PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10: Solutions
|
|
- Sydney Stephens
- 6 years ago
- Views:
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 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 information2. 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 informationAngular 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 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 informationQuantization 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 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 informationComputational 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 informationSt 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 informationLecture 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 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 information9 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 informationFor 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 informationPhysics 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 informationRotations 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 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 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 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 informationPHYS852 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 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 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 informationTight-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 informationSummary: 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 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 informationG : 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 informationAppendix 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 informationAppendix: 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 informationAngular 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 informationIntroduction 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 informationIn 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 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 informationNuclear 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 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 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 informationAngular 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 informationHomework 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 informationPage 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 informationPhysics 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 informationThe 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 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 information26 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 informationNIU 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 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 informationTheory 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 information1. 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 informationNotes 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 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 information1. 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 informationLesson 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 informationPH 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 informationPhysics 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 informationCHM 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 informationHyperfine 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 informationQUALIFYING 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 informationAngular 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 informationQuantum 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 informationSolution 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 informationCHEM 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 informationQMI 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 informationSecond 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 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 informationSystems 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 informationPhysics 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 informationAre 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 informationAngular 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 informationRotations 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 information2m 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 informationLecture 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 informationAngular 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 informationd 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 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 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 informationA 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 informationPlane 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 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 informationPath 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 informationG : 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 informationLesson 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 information5.5. 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 informationCrystal 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 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 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 informationFun 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 information3.3 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 informationSpin 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 informationMath 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 informationREFRESHER. 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 informationPhysics 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,
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 informationBasic 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 information1. 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 informationRotation 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 informationGenerators 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.
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 informationarxiv: 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 informationOne-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 information4. 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 informationAngular 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 informationThe 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 informationTrigonometry 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 informationSome 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