Angular Momentum. Classical. J r p. radius vector from origin. linear momentum. determinant form of cross product iˆ xˆ J J J J J J
|
|
- Beverly Shelton
- 5 years ago
- Views:
Transcription
1 Angular Momentum Classical r p p radius vector from origin linear momentum r iˆ ˆj kˆ x y p p p x y determinant form of cross product iˆ xˆ ˆj yˆ kˆ ˆ y p p x y p x p y x x p y p y x x y Copyright Michael D. Fayer, 07
2 Q.M. Angular Momentum In the Schrödinger Representation, use Q.M. operators for x and p, etc. P x = i x = x x Substituting i ˆ ˆ ˆ i j k x y x y x i y y ix y y x y i x x x y Copyright Michael D. Fayer, 07
3 Commutators Consider, x y x y y x substituting operators in units of x y y x y x Keep track of what commutes. y y x x x y x y Similarly y x y x y x x x y y Subtracting x, y y x x y Copyright Michael D. Fayer, 07
4 x, y y x x y y x, x y x y, y x i, But, because P i Therefore,, i x y i i P,, P i i Using P, i, i x y in conventional units Copyright Michael D. Fayer, 07
5 The commutators in units of are, i x y, i y x, i. x y Using these it is found that,, x, y 0 Components of angular momentum do not commute. commutes with all components. Copyright Michael D. Fayer, 07
6 Therefore, and one component of angular momentum can be measured simultaneously. Call this component. Therefore, and can be simultaneously diagonalied by the same unitary transformation. Furthermore, Therefore, H, 0 H, 0 ( looks like rotation) H,, are all simultaneous observables. Copyright Michael D. Fayer, 07
7 Diagonaliation of and and commute. Therefore, set of vectors m are eigenvectors of both operators. Labeling kets with eigenvalues. and are simultaneously diagonal in the basis m m m m m m (in units of ) Copyright Michael D. Fayer, 07
8 Form operators i x i y x y From the definitions of and and the angular momentum commutators, the following commutators and identities can be derived. Commutators Identities,,, Copyright Michael D. Fayer, 07
9 Expectation value m m m m Because x y m m m m m m m m Positive numbers because s are Hermitian give real numbers. Square of real numbers positive. Therefore, the sum of three positive numbers is greater than or equal to one of them. Now m m m m m Therefore, m Eigenvalues of are greater than or equal to square of eigenvalues of. Copyright Michael D. Fayer, 07
10 Using, Consider m m m m m m m m eigenvalue eigenvector Furthermore,, 0 commutes with + because it commutes with x and y. Then m m m eigenvalue eigenvector Copyright Michael D. Fayer, 07
11 Thus, m is eigenvector of with eigenvalue m + and of with eigenvalue. + is a raising operator. It increases m by and leaves unchanged. Copyright Michael D. Fayer, 07
12 Repeated applications of to m gives new eigenvectors of values of m. (and ) with larger and larger But, because this must stop at a largest value of m, m max m. (m increases, doesn t change) Call largest value of m (m max ) j. m max =j For this value of m, that is, m = j j 0 with j 0 Can t raise past max value. Copyright Michael D. Fayer, 07
13 In similar manner can prove m is an eigenvector of with eigenvalues m and of with eigenvalues. Therefore, is a lowering operator. It reduces the value of m by and leaves unchanged. Operating repeatedly on j j m j, j, j, largest value of m gives eigenvectors with sequence of m eigenvalues Copyright Michael D. Fayer, 07
14 But, m Therefore, can t lower indefinitely. Must be some j such that j 0 with j 0 Smallest value of m. Can t lower below smallest value. Thus, largest value of m j = j' + an integer. smallest value of m Went from largest value to smallest value in unit steps. Copyright Michael D. Fayer, 07
15 We have largest value of m j 0 j 0 smallest value of m Left multiplying top equation by and bottom equation by j j 0 0 identities Then and operating 0 j j 0 j j j 0 j j j 0 j j j j Copyright Michael D. Fayer, 07
16 0 j j 0 j j j j Because j 0 and j 0 the coefficients of the kets must equal 0. Therefore, Because j > j' and j j j( j ) and ( j)( j ) j j j = an integer j = integer/; j can have integer or half integer values. because we go from j to j' =-jin unit steps with lowering operator. Thus, the eigenvalues of are 3 j( j) and j 0,,,, (largest m for a ) The eigenvalues of are m j, j,, j, j largest m change by unit steps smallest value of m Copyright Michael D. Fayer, 07
17 Final results jm j( j) jm jm m jm There are (j + ) m-states for a given j, going from j to j in integer steps. Can derive jm jm jm jm jm jm jm jm Copyright Michael D. Fayer, 07
18 Angular momentum states can be grouped by the value of j. Eigenvalues of, = j(j + ). j 0, /,, 3/,, j 0 m 0 00 j j j / m /, / m, 0, / m 3/, /, /, 3/ j m,, 0,, 0 etc. Copyright Michael D. Fayer, 07
