Two and Three-Dimensional Systems
|
|
- Gwen Cynthia Long
- 6 years ago
- Views:
Transcription
1 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 harmonic oscillator: ( p 2 H = 2m + ) ( p 2 mω2 q m + ) 2 mω2 2 q2 2 ; ω ω 2. a) Exploit the fact that the Schrödinger eigenvalue equation can be solved by separating the variables and find a complete set of eigenfunctions of H and the corresponding eigenvalues. b) Assume that ω /ω 2 =3/4. Find the first two degenerate energy levels. What can one say about the degeneracy of energy levels when the ratio between ω and ω 2 is not a rational number? c) Write the eigenfunctions of the Hamiltonian in the case ω 2 =0. Consider now a particle of mass m in two dimensions subject to the potential: V (q,q 2 )=mω 2( q 2 q q 2 + q 2 2). d) Say whether the problem of finding the eigenvalues of the Hamiltonian H =(p 2 + p 2 2)/2m + V (q,q 2 )canbesolvedbythemethodofseparation of variables. 0.2 A particle of mass m in two dimensions is constrained inside a square whose edge is 2a: x a, y a. a) Write the Schrödinger equation, separate the variables and find a complete set of eigenfunctions of the Hamiltonian. b) Find the energy levels of the system and say whether there is degeneracy. c) Say whether there exist operators (i.e. transformations) that commute with the Hamiltonian but do not commute among themselves. In the affirmative case, give one or more examples. E. d Emilio and L.E. Picasso, Problems in Quantum Mechanics: with solutions, UNITEXT, DOI 0.007/ _0, Springer-Verlag Italia 20 99
2 200 0 Two and Three-Dimensional Systems Assume now that within the square the potential: V a (x, y) =V 0a cos(πx/2a) cos(πy/2a) is present. d) Is it still possible to separate the variables in the Schrödinger equation? Do degenerate energy levels exist? e) Say whether it is possible to guarantee the existence of degenerate energy levels if, instead, the potential is V b (x, y) =V 0b sin(πx/a) sin(πy/a). Is any relationship between the eigenfunctions of the Hamiltonian ψ E (x, y) and ψ E (y, x) expected?(namely,aretheyequal?aretheydifferent?...) 0.3 A particle of mass m in two dimensions is constrained inside the triangle whose vertices have the coordinates (x = 0, y =0); (x = a, y =0); (x = a, y = a) (a half of the square with edge a ). a) Find eigenvectors and eigenvalues of the energy. b) For the same system and exploiting the results of the previous question, find a complete set of eigenvectors of the operator that implements the reflection through the straight line x + y = a (the dotted line in the figure). 0.4 A particle of mass m in three dimensions is confined within an infinite rectilinear guide with a cross section that is a square of edge a. y a) Find eigenfunctions and eigenvalues of the Hamiltonian. What is the minimum energy (threshold energy) the particle must have in order to propagate along the guide? Consider the wavefunctions: ψ (x, y, z) =A sin(2πx/a)sin(πy/a)e i k z ψ 2 (x, y, z) =B sin(πx/a)sin(πy/a)e i k2 z. b) Determine the normalization coefficients A and B in such a way that the integral of the densities ρ,2 over a slice of the guide of unit volume equals (normalization oneparticleperunitvolume ). c) Calculate the probability current densities: ȷ,2 (x, y, z) = h m Im ( ψ,2(x, y, z) ψ,2 (x, y, z) ) x
3 Problems 20 for the states represented by the wavefunctions ψ and ψ 2 normalized as above, and verify that div ȷ,2 (x, y, z) =0. d) Say for which values of k 2 the probability current associated to the state represented by ψ(x, y, z, t) withψ(x, y, z, 0) = ψ (x, y, z)+ψ 2 (x, y, z), is divergenceless. 