Lecture Notes 2: Review of Quantum Mechanics
|
|
- Anissa Carson
- 5 years ago
- Views:
Transcription
1 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 of quantum mechanics, with special emphasis on the problems arising from infinite-dimensional spaces. In introductory courses of quantum mechanics these issues are often swept under the rug to keep the mathematics simple. Also, much of the categorical language developed here in Oxford is suited only for finite-dimensional spaces. However, being on our way to Quantum Field Theory, it is important to notice the differences and the problems they cause. 1 Basic Elements of Quantum Mechanics In quantum mechanics, the state space of a physical system is represented by a Hilbert space H. Two nonzero vectors of H will represent the same physical state if they are linearly dependent. Thus the state space is really the projective space of H. This is often underemphasised and H is still called the state space. 1.1 Tensor Product A vital feature of quantum mechanics is that the combined system, consisting of the systems represented by Hilbert spaces H 1 and H 2, is represented by the tensor product H 1 H 2. Finite-dimensional Hilbert spaces with the tensor product form a compact-closed monoidal category, indeed, a -compact category. However, this fails in the infinite-dimensional case. The usual formulation of the universal property of the tensor product also fails in the infinite-dimensional case. Recall the universal property of the tensor product of two vector spaces U and V : for any vector space W and any bilinear map λ : U V W, there exists a unique linear map ˆλ : U V W such that λ = ˆλ ǫ, where ǫ : U V U V is the natural embedding (u,v) u v. Since the tensor product of two Hilbert spaces is the completion of their usual vector-space tensor product, it turns out that this universal property only holds for Hilbert-Schmidt maps λ. It may seem that considering a category of Hilbert spaces and Hilbert-Schmidt maps would solve the problem, but the identity map is not Hilbert-Schmidt, so the issue is more intricate. This issue is addressed in [1], introducing nuclearity and other concepts that may be useful in axiomatising infinite-dimensional quantum mechanics. 2-1
2 1.2 Time Evolution The time evolution is described by a family of unitary operators U(t) : H H, so that the state vector at time t, ψ(t), is given by ψ(t) = U(t) ψ(0). It is a fundamental law of quantum mechanics that U(t) = e iht/ for a Hermitian operator H, called the Hamiltonian, which depends on the physics of the specific system considered. The infinitesimal version of the time-evolution equation then reads and is called the Schrödinger equation. i ψ(t) = H ψ(t) t Consider the eigenstates E n of the Hamiltonian and their corresponding eigenvalues E n. The time evolution of a state initalised by E n is given by ψ(t) = e iht/ E n = e ient/ E n, so it is a persistent state, only picking up a phase as time passes. Since the time evolution is linear, if we can express an initial state ψ(0) as a linear combination of E n s, it is then easy to express the state ψ(t) at any later time t. 2 Free Particle in 1D A classic example for an infinite-dimensional quantum-mechanical system, exposing many of the issues of infinite-dimensionality, is a free particle moving on a line in a potential V (x). The state space of the particle is the Hilbert space L 2 (R), whose elements are the (equivalence classes of) square-integrable complex-valued functions on R. Setting = 1 in the following and in the rest of this lecture, the Hamiltonian of the system is H = 1 2m 2 + V (x), so the equation of motion is i ψ t = 1 2m 2 ψ + V (x)ψ. 2-2
