenstates of J and J z exist? Yes, because J and J z commute. (c) Do simultaneous eigenstates of J and J y exist? Yes, because if J commutes with J z, then by symmetry, it must commute with J y. (d) Do simultaneous eigenstates of J, J z, and J y exist? No, because J z and J y do not commute. (e) Do simultaneous eigenstates of J and J x exist? Yes, simultaneous eigenstates of J and J x exist because J and J x commute. Clearly an eigenstate of J x is also an eigenstate of J x. (f) Do simultaneous eigenstates of J z and J x + J y exist? Yes, because J x + J y = J J z, and J z commutes with J and J z. 4

5 5. [0 pts] With J ± = J x ± ij y, express J + J and J J + in terms of the operators J and J z, then compute the the following commutators: J + J = (J x + ij y )(J x ij y ) = J x + J y + i[j y,j x ] = J J z + J z likewise J J + = J J z J z (a) [J +,J ] [J +,J ] = J + J J J + = J z (b) [J ±,J ] [J ±,J ] = [J x,j ] ± i[j y,j ] = 0 (c) [J ±,J z ] [J ±,J z ] = [J x,j z ] ± i[j y,j z ] = i J y ± ( J x ) = J ± (d) [J ±,J x ] [J ±,J x ] = ±i[j y,j x ] = ± J z (e) [J ±,J y ] [J ±,J y ] = [J x,j y ] = i J z 5

6 6. [0 pts] Let j,m be the standard simultaneous eigenstate of J and J z. (a) What are J j,m and J z j,m in terms of j and m? (b) What are the allowed values of j? (c) For a given j-value, what are the allowed values of m? First, re-write your answers to (a), (b), and (c) ten times, then compute the following matrix elements: (a) J j,m = j(j + 1) j,m and J z j,m = m j,m (b) j {0, 1,1, 3,, 5,...} (c) m { j, j+1,...,j} Optional: J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J j, m = j(j + 1) j, m J z j, m = m j, m J z j, m = m j, m J z j, m = m j, m J z j, m = m j, m J z j, m = m j, m J z j, m = m j, m J z j, m = m j, m J z j, m = m j, m J z j, m = m j, m J z j, m = m j, m j {0, 1, 1, 3,, 5,...} j {0, 1, 1, 3,, 5,...} j {0, 1, 1, 3,, 5,...} j {0, 1, 1, 3,, 5,...} j {0, 1,1, 3,, 5,...} j {0, 1,1, 3,, 5,...} j {0, 1, 1, 3,, 5,...} j {0, 1,1, 3,, 5,...} j {0, 1, 1, 3,, 5,...} j {0, 1,1, 3,, 5,...} m { j, j+1,..., j} m { j, j+1,..., j} m { j, j+1,..., j} m { j, j+1,..., j} m { j, j+1,..., j} m { j, j+1,..., j} m { j, j+1,..., j} m { j, j+1,..., j} m { j, j+1,..., j} m { j, j+1,..., j} (d) j,m J j,m = j(j + 1)δ j,j δ m,m (e) j,m J z j,m = m δ j,j δ m,m (f) j,m J ± j m = j (j +1) m (m ±1)δ j,j δ m,m ±1 (g) j,m J x j m = 1 j,m (J + + J ) j m = δ j,j ( j (j +1) m (m +1)δ m,m +1 + j (j +1) m (m 1)δ m,m 1 ) (h) j,m J y j m = 1 i j,m (J + J ) j m = i δ j,j ( j (j +1) m (m +1)δ m,m +1 j (j +1) m (m 1)δ m,m 1 ) 6

7 7. [10 pts] Consider a system described by the Hamiltonian H = J z + U(J x + J y) (1) where J x, J y, and J z are the three components of a generalized angular momentum operator, and and U are constants. What are the energy levels of this system? With J x + J y = J J z, the Hamiltonian becomes H = U(J J z ) + J z The eigenstates of this are the standard j,m states, with eigenvalues given by H j,m = ( U(j + j m ) + m ) j,m 7

Angular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z.