0.5 Aparticleissubjecttothepotential V = V (q 2 + q 2 2,q 3 ). a) Show that the Hamiltonian H 0 = p 2 /2m + V commutes with the angular momentum operator L z = q p 2 q 2 p. b) Use the degeneracy theorem to show that there exist degenerate energy levels. c) Say whether and how the degeneracy is removed if the system is on a platform rotating around the z axis with constant angular velocity ω. 0.6 The Hamiltonian of a two-dimensional isotropic harmonic oscillator of mass m and angular frequency ω is H = ( p 2 2m + p 2 2) + 2 mω2( q 2 + q2 2 ) = H (q,p )+H 2 (q 2,p 2 ). a) Exploit the separation of variables ( H = H +H 2 )andfindtheeigenvalues of H and their degeneracies. b) Write the eigenfunctions of the Hamiltonian in the Schrödinger representation, in the basis in which both H and H 2 are diagonal. c) Is the degeneracy found in a) in agreement with the result established in Problem 0.5? Find the maximum and the minimum of the eigenvalues m of L 3 = q p 2 q 2 p within each energy level. Do all its possible values ranging between m max and m min occur? d) For each of the first three energy levels, say which eigenvalues of L 3 do occur and explicitly write the wavefunctions relative to the states E,m (simultaneous eigenstates of H and L 3 ). 0.7 This problem is devoted to establish a priori the degeneracies of the two-dimensional isotropic harmonic oscillator found in Problem 0.6. Set η a = (p a i mωq a ), a =, 2. 2mω h a) Write the Hamiltonian H of the two-dimensional oscillator in terms of the operators η a and η a and the commutation rules [η a,η b ], a, b =, 2. b) Show that the four operators η a η b commute with the Hamiltonian H. Consider the operators: j = 2 (η η 2 + η 2 η ), j 2 = 2i (η η 2 η 2 η ), j 3 = 2 (η η η 2 η 2 ).
4 202 0 Two and Three-Dimensional Systems c) Show that the operators j a have the same commutation rules as the angular momentum (divided by h). Write j 2 and j 3 in terms of the q s and p s and show that the angular momentum operators j a have both integer and half-integer eigenvalues. Setting h 0 = H/ hω, the identity j 2 j 2 + j2 2 + j3 2 = h ( 0 2 h0 ) 2 + holds (it may be verified using the commutation rules). d) Exploit the theory of angular momentum (all the properties of the angular momentum follow uniquely from the commutation relations) and the above identity to find the eigenvalues of H and the relative degeneracies. Say which eigenvalues of L 3 do occur in each energy level. 0.8 In Problem 0.7 the energy levels of a two-dimensional isotropic harmonic oscillator and their degeneracies have been found starting from the commutation rules of the three constants of motion j,j 2,j 3 given from the outside. We now want to establish both the existence and the form of such constants of motion starting from the invariance group of the Hamiltonian. Adopting the notation of Problem 0.7 one has: ( 2 ) H = hω η a η a + ; [η a,η b ]=0, [η a,η b ]=δ ab; a, b =, 2. a= Consider the linear transformation: η a = b u ab η b. () a) Show that () is an invariance transformation both for the Hamiltonian and for the commutation rules if and only if u is a unitary 2 2matrix. We shall consider only the transformations that fulfill det u =. b) Show that all the unitary 2 2matrices,whosedeterminantis,maybe written as: ( ) z z 2 z2 z, z 2 + z 2 2 =, z,z 2 C. They, therefore, form a continuous, 3 parameter group the group SU(2). The transformation () in a neighborhood of the identity takes the form: η a = η a +iϵ b g ab η b, u l+iϵg, ϵ. (2) c) Show that the matrix g is Hermitian and traceless.
5 Problems 203 Thanks to the von Neumann theorem, for any transformation () there exists aunitaryoperatorthatimplementsit: η a = U(u) η a U (u). d) Let U(g, ϵ) =e i ϵgg be the unitary operator that implements the infinitesimal transformation (2) (the operators G g = G g are the generators of the group). Compare η a = U(g, ϵ) η a U (g, ϵ), expanded to the first order in ϵ, and(2)andshowthat [G g,η a ]= b g ab η b.findtheexpression for G g and show that [G g,h]=0. e) Show that any traceless Hermitian 2 2matrix g may be written in the form (the factor 2 is there only for the sake of convenience): g = a 2 σ + a 2 2 σ 2 + a 3 2 σ 3, a i R where σ i are the Pauli matrices ( ) ( 0 0 i σ = ; σ 0 2 = i 0 ) ; σ 3 = ( ) 0. 0 f) Write the expressions for G g in the three particular cases when only one of the a i equals and the other two are vanishing: compare the generators G,G 2,G 3 so obtained with the operators j,j 2,j 3 of Problem 0.7. Show that [G g,g g ]= ab η a [g,g ] ab η b and make use of the commutation relations of the Pauli matrices: [ 2 σ a, 2 σ b]=iϵ abc 2 σ c to find the commutation rules of the generators: [G a,g b ]=iϵ abc G c.