3 2.1 Observables The momentum operator p is defined as P = i d dx and its eigenfunctions are wave functions e ipx, where p is the eigenvalue (value of momentum). However, note that p as defined here is not really a total operator L 2 (R) L 2 (R) and none of the functions e ipx is an element of L 2 (R). Similarly, the position operator x is defined by (x ψ )(x) = x ψ (x), but again, its eigenvalues are the Dirac delta-functions δ x, which are not functions at all, let alone elements of L 2 (R). These issues are often neglected in introductory treatments. To treat them formally, the idea of rigged Hilbert spaces is used. The idea is that we pick a nuclear subspace Ω of our Hilbert space H, which is dense in but not equal to H, and form its dual Ω. Since H is isomorphic to H, we get Ω H Ω. The needed eigenfunctions of our operators will now reside in Ω, and in fact enough spectral theory can be mimicked in this context to allow for the development of quantum-mechanical constructions. More details can be found in the much recommended book [2]. 2.2 Quantisation Where did we get the Hamiltonian for the free particle? As said in the previous lecture, there is a (heuristic) process called quantisation which tries to construct a quantum theory from a given classical theory, in particular, for each classical observable we need to produce a corresponding quantal observable. Classical Quantal State space Manifold (Projective) Hilbert space Observable Real-valued function quantisation Hermitian operator Time evolution Hamiltonian Hamiltonian In our simple case, we just replace the quantities x and p in the classical Hamiltonian by the quantum mechanical operators x and p. Thus from the classical Hamiltonian H = p 2 2m + V (x) we obtain the quantum mechanical Hamiltonian H = P2 2m + V (x) = 1 2m 2-3 d 2 dx2ψ + V (x),
4 which is correct up to setting = 1. However, note that the operators x and p do not commute, in fact, they satisfy the canonical commutation relations 1 [x,p] = ii. Thus the above prescription is sometimes ambiguous, i.e. classically we have x p = xpx = px 2, while for quantal observables we have x 2 p xpx px 2. In such cases we just have to guess the correct expression. 3 Symmetry in Quantum Mechanics Let G be a Lie group acting on a Hilbert space H, let G 0 be a one-parameter subgroup of G, and let U(θ) be a representation of G 0. Theorem (Stone): There exists a Hermitian operator A such that U(θ) = e iθa. Moreover, if the group action commutes with the Hamiltonian H, it follows that A is conserved during time evolution. This is a quantum analogue of Nöther s theorem from classical mechanics. 4 Harmonic Oscillator In a quadratic potential V (x) = 1 2 kx2, we get a Hamiltonian H = p2 2m kx2. The solution of the corresponding classical system is x = Asin(ωt + φ), where ω = k/m. In anticipation of analogous solutions we write the Hamiltonian as H = p2 2m mω2 x 2. While the eigenvectors of H can be found using Hermite polynomials and other tricks for dealing with the differential equations involved, we will present a nicer, algebraic solution. 1 Note that in the finite-dimensional case, there are no operators X, P such that [X, P] = I. Further, it is a theorem of von Neumann that the position and momentum operators are essentially the only realisation of operators such that [X, P] = i. 2-4
5 Introduce the operator and its adjoint so that mω 1 a = x 2 + ip 2mω mω 1 a = x 2 ip 2mω, 1 x = [a + a], 2mω mω p = i 2 [a a] and hence ( ) H = ω a a Defining N = a a, we have H = ω ( N ) and hence to find the eigenvalues of H, it is sufficient to find the eigenvalues of N. Also, it is straightforward to check that, [a,a ] = 1, [N,a] = a and [N,a ] = a. Let n be an eigenvector of N with an eigenvalue n. (Anticipating, but not supposing, that n will be an integer.) N n = n n implies that N(a n ) = (a + a N) n = (n + 1)(a n ), i.e. that a n is an eigenvector of N with eigenvalue n + 1, unless it is the zero vector. Similarly observe that N(a n ) = ( a + an) n = (n 1)(a n ), i.e. that a n is an eigenvector of N with eigenvalue n 1, unless it is the zero vector. We call a the raising operator and a the lowering operator. Note that ψ N ψ = ψ a a ψ = a ψ 2 0, so the spectrum of N is bounded below by zero. 2 By repeated application of the lowering operator a to n, we obtain eigenvectors with lower and lower eigenvalues, or the zero vector. Since the spectrum of N is bounded below, after a finite number (at most n + 1) of applications, we must have obtained a zero 2 This implies that the spectrum of H is bounded below by 1 ω, i.e. the lowest possible energy is positive