19 Eigenvalues of are the square of the total angular momentum. The length of the angular momentum vector is j( j ) or in conventional units j( j ) Example m = j = Eigenvalues of are the projections of the angular momentum on the axis. m = 0 m = - Copyright Michael D. Fayer, 07
20 The matrix elements of are ( ) jj m, m jm jm j j jm jm m jj m, m jj m, m jm jm jm jm jj m, m jm jm jm jm The matrices for the first few values of j are (in units of ) j = 0 j = / (0) (0) (0) (0) / / Copyright Michael D. Fayer, 07
21 j = The jm are eigenkets of the and operators diagonal matrices. The raising and lowering operators and have matrix elements one step above and one step below the principal diagonal, respectively. Copyright Michael D. Fayer, 07
22 Particles such as atoms m RrY ( ) (, ) spherical harmonics from solution of H atom m The Y (, ) are the eigenvectors of the operators L and L. The m Y (, ) jm m LY LY m m m (, ) ( ) Y (, ) m (, ) my (, ) Copyright Michael D. Fayer, 07
23 Addition of Angular Momentum Examples Orbital and spin angular momentum - and s. These are really coupled spin-orbit coupling. ESR electron spins coupled to nuclear spins Inorganic spectroscopy unpaired d electrons Molecular excited triplet states two unpaired electrons Could consider separate angular momentum vectors j and j. These are distinct. But will see, that when they are coupled, want to combine the angular momentum vectors into one resultant vector. Copyright Michael D. Fayer, 07
24 Specific Case j m j m Four product states j m jm mm j and j omitted because they are always the same. Called the m m representation The two angular momenta are considered separately. Copyright Michael D. Fayer, 07
25 mm jjmm m m representation Want different representation Unitary Transformation to coupled rep. Angular momentum vectors added. New States labeled jm jjjm jm jm representation Copyright Michael D. Fayer, 07
26 jm where Eigenkets of operators in jm representation. and jm j j jm jm m jm vector sum of j and j Want unitary transformation from the m m representation to the jm representation. Copyright Michael D. Fayer, 07
27 Want jm Cmm m m mm C mm mm jm Cmm mm are the Clebsch-Gordan coefficients; Wigner coefficients; vector coupling coefficients are the basis vectors N states in the m m representation N states in the jm representation. N ( j )( j ) and obey the normal commutator relations. Prove by using and cranking through commutator relations using the fact that and and their components commute. Operators operating on different state spaces commute. Copyright Michael D. Fayer, 07
28 Finding the transformation m m m or coupling coefficient vanishes. To see this consider jm Cmm m m mm Operate with equal jm m jm Cmm m m mm These must be equal. Other terms 0 C mm if m m m mm mm m m C m m Copyright Michael D. Fayer, 07
29 Largest value of m since largest max m j + j m m max m j and m j Then the largest value of j is j j j because the largest value of j equals the largest value of m. There is only one state with the largest j and m. There are a total of (j + ) m states associated with the largest j j j. Copyright Michael D. Fayer, 07
30 Next largest m (m ) But m j j m m m Two ways to get m - m j and m j m j and m j Can form two orthogonal and normalied combinations. One of the combinations belongs to j j j Because this value of j has m values m ( j j ),( j j ),,( j j ) Other combination with m j j with j j j m ( j j ), ( j j ),,( j j ) largest smallest Copyright Michael D. Fayer, 07
31 Doing this repeatedly j values from j j j to j j in unit steps Each j has associated with it, its j + m values. Copyright Michael D. Fayer, 07
32 Example j values j, j j j j to j j in unit steps. j j 0 j j m, 0, 0 m 0 jm rep. kets m m rep. kets Know jm kets,0,00,,,, still need correct combo s of m m rep. kets Copyright Michael D. Fayer, 07
33 Generating procedure Start with the jm ket with the largest value of j and the largest value of m. m = But m m m Therefore, m m m because this is the only way to get m Then jm m m Clebsch-Gordan coefficient = Copyright Michael D. Fayer, 07
34 Use lowering operators jm m m jm m m from lowering op. expression Then 0 0 Clebsch-Gordan Coefficients from lowering op. expression (Use correct j i and m i values.) Copyright Michael D. Fayer, 07
35 Plug into raising and lowering op. formulas correctly. jm jm jm jm jm jm jm jm For jm rep. plug in j and m. jm For m m rep. mm mm means j jmm For must put in an d j when operating with and m and j and mwhen operating with Copyright Michael D. Fayer, 07
36 Lowering again 0 m m m m 00 Therefore, jm m m Have found the three m states for j = in terms of the m m states. Still need 00 m 0 m m Copyright Michael D. Fayer, 07