5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:
5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ 1 0 0 0 ξ 1 0 0 0 ξ 3, ξ
More informationProblem 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 informationIsotropic 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 informationStatistical 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 informationTime 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 informationProblems and Multiple Choice Questions
Problems and Multiple Choice Questions 1. A momentum operator in one dimension is 2. A position operator in 3 dimensions is 3. A kinetic energy operator in 1 dimension is 4. If two operator commute, a)
More informationPHY413 Quantum Mechanics B Duration: 2 hours 30 minutes
BSc/MSci Examination by Course Unit Thursday nd May 4 : - :3 PHY43 Quantum Mechanics B Duration: hours 3 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO
More informationPHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep
Problem set 1 Due Sep 15 2005 1. Let V be the set of all complex valued functions of a real variable θ, that are periodic with period 2π. That is u(θ + 2π) = u(θ), for all u V. (1) (i) Show that this V
More informationQuantum Mechanics Solutions. λ i λ j v j v j v i v i.
Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =
More 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 informationSymmetries for fun and profit
Symmetries for fun and profit Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 August 28, 2008 Sourendu Gupta (TIFR Graduate School) Symmetries for fun and profit QM I 1 / 20 Outline 1 The isotropic
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 informationGeneral Exam Part II, Fall 1998 Quantum Mechanics Solutions
General Exam Part II, Fall 1998 Quantum Mechanics Solutions Leo C. Stein Problem 1 Consider a particle of charge q and mass m confined to the x-y plane and subject to a harmonic oscillator potential V
More 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 informationQuantum 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 informationa = ( a σ )( b σ ) = a b + iσ ( a b) mω 2! x + i 1 2! x i 1 2m!ω p, a = mω 2m!ω p Physics 624, Quantum II -- Final Exam
Physics 624, Quantum II -- Final Exam Please show all your work on the separate sheets provided (and be sure to include your name). You are graded on your work on those pages, with partial credit where
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More 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 information1 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 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 informationPage 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 informationQuantum Theory and Group Representations
Quantum Theory and Group Representations Peter Woit Columbia University LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017 Peter Woit (Columbia University)
More informationQualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!
Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics
More 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 informationHarmonic 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 informationNotes on SU(3) and the Quark Model
Notes on SU() and the Quark Model Contents. SU() and the Quark Model. Raising and Lowering Operators: The Weight Diagram 4.. Triangular Weight Diagrams (I) 6.. Triangular Weight Diagrams (II) 8.. Hexagonal
More informationIntroduction 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 informationQuantum 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 informationThe Postulates of Quantum Mechanics Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case: 3-D case
The Postulates of Quantum Mechanics Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about the
More informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
More informationJoint Entrance Examination for Postgraduate Courses in Physics EUF
Joint Entrance Examination for Postgraduate Courses in Physics EUF Second Semester 013 Part 1 3 April 013 Instructions: DO NOT WRITE YOUR NAME ON THE TEST. It should be identified only by your candidate
More information3.5 Finite Rotations in 3D Euclidean Space and Angular Momentum in QM
3.5 Finite Rotations in 3D Euclidean Space and Angular Momentum in QM An active rotation in 3D position space is defined as the rotation of a vector about some point in a fixed coordinate system (a passive
More informationSymmetries, Groups, and Conservation Laws
Chapter Symmetries, Groups, and Conservation Laws The dynamical properties and interactions of a system of particles and fields are derived from the principle of least action, where the action is a 4-dimensional
More informationNANOSCALE SCIENCE & TECHNOLOGY
. NANOSCALE SCIENCE & TECHNOLOGY V Two-Level Quantum Systems (Qubits) Lecture notes 5 5. Qubit description Quantum bit (qubit) is an elementary unit of a quantum computer. Similar to classical computers,