6 vector. If m is the least integer such that a m n is nonzero, we get a(a m n ) = 0, so also N(a m n ) = (a a)(a m n ) = 0, which means that a m n is an eigenvector of N with eigenvalue zero. But we also know that a m n is an eigenvector of N with eigenvalue n m, therefore n = m is a nonnegative integer. Furthermore, if n = 1, then a n 2 = n aa n = n (1 + a a) n = n (1 + N) n = (n + 1), so in particular, we never get a zero vector when applying a to an eigenvector of N. This means that by repeated application of the raising operator we obtain eigenvalues with arbitrarily large eigenvalues. Therefore, all nonnegative integers are eigenvalues of N. It is left as an exercise to the reader to show that all the eigenspaces are one-dimensional, so up to physically unimportant phase, there is a unique normalised n-eigenvector n for each nonnegative integer n. From this and the calculation above it then follows that a n = n + 1 n + 1 and similarly a n = n n 1. We have found all the eigenvalues of N and hence of H. Notice however that it takes more work to establish that there are no elements in the spectrum of N or H other than their respective eigenvalues. References [1] S. Abramsky, R. Blute and P. Panangaden, Nuclear and trace ideals in tensor-* categories, Journal of Pure and Applied Algebra (1999). [2] N. N. Bogoliubov, A. A. Logunov and I. T. Todorov (1975): Introduction to Axiomatic Quantum Field Theory. Reading, Mass.: W. A. Benjamin, Advanced Book Program. 2-6
Simple Harmonic Oscillator
Classical harmonic oscillator Linear force acting on a particle (Hooke s law): F =!kx From Newton s law: F = ma = m d x dt =!kx " d x dt + # x = 0, # = k / m Position and momentum solutions oscillate in
More 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 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 mechanics. Chapter The quantum mechanical formalism
Chapter 5 Quantum mechanics 5.1 The quantum mechanical formalism The realisation, by Heisenberg, that the position and momentum of a quantum mechanical particle cannot be measured simultaneously renders
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 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 information3 The Harmonic Oscillator
3 The Harmonic Oscillator What we ve laid out so far is essentially just linear algebra. I now want to show how this formalism can be used to do some physics. To do this, we ll study a simple system the
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 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 information4.3 Lecture 18: Quantum Mechanics
CHAPTER 4. QUANTUM SYSTEMS 73 4.3 Lecture 18: Quantum Mechanics 4.3.1 Basics Now that we have mathematical tools of linear algebra we are ready to develop a framework of quantum mechanics. The framework
More informationTopics in Representation Theory: Cultural Background
Topics in Representation Theory: Cultural Background This semester we will be covering various topics in representation theory, see the separate syllabus for a detailed list of topics, including some that
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 informationRecitation 1 (Sep. 15, 2017)
Lecture 1 8.321 Quantum Theory I, Fall 2017 1 Recitation 1 (Sep. 15, 2017) 1.1 Simultaneous Diagonalization In the last lecture, we discussed the situations in which two operators can be simultaneously
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 informationLecture 3 Dynamics 29
Lecture 3 Dynamics 29 30 LECTURE 3. DYNAMICS 3.1 Introduction Having described the states and the observables of a quantum system, we shall now introduce the rules that determine their time evolution.