37 Need jm 00 m 0 m m 0 Two m m kets with mm 0, The 00 is a superposition of these. Have already used one superposition of these to form orthogonal to 0 and normalied. Find combination of normalied and orthogonal to 0. 00, Clebsch-Gordan Coefficients Copyright Michael D. Fayer, 07
38 Table of Clebsch-Gordan Coefficients j m m m j =/ j =/ Copyright Michael D. Fayer, 07
39 Next largest system j m j,0, m, m m kets 0 0 jm states j j j m,,, j j j m, jm kets Copyright Michael D. Fayer, 07
40 0 3 0 m m j m j = j = / Table of Clebsch-Gordan Coefficients jm m m m m Example Copyright Michael D. Fayer, 07
Alkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split
Electron Spin Electron spin hypothesis Solution to H atom problem gave three quantum numbers, n,, m. These apply to all atoms. Experiments show not complete description. Something missing. Alkali metals
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 informationPhysics 221A Fall 1996 Notes 14 Coupling of Angular Momenta
Physics 1A Fall 1996 Notes 14 Coupling of Angular Momenta In these notes we will discuss the problem of the coupling or addition of angular momenta. It is assumed that you have all had experience with
More informationMathematical Formulation of the Superposition Principle
Mathematical Formulation of the Superposition Principle Superposition add states together, get new states. Math quantity associated with states must also have this property. Vectors have this property.
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 informationMatrix Representation
Matrix Representation Matrix Rep. Same basics as introduced already. Convenient method of working with vectors. Superposition Complete set of vectors can be used to express any other vector. Complete set
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 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 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 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 informationWigner 3-j Symbols. D j 2. m 2,m 2 (ˆn, θ)(j, m j 1,m 1; j 2,m 2). (7)
Physics G6037 Professor Christ 2/04/2007 Wigner 3-j Symbols Begin by considering states on which two angular momentum operators J and J 2 are defined:,m ;,m 2. As the labels suggest, these states are eigenstates
More informationRepresentations of Lorentz Group
Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the
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 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 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 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 informationPlan for the rest of the semester. ψ a
Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and
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 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 information16.2 Coupling of angular momentum operators: The Clebsch Gordon coefficients and the Wigner 3j symbols. J(i) (16.62) J = i
16.2 Coupling of angular momentum operators: The Clebsch Gordon coefficients and the Wigner 3j symbols 31. In the previous section we have talked about the the orbital angular momentum and the spin angular
More informationEFFICIENT COMPUTATION OF CLEBSCH-GORDAN COEFFICIENTS
EFFICIENT COMPUTATION OF CLEBSCH-GORDAN COEFFICIENTS c William O. Straub, PhD Pasadena, California Here s a paper I wrote many years ago, back when the calculation of Clebsch-Gordan coefficients was giving
More informationSolutions to exam : 1FA352 Quantum Mechanics 10 hp 1
Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem (4 p): (a) Define the concept of unitary operator and show that the operator e ipa/ is unitary (p is the momentum operator in one dimension) (b)
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 informationAdding angular momenta
Adding angular momenta Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 Eleventh Lecture Outline 1 Outline 2 Some definitions 3 The simplest example: summing two momenta 4 Interesting physics: summing
More informationQM and Angular Momentum
Chapter 5 QM and Angular Momentum 5. Angular Momentum Operators In your Introductory Quantum Mechanics (QM) course you learned about the basic properties of low spin systems. Here we want to review that
More 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 informationAngular Momentum in Quantum Mechanics
Angular Momentum in Quantum Mechanics In classical mechanics the angular momentum L = r p of any particle moving in a central field of force is conserved. For the reduced two-body problem this is the content
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 informationSymmetries in Quantum Physics
Symmetries in Quantum Physics U. Fano Department