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 informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
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 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 informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More informationLinear Algebra in Hilbert Space
Physics 342 Lecture 16 Linear Algebra in Hilbert Space Lecture 16 Physics 342 Quantum Mechanics I Monday, March 1st, 2010 We have seen the importance of the plane wave solutions to the potentialfree Schrödinger
More informationLecture 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 informationLecture 7. More dimensions
Lecture 7 More dimensions 67 68 LECTURE 7. MORE DIMENSIONS 7.1 Introduction In this lecture we generalize the concepts introduced so far to systems that evolve in more than one spatial dimension. While
More informationPhysics 137A Quantum Mechanics Fall 2012 Midterm II - Solutions
Physics 37A Quantum Mechanics Fall 0 Midterm II - Solutions These are the solutions to the exam given to Lecture Problem [5 points] Consider a particle with mass m charge q in a simple harmonic oscillator
More informationwhich 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-state problems and an application to the free particle
-state problems and an application to the free particle Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 2013 3 September, 2013 Outline 1 Outline 2 The Hilbert space 3 A free particle 4 Keywords
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 informationLecture #1. Review. Postulates of quantum mechanics (1-3) Postulate 1
L1.P1 Lecture #1 Review Postulates of quantum mechanics (1-3) Postulate 1 The state of a system at any instant of time may be represented by a wave function which is continuous and differentiable. Specifically,
More informationKet space as a vector space over the complex numbers
Ket space as a vector space over the complex numbers kets ϕ> and complex numbers α with two operations Addition of two kets ϕ 1 >+ ϕ 2 > is also a ket ϕ 3 > Multiplication with complex numbers α ϕ 1 >
More informationThe 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 informationLecture 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
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 The Eigenvector-Eigenvalue Problem of L z and L 2 Section
More information8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Friday May 24, Final Exam
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Friday May 24, 2012 Final Exam Last Name: First Name: Check Recitation Instructor Time R01 Barton Zwiebach 10:00 R02
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 informationThe Simple Harmonic Oscillator
The Simple Harmonic Oscillator Asaf Pe er 1 November 4, 215 This part of the course is based on Refs [1] [3] 1 Introduction We return now to the study of a 1-d stationary problem: that of the simple harmonic
More information+E v(t) H(t) = v(t) E where v(t) is real and where v 0 for t ±.
. Brick in a Square Well REMEMBER: THIS PROBLEM AND THOSE BELOW SHOULD NOT BE HANDED IN. THEY WILL NOT BE GRADED. THEY ARE INTENDED AS A STUDY GUIDE TO HELP YOU UNDERSTAND TIME DEPENDENT PERTURBATION THEORY
More informationPhysics 828 Problem Set 7 Due Wednesday 02/24/2010
Physics 88 Problem Set 7 Due Wednesday /4/ 7)a)Consider the proton to be a uniformly charged sphere of radius f m Determine the correction to the s ground state energy 4 points) This is a standard problem
More informationSimple one-dimensional potentials
Simple one-dimensional potentials Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 Ninth lecture Outline 1 Outline 2 Energy bands in periodic potentials 3 The harmonic oscillator 4 A charged particle
More informationDegenerate Perturbation Theory. 1 General framework and strategy
Physics G6037 Professor Christ 12/22/2015 Degenerate Perturbation Theory The treatment of degenerate perturbation theory presented in class is written out here in detail. The appendix presents the underlying
More informationPhysics 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 informationQuantum Mechanics for Scientists and Engineers
Quantum Mechanics for Scientists and Engineers Syllabus and Textbook references All the main lessons (e.g., 1.1) and units (e.g., 1.1.1) for this class are listed below. Mostly, there are three lessons
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationLecture 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 informationEigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator.
PHYS208 spring 2008 Eigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator. 07.02.2008 Adapted from the text Light - Atom Interaction PHYS261 autumn 2007 Go to list of topics
More informationSection 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 informationLecture 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 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 informationAn operator is a transformation that takes a function as an input and produces another function (usually).