More informationPLEASE LET ME KNOW IF YOU FIND TYPOS (send to
Teoretisk Fysik KTH Advanced QM (SI2380), Lecture 2 (Summary of concepts) 1 PLEASE LET ME KNOW IF YOU FIND TYPOS (send email to langmann@kth.se) The laws of QM 1. I now discuss the laws of QM and their
More informationLecture 4: Postulates of quantum mechanics
Lecture 4: Postulates of quantum mechanics Rajat Mittal IIT Kanpur The postulates of quantum mechanics provide us the mathematical formalism over which the physical theory is developed. For people studying
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 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 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 informationPhysics 221A Fall 2010 Notes 1 The Mathematical Formalism of Quantum Mechanics
Physics 221A Fall 2010 Notes 1 The Mathematical Formalism of Quantum Mechanics 1. Introduction The prerequisites for Physics 221A include a full year of undergraduate quantum mechanics. In this semester
More informationThe geometrical semantics of algebraic quantum mechanics
The geometrical semantics of algebraic quantum mechanics B. Zilber University of Oxford November 29, 2016 B.Zilber, The semantics of the canonical commutation relations arxiv.org/abs/1604.07745 The geometrical
More informationwhere P a is a projector to the eigenspace of A corresponding to a. 4. Time evolution of states is governed by the Schrödinger equation
1 Content of the course Quantum Field Theory by M. Srednicki, Part 1. Combining QM and relativity We are going to keep all axioms of QM: 1. states are vectors (or rather rays) in Hilbert space.. observables
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 informationQuantum Mechanics for Mathematicians: Energy, Momentum, and the Quantum Free Particle
Quantum Mechanics for Mathematicians: Energy, Momentum, and the Quantum Free Particle Peter Woit Department of Mathematics, Columbia University woit@math.columbia.edu November 28, 2012 We ll now turn to
More information1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
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 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 informationHarmonic Oscillator. Robert B. Griffiths Version of 5 December Notation 1. 3 Position and Momentum Representations of Number Eigenstates 2
qmd5 Harmonic Oscillator Robert B. Griffiths Version of 5 December 0 Contents Notation Eigenstates of the Number Operator N 3 Position and Momentum Representations of Number Eigenstates 4 Coherent States
More informationin terms of the classical frequency, ω = , puts the classical Hamiltonian in the form H = p2 2m + mω2 x 2
One of the most important problems in quantum mechanics is the simple harmonic oscillator, in part because its properties are directly applicable to field theory. The treatment in Dirac notation is particularly
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 informationThe Position Representation in Quantum Mechanics
The Position Representation in Quantum Mechanics Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 13, 2007 The x-coordinate of a particle is associated with
More informationFormalism of Quantum Mechanics
Dirac Notation Formalism of Quantum Mechanics We can use a shorthand notation for the normalization integral I = "! (r,t) 2 dr = "! * (r,t)! (r,t) dr =!! The state! is called a ket. The complex conjugate
More informationPY 351 Modern Physics - Lecture notes, 3
PY 351 Modern Physics - Lecture notes, 3 Copyright by Claudio Rebbi, Boston University, October 2016. These notes cannot be duplicated and distributed without explicit permission of the author. Time dependence
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 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7
Lecture 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7 In Lecture 1 and 2, we have discussed how to represent the state of a quantum mechanical system based the superposition
More informationProperties of Commutators and Schroedinger Operators and Applications to Quantum Computing
International Journal of Engineering and Advanced Research Technology (IJEART) Properties of Commutators and Schroedinger Operators and Applications to Quantum Computing N. B. Okelo Abstract In this paper
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 informationThe quantum state as a vector
The quantum state as a vector February 6, 27 Wave mechanics In our review of the development of wave mechanics, we have established several basic properties of the quantum description of nature:. A particle
More informationHarmonic Oscillator with raising and lowering operators. We write the Schrödinger equation for the harmonic oscillator in one dimension as follows:
We write the Schrödinger equation for the harmonic oscillator in one dimension as follows: H ˆ! = "!2 d 2! + 1 2µ dx 2 2 kx 2! = E! T ˆ = "! 2 2µ d 2 dx 2 V ˆ = 1 2 kx 2 H ˆ = ˆ T + ˆ V (1) where µ is
More informationPhysics 221A Fall 2017 Notes 1 The Mathematical Formalism of Quantum Mechanics
Copyright c 2017 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 1 The Mathematical Formalism of Quantum Mechanics 1. Introduction The prerequisites for Physics 221A include a full year of undergraduate
More informationmsqm 2011/8/14 21:35 page 189 #197
msqm 2011/8/14 21:35 page 189 #197 Bibliography Dirac, P. A. M., The Principles of Quantum Mechanics, 4th Edition, (Oxford University Press, London, 1958). Feynman, R. P. and A. P. Hibbs, Quantum Mechanics
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 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 informationSample Quantum Chemistry Exam 2 Solutions
Chemistry 46 Fall 7 Dr. Jean M. Standard Name SAMPE EXAM Sample Quantum Chemistry Exam Solutions.) ( points) Answer the following questions by selecting the correct answer from the choices provided. a.)