Angular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z. Angular momentum is an important concept in quantum theory, necessary for analyzing motion in 3D as well as intrinsic properties such as spin Classically the orbital angular momentum with respect to a

More information

2. Griffith s 4.19 a. The commutator of the z-component of the angular momentum with the coordinates are: [L z. + xyp x. x xxp y.

2. Griffith s 4.19 a. The commutator of the z-component of the angular momentum with the coordinates are: [L z. + xyp x. x xxp y. Physics Homework #8 Spring 6 Due Friday 5/7/6. Download the Mathematica notebook from the notes page of the website (http://minerva.union.edu/labrakes/physics_notes_s5.htm) or the homework page of the

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

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

Quantization of the Spins

Quantization of the Spins Chapter 5 Quantization of the Spins As pointed out already in chapter 3, the external degrees of freedom, position and momentum, of an ensemble of identical atoms is described by the Scödinger field operator.

More information

Physics 215 Quantum Mechanics I Assignment 8

Physics 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 information

Computational Spectroscopy III. Spectroscopic Hamiltonians

Computational Spectroscopy III. Spectroscopic Hamiltonians Computational Spectroscopy III. Spectroscopic Hamiltonians (e) Elementary operators for the harmonic oscillator (f) Elementary operators for the asymmetric rotor (g) Implementation of complex Hamiltonians

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 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

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

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

For example, in one dimension if we had two particles in a one-dimensional infinite potential well described by the following two wave functions.

For example, in one dimension if we had two particles in a one-dimensional infinite potential well described by the following two wave functions. Identical particles In classical physics one can label particles in such a way as to leave the dynamics unaltered or follow the trajectory of the particles say by making a movie with a fast camera. Thus

More information

Physics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics

Physics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics Physics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics We now consider the spatial degrees of freedom of a particle moving in 3-dimensional space, which of course is an important

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

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

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

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

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

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

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

Tight-Binding Model of Electronic Structures

Tight-Binding Model of Electronic Structures Tight-Binding Model of Electronic Structures Consider a collection of N atoms. The electronic structure of this system refers to its electronic wave function and the description of how it is related to

More information

Summary: angular momentum derivation

Summary: angular momentum derivation Summary: angular momentum derivation L = r p L x = yp z zp y, etc. [x, p y ] = 0, etc. (-) (-) (-3) Angular momentum commutation relations [L x, L y ] = i hl z (-4) [L i, L j ] = i hɛ ijk L k (-5) Levi-Civita

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

G : Quantum Mechanics II

G : Quantum Mechanics II G5.666: Quantum Mechanics II Notes for Lecture 7 I. A SIMPLE EXAMPLE OF ANGULAR MOMENTUM ADDITION Given two spin-/ angular momenta, S and S, we define S S S The problem is to find the eigenstates of the

More information

Appendix to Lecture 2

Appendix to Lecture 2 PHYS 652: Astrophysics 1 Appendix to Lecture 2 An Alternative Lagrangian In class we used an alternative Lagrangian L = g γδ ẋ γ ẋ δ, instead of the traditional L = g γδ ẋ γ ẋ δ. Here is the justification

More information

Appendix: SU(2) spin angular momentum and single spin dynamics

Appendix: SU(2) spin angular momentum and single spin dynamics Phys 7 Topics in Particles & Fields Spring 03 Lecture v0 Appendix: SU spin angular momentum and single spin dynamics Jeffrey Yepez Department of Physics and Astronomy University of Hawai i at Manoa Watanabe

More information

Angular Momentum set II

Angular Momentum set II Angular Momentum set II PH - QM II Sem, 7-8 Problem : Using the commutation relations for the angular momentum operators, prove the Jacobi identity Problem : [ˆL x, [ˆL y, ˆL z ]] + [ˆL y, [ˆL z, ˆL x