Formalism of Quantum Mechanics Operators Engel 3.2 An operator is a transformation that takes a function as an input and produces another function (usually). Example: In QM, most operators are linear:
More informationSolution to Problem Set No. 6: Time Independent Perturbation Theory
Solution to Problem Set No. 6: Time Independent Perturbation Theory Simon Lin December, 17 1 The Anharmonic Oscillator 1.1 As a first step we invert the definitions of creation and annihilation operators
More informationLecture 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 informationChemistry 432 Problem Set 4 Spring 2018 Solutions
Chemistry 4 Problem Set 4 Spring 18 Solutions 1. V I II III a b c A one-dimensional particle of mass m is confined to move under the influence of the potential x a V V (x) = a < x b b x c elsewhere and
More informationQuantum Mechanics (Draft 2010 Nov.)
Quantum Mechanics (Draft 00 Nov) For a -dimensional simple harmonic quantum oscillator, V (x) = mω x, it is more convenient to describe the dynamics by dimensionless position parameter ρ = x/a (a = h )
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 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 informationQuantum Physics 2006/07
Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.
PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms
More informationPhysics 221A Fall 1996 Notes 16 Bloch s Theorem and Band Structure in One Dimension
Physics 221A Fall 1996 Notes 16 Bloch s Theorem and Band Structure in One Dimension In these notes we examine Bloch s theorem and band structure in problems with periodic potentials, as a part of our survey
More informationPhysics 215 Quantum Mechanics 1 Assignment 5
Physics 15 Quantum Mechanics 1 Assignment 5 Logan A. Morrison February 10, 016 Problem 1 A particle of mass m is confined to a one-dimensional region 0 x a. At t 0 its normalized wave function is 8 πx
More information04. Five Principles of Quantum Mechanics
04. Five Principles of Quantum Mechanics () States are represented by vectors of length. A physical system is represented by a linear vector space (the space of all its possible states). () Properties
More informationLecture 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 information11.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 informationThe Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case:
The Schrodinger Equation and Postulates Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about
More informationTime-Independent Perturbation Theory
4 Phys46.nb Time-Independent Perturbation Theory.. Overview... General question Assuming that we have a Hamiltonian, H = H + λ H (.) where λ is a very small real number. The eigenstates of the Hamiltonian
More informationP3317 HW from Lecture and Recitation 7
P3317 HW from Lecture 1+13 and Recitation 7 Due Oct 16, 018 Problem 1. Separation of variables Suppose we have two masses that can move in 1D. They are attached by a spring, yielding a Hamiltonian where
More information8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Wednesday April Exam 2
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Wednesday April 18 2012 Exam 2 Last Name: First Name: Check Recitation Instructor Time R01 Barton Zwiebach 10:00 R02
More informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation
Lecture 17 Page 1 Lecture 17 L17.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent
More informationRepresentation 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 informationMatrices of Dirac Characters within an irrep
Matrices of Dirac Characters within an irrep irrep E : 1 0 c s 2 c s D( E) D( C ) D( C ) 3 3 0 1 s c s c 1 0 c s c s D( ) D( ) D( ) a c b 0 1 s c s c 2 1 2 3 c cos( ), s sin( ) 3 2 3 2 E C C 2 3 3 2 3
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 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 informationProblem 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 informationSymmetry and degeneracy
Symmetry and degeneracy Let m= degeneracy (=number of basis functions) of irrep i: From ( irrep) 1 one can obtain all the m irrep j by acting with off-diagonal R and orthogonalization. For instance in
More informationSelection rules - electric dipole
Selection rules - electric dipole As an example, lets take electric dipole transitions; when is j, m z j 2, m 2 nonzero so that j 1 = 1 and m 1 = 0. The answer is equivalent to the question when can j
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationPhysics 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 informationColumbia 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 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 informationSection 11: Review. µ1 x < 0
Physics 14a: Quantum Mechanics I Section 11: Review Spring 015, Harvard Below are some sample problems to help study for the final. The practice final handed out is a better estimate for the actual length
More informationJ10M.1 - Rod on a Rail (M93M.2)
Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight
More informationQuantum 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 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 information