More 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 information8.05, Quantum Physics II, Fall 2013 TEST Wednesday October 23, 12:30-2:00pm You have 90 minutes.
8.05, Quantum Physics II, Fall 03 TEST Wednesday October 3, :30-:00pm You have 90 minutes. Answer all problems in the white books provided. Write YOUR NAME and YOUR SECTION on your white books). There
More informationC/CS/Phys 191 Quantum Mechanics in a Nutshell I 10/04/05 Fall 2005 Lecture 11
C/CS/Phys 191 Quantum Mechanics in a Nutshell I 10/04/05 Fall 2005 Lecture 11 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
More informationPHYS Handout 6
PHYS 060 Handout 6 Handout Contents Golden Equations for Lectures 8 to Answers to examples on Handout 5 Tricks of the Quantum trade (revision hints) Golden Equations (Lectures 8 to ) ψ Â φ ψ (x)âφ(x)dxn
More informationThe semantics of non-commutative geometry and quantum mechanics.
The semantics of non-commutative geometry and quantum mechanics. B. Zilber University of Oxford July 30, 2014 1 Dualities in logic and geometry 2 The Weyl-Heisenberg algebra 3 Calculations Tarskian duality
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 informationLecture 6. Four postulates of quantum mechanics. The eigenvalue equation. Momentum and energy operators. Dirac delta function. Expectation values
Lecture 6 Four postulates of quantum mechanics The eigenvalue equation Momentum and energy operators Dirac delta function Expectation values Objectives Learn about eigenvalue equations and operators. Learn
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 informationQuantum Mechanics I Physics 5701
Quantum Mechanics I Physics 5701 Z. E. Meziani 02/24//2017 Physics 5701 Lecture Commutation of Observables and First Consequences of the Postulates Outline 1 Commutation Relations 2 Uncertainty Relations
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 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 informationChapter 2 The Group U(1) and its Representations
Chapter 2 The Group U(1) and its Representations The simplest example of a Lie group is the group of rotations of the plane, with elements parametrized by a single number, the angle of rotation θ. It is
More informationTwo and Three-Dimensional Systems
0 Two and Three-Dimensional Systems Separation of variables; degeneracy theorem; group of invariance of the two-dimensional isotropic oscillator. 0. Consider the Hamiltonian of a two-dimensional anisotropic
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
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 informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More information0.1 Schrödinger Equation in 2-dimensional system
0.1 Schrödinger Equation in -dimensional system In HW problem set 5, we introduced a simpleminded system describing the ammonia (NH 3 ) molecule, consisting of a plane spanned by the 3 hydrogen atoms and
More information1 Revision to Section 17.5: Spin
1 Revision to Section 17.5: Spin We classified irreducible finite-dimensional representations of the Lie algebra so(3) by their spin l, where l is the largest eigenvalue for the operator L 3 = iπ(f 3 ).