More information

Introduction to Modern Quantum Field Theory

Introduction to Modern Quantum Field Theory Department of Mathematics University of Texas at Arlington Arlington, TX USA Febuary, 2016 Recall Einstein s famous equation, E 2 = (Mc 2 ) 2 + (c p) 2, where c is the speed of light, M is the classical

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

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

Nuclear models: Collective Nuclear Models (part 2)

Nuclear models: Collective Nuclear Models (part 2) Lecture 4 Nuclear models: Collective Nuclear Models (part 2) WS2012/13: Introduction to Nuclear and Particle Physics,, Part I 1 Reminder : cf. Lecture 3 Collective excitations of nuclei The single-particle

More information

16.1. PROBLEM SET I 197

16.1. 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 information

20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R

20 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 information

Angular Momentum - set 1

Angular Momentum - set 1 Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x = 0,

More information

Homework assignment 3: due Thursday, 10/26/2017

Homework assignment 3: due Thursday, 10/26/2017 Homework assignment 3: due Thursday, 10/6/017 Physics 6315: Quantum Mechanics 1, Fall 017 Problem 1 (0 points The spin Hilbert space is defined by three non-commuting observables, S x, S y, S z. These

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

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

The first change comes in how we associate operators with classical observables. In one dimension, we had. p p ˆ

The first change comes in how we associate operators with classical observables. In one dimension, we had. p p ˆ VI. Angular momentum Up to this point, we have been dealing primaril with one dimensional sstems. In practice, of course, most of the sstems we deal with live in three dimensions and 1D quantum mechanics

More information

Graduate Quantum Mechanics I: Prelims and Solutions (Fall 2015)

Graduate 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 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

NIU PHYS 500, Fall 2006 Classical Mechanics Solutions for HW6. Solutions

NIU PHYS 500, Fall 2006 Classical Mechanics Solutions for HW6. Solutions NIU PHYS 500, Fall 006 Classical Mechanics Solutions for HW6 Assignment: HW6 [40 points] Assigned: 006/11/10 Due: 006/11/17 Solutions P6.1 [4 + 3 + 3 = 10 points] Consider a particle of mass m moving in

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

Theory of Angular Momentum

Theory of Angular Momentum Chapter 3 Theory of Angular Momentum HW#3: 3.1, 3., 3.5, 3.9, 3.10, 3.17, 3.0, 3.1, 3.4, 3.6, 3.7 3.1 Transformations of vectors Clearly the choice of the reference frame or coordinates system i.e. origin,

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

Notes on Spin Operators and the Heisenberg Model. Physics : Winter, David G. Stroud

Notes on Spin Operators and the Heisenberg Model. Physics : Winter, David G. Stroud Notes on Spin Operators and the Heisenberg Model Physics 880.06: Winter, 003-4 David G. Stroud In these notes I give a brief discussion of spin-1/ operators and their use in the Heisenberg model. 1. Spin

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

1. Rotations in 3D, so(3), and su(2). * version 2.0 *

1. Rotations in 3D, so(3), and su(2). * version 2.0 * 1. Rotations in 3D, so(3, and su(2. * version 2.0 * Matthew Foster September 5, 2016 Contents 1.1 Rotation groups in 3D 1 1.1.1 SO(2 U(1........................................................ 1 1.1.2

More information

Lesson 33 - Trigonometric Identities. Pre-Calculus

Lesson 33 - Trigonometric Identities. Pre-Calculus Lesson 33 - Trigonometric Identities Pre-Calculus 1 (A) Review of Equations An equation is an algebraic statement that is true for only several values of the variable The linear equation 5 = 2x 3 is only

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

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

CHM 671. Homework set # 6. 2) Do problems 3.4, 3.7, 3.10, 3.14, 3.15 and 3.16 in the book.

CHM 671. Homework set # 6. 2) Do problems 3.4, 3.7, 3.10, 3.14, 3.15 and 3.16 in the book. CHM 67 Homework set # 6 Due: Thursday, October 9 th ) Read Chapter 3 in the 4 th edition Atkins & Friedman's Molecular Quantum Mechanics book. 2) Do problems 3.4, 3.7, 3., 3.4, 3.5 and 3.6 in the book.

More information

Hyperfine interaction

Hyperfine interaction Hyperfine interaction The notion hyperfine interaction (hfi) comes from atomic physics, where it is used for the interaction of the electronic magnetic moment with the nuclear magnetic moment. In magnetic

More information

QUALIFYING EXAMINATION, Part 2. Solutions. Problem 1: Quantum Mechanics I

QUALIFYING EXAMINATION, Part 2. Solutions. Problem 1: Quantum Mechanics I QUALIFYING EXAMINATION, Part Solutions Problem 1: Quantum Mechanics I (a) We may decompose the Hamiltonian into two parts: H = H 1 + H, ( ) where H j = 1 m p j + 1 mω x j = ω a j a j + 1/ with eigenenergies

More information

Angular Momentum - set 1

Angular Momentum - set 1 Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x ] =

More information

Quantum Field Theory II

Quantum Field Theory II Quantum Field Theory II T. Nguyen PHY 391 Independent Study Term Paper Prof. S.G. Rajeev University of Rochester April 2, 218 1 Introduction The purpose of this independent study is to familiarize ourselves

More information

Solution Set of Homework # 6 Monday, December 12, Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second Volume

Solution Set of Homework # 6 Monday, December 12, Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second Volume Department of Physics Quantum II, 570 Temple University Instructor: Z.-E. Meziani Solution Set of Homework # 6 Monday, December, 06 Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second

More information

CHEM 301: Homework assignment #5

CHEM 301: Homework assignment #5 CHEM 30: Homework assignment #5 Solutions. A point mass rotates in a circle with l =. Calculate the magnitude of its angular momentum and all possible projections of the angular momentum on the z-axis.

More information

QMI PRELIM Problem 1. All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work.

QMI PRELIM Problem 1. All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work. QMI PRELIM 013 All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work. Problem 1 L = r p, p = i h ( ) (a) Show that L z = i h y x ; (cyclic

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

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

Systems of Identical Particles

Systems of Identical Particles qmc161.tex Systems of Identical Particles Robert B. Griffiths Version of 21 March 2011 Contents 1 States 1 1.1 Introduction.............................................. 1 1.2 Orbitals................................................

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

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

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

Rotations and Angular Momentum

Rotations and Angular Momentum qmd1 Rotations and Angular Momentum Robert B. Griffiths Version of 7 September 01 Contents 1 Unitary Transformations 1 Rotations in Space.1 Two dimensions............................................. Three

More information

2m r2 (~r )+V (~r ) (~r )=E (~r )

2m r2 (~r )+V (~r ) (~r )=E (~r ) Review of the Hydrogen Atom The Schrodinger equation (for 1D, 2D, or 3D) can be expressed as: ~ 2 2m r2 (~r, t )+V (~r ) (~r, t )=i~ @ @t The Laplacian is the divergence of the gradient: r 2 =r r The time-independent

More information

Lecture 5: Orbital angular momentum, spin and rotation

Lecture 5: Orbital angular momentum, spin and rotation Lecture 5: Orbital angular momentum, spin and rotation 1 Orbital angular momentum operator According to the classic expression of orbital angular momentum L = r p, we define the quantum operator L x =

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

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 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom

Physics 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 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

A Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor

A Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule

More information

Plane wave solutions of the Dirac equation

Plane wave solutions of the Dirac equation Lecture #3 Spherical spinors Hydrogen-like systems again (Relativistic version) irac energy levels Chapter, pages 48-53, Lectures on Atomic Physics Chapter 5, pages 696-76, Bransden & Joachain,, Quantum

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

Path Integral for Spin

Path Integral for Spin Path Integral for Spin Altland-Simons have a good discussion in 3.3 Applications of the Feynman Path Integral to the quantization of spin, which is defined by the commutation relations [Ŝj, Ŝk = iɛ jk

More information

G : Quantum Mechanics II

G : Quantum Mechanics II G5.666: Quantum Mechanics II Notes for Lecture 5 I. REPRESENTING STATES IN THE FULL HILBERT SPACE Given a representation of the states that span the spin Hilbert space, we now need to consider the problem

More information

Lesson 22 - Trigonometric Identities

Lesson 22 - Trigonometric Identities POP QUIZ Lesson - Trigonometric Identities IB Math HL () Solve 5 = x 3 () Solve 0 = x x 6 (3) Solve = /x (4) Solve 4 = x (5) Solve sin(θ) = (6) Solve x x x x (6) Solve x + = (x + ) (7) Solve 4(x ) = (x

More information

5.5. Representations. Phys520.nb Definition N is called the dimensions of the representations The trivial presentation

5.5. Representations. Phys520.nb Definition N is called the dimensions of the representations The trivial presentation Phys50.nb 37 The rhombohedral and hexagonal lattice systems are not fully compatible with point group symmetries. Knowing the point group doesn t uniquely determine the lattice systems. Sometimes we can

More information

Crystal field effect on atomic states

Crystal field effect on atomic states Crystal field effect on atomic states Mehdi Amara, Université Joseph-Fourier et Institut Néel, C.N.R.S. BP 66X, F-3842 Grenoble, France References : Articles - H. Bethe, Annalen der Physik, 929, 3, p.

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

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

Fun With Carbon Monoxide. p. 1/2

Fun With Carbon Monoxide. p. 1/2 Fun With Carbon Monoxide p. 1/2 p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results C V (J/K-mole) 35 30 25

More information

3.3 Lagrangian and symmetries for a spin- 1 2 field

3.3 Lagrangian and symmetries for a spin- 1 2 field 3.3 Lagrangian and symmetries for a spin- 1 2 field The Lagrangian for the free spin- 1 2 field is The corresponding Hamiltonian density is L = ψ(i/ µ m)ψ. (3.31) H = ψ( γ p + m)ψ. (3.32) The Lagrangian

More information

Spin Dynamics Basic Theory Operators. Richard Green SBD Research Group Department of Chemistry

Spin Dynamics Basic Theory Operators. Richard Green SBD Research Group Department of Chemistry Spin Dynamics Basic Theory Operators Richard Green SBD Research Group Department of Chemistry Objective of this session Introduce you to operators used in quantum mechanics Achieve this by looking at:

More information

Math review. Math review

Math review. Math review Math review 1 Math review 3 1 series approximations 3 Taylor s Theorem 3 Binomial approximation 3 sin(x), for x in radians and x close to zero 4 cos(x), for x in radians and x close to zero 5 2 some geometry

More information

REFRESHER. William Stallings

REFRESHER. William Stallings BASIC MATH REFRESHER William Stallings Trigonometric Identities...2 Logarithms and Exponentials...4 Log Scales...5 Vectors, Matrices, and Determinants...7 Arithmetic...7 Determinants...8 Inverse of a Matrix...9

More information

Physics 221A Fall 2017 Notes 15 Orbital Angular Momentum and Spherical Harmonics

Physics 221A Fall 2017 Notes 15 Orbital Angular Momentum and Spherical Harmonics Copyright c 2018 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 15 Orbital Angular Momentum and Spherical Harmonics 1. Introduction In Notes 13, we worked out the general theory of the representations