More informationThe Principles of Quantum Mechanics: Pt. 1
The Principles of Quantum Mechanics: Pt. 1 PHYS 476Q - Southern Illinois University February 15, 2018 PHYS 476Q - Southern Illinois University The Principles of Quantum Mechanics: Pt. 1 February 15, 2018
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationQuantum Mechanics I Physics 5701
Quantum Mechanics I Physics 5701 Z. E. Meziani 02/23//2017 Physics 5701 Lecture Outline 1 General Formulation of Quantum Mechanics 2 Measurement of physical quantities and observables 3 Representations
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 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 informationSymmetries and particle physics Exercises
Symmetries and particle physics Exercises Stefan Flörchinger SS 017 1 Lecture From the lecture we know that the dihedral group of order has the presentation D = a, b a = e, b = e, bab 1 = a 1. Moreover
More information1 The postulates of quantum mechanics
1 The postulates of quantum mechanics The postulates of quantum mechanics were derived after a long process of trial and error. These postulates provide a connection between the physical world and the
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
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 information8.04 Spring 2013 March 12, 2013 Problem 1. (10 points) The Probability Current
Prolem Set 5 Solutions 8.04 Spring 03 March, 03 Prolem. (0 points) The Proaility Current We wish to prove that dp a = J(a, t) J(, t). () dt Since P a (t) is the proaility of finding the particle in the
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationTopics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem
Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem 1 Fourier Analysis, a review We ll begin with a short review of simple facts about Fourier analysis, before going on to interpret
More informationLecture 6 Quantum Mechanical Systems and Measurements
Lecture 6 Quantum Mechanical Systems and Measurements Today s Program: 1. Simple Harmonic Oscillator (SHO). Principle of spectral decomposition. 3. Predicting the results of measurements, fourth postulate
More informationEigenvectors and Hermitian Operators
7 71 Eigenvalues and Eigenvectors Basic Definitions Let L be a linear operator on some given vector space V A scalar λ and a nonzero vector v are referred to, respectively, as an eigenvalue and corresponding
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 informationWOMP 2001: LINEAR ALGEBRA. 1. Vector spaces
WOMP 2001: LINEAR ALGEBRA DAN GROSSMAN Reference Roman, S Advanced Linear Algebra, GTM #135 (Not very good) Let k be a field, eg, R, Q, C, F q, K(t), 1 Vector spaces Definition A vector space over k is
More informationThe Postulates of Quantum Mechanics
p. 1/23 The Postulates of Quantum Mechanics We have reviewed the mathematics (complex linear algebra) necessary to understand quantum mechanics. We will now see how the physics of quantum mechanics fits
More 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 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 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 informationThe Spinor Representation
The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)
More informationPhysics 505 Homework No. 1 Solutions S1-1
Physics 505 Homework No s S- Some Preliminaries Assume A and B are Hermitian operators (a) Show that (AB) B A dx φ ABψ dx (A φ) Bψ dx (B (A φ)) ψ dx (B A φ) ψ End (b) Show that AB [A, B]/2+{A, B}/2 where
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationSupplementary information I Hilbert Space, Dirac Notation, and Matrix Mechanics. EE270 Fall 2017
Supplementary information I Hilbert Space, Dirac Notation, and Matrix Mechanics Properties of Vector Spaces Unit vectors ~xi form a basis which spans the space and which are orthonormal ( if i = j ~xi
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 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 informationLinear Algebra and Dirac Notation, Pt. 2
Linear Algebra and Dirac Notation, Pt. 2 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 2 February 1, 2017 1 / 14
More informationMathematical Structures of Quantum Mechanics
msqm 2011/8/14 21:35 page 1 #1 Mathematical Structures of Quantum Mechanics Kow Lung Chang Physics Department, National Taiwan University msqm 2011/8/14 21:35 page 2 #2 msqm 2011/8/14 21:35 page i #3 TO
More information5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:
5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ 1 0 0 0 ξ 1 0 0 0 ξ 3, ξ
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler March 8, 011 1 Lecture 19 1.1 Second Quantization Recall our results from simple harmonic oscillator. We know the Hamiltonian very well so no need to repeat here.
More informationTutorial 5 Clifford Algebra and so(n)
Tutorial 5 Clifford Algebra and so(n) 1 Definition of Clifford Algebra A set of N Hermitian matrices γ 1, γ,..., γ N obeying the anti-commutator γ i, γ j } = δ ij I (1) is the basis for an algebra called
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationThe Framework of Quantum Mechanics
The Framework of Quantum Mechanics We now use the mathematical formalism covered in the last lecture to describe the theory of quantum mechanics. In the first section we outline four axioms that lie at
More information