More information

.1 "patedl-righl" timti tame.nto our oai.c iii C. W.Fiak&Co. She ftowtt outnal,

.1 patedl-righl timti tame.nto our oai.c iii C. W.Fiak&Co. She ftowtt outnal, J 2 X Y J Y 3 : > Y 6? ) Q Y x J Y Y // 6 : : \ x J 2 J Q J Z 3 Y 7 2 > 3 [6 2 : x z (7 :J 7 > J : 7 (J 2 J < ( q / 3 6 q J $3 2 6:J : 3 q 2 6 3 2 2 J > 2 :2 : J J 2 2 J 7 J 7 J \ : q 2 J J Y q x ( ) 3:

More information

Basic Physical Chemistry Lecture 2. Keisuke Goda Summer Semester 2015

Basic Physical Chemistry Lecture 2. Keisuke Goda Summer Semester 2015 Basic Physical Chemistry Lecture 2 Keisuke Goda Summer Semester 2015 Lecture schedule Since we only have three lectures, let s focus on a few important topics of quantum chemistry and structural chemistry

More information

1. We want to solve the time independent Schrödinger Equation for the hydrogen atom.

1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 16 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom.. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian

More information

Rotation Matrices. Chapter 21

Rotation Matrices. Chapter 21 Chapter 1 Rotation Matrices We have talked at great length last semester and this about unitary transformations. The putting together of successive unitary transformations is of fundamental importance.

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

( a ', jm lj - V l a, jm ) jm 'l ] Ijm ), ( ) Ii 2j ( j + 1 ) We will give applications of the theorem in subsequent sections.

( a ', jm lj - V l a, jm ) jm 'l ] Ijm ), ( ) Ii 2j ( j + 1 ) We will give applications of the theorem in subsequent sections. Theory of Angular Momentum 4 But the righthand side of 3144) is the same as a', jm lj V la, jm ) / a, jm lj 1 a, jm ) by 314) and 3143) Moreover, the lefthand side of 3143) is just j j 1 ) 1i So a ', jm

More information

arxiv: v4 [quant-ph] 9 Jun 2016

arxiv: v4 [quant-ph] 9 Jun 2016 Applying Classical Geometry Intuition to Quantum arxiv:101.030v4 [quant-ph] 9 Jun 016 Spin Dallin S. Durfee and James L. Archibald Department of Physics and Astronomy, Brigham Young University, Provo,

More information

One-electron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation:

One-electron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation: One-electron Atom The atomic orbitals of hydrogen-like atoms are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's

More information

4. Two-level systems. 4.1 Generalities

4. Two-level systems. 4.1 Generalities 4. Two-level systems 4.1 Generalities 4. Rotations and angular momentum 4..1 Classical rotations 4.. QM angular momentum as generator of rotations 4..3 Example of Two-Level System: Neutron Interferometry

More information

Angular Momentum Properties

Angular Momentum Properties Cheistry 460 Fall 017 Dr. Jean M. Standard October 30, 017 Angular Moentu Properties Classical Definition of Angular Moentu In classical echanics, the angular oentu vector L is defined as L = r p, (1)

More information

The one and three-dimensional particle in a box are prototypes of bound systems. As we

The one and three-dimensional particle in a box are prototypes of bound systems. As we 6 Lecture 10 The one and three-dimensional particle in a box are prototypes of bound systems. As we move on in our study of quantum chemistry, we'll be considering bound systems that are more and more

More information

Trigonometry Final Exam Review

Trigonometry Final Exam Review Name Period Trigonometry Final Exam Review 2014-2015 CHAPTER 2 RIGHT TRIANGLES 8 1. Given sin θ = and θ terminates in quadrant III, find the following: 17 a) cos θ b) tan θ c) sec θ d) csc θ 2. Use a calculator

More information

Some elements of vector and tensor analysis and the Dirac δ-function

Some elements of vector and tensor analysis and the Dirac δ-function Chapter 1 Some elements of vector and tensor analysis and the Dirac δ-function The vector analysis is useful in physics formulate the laws of physics independently of any preferred direction in space experimentally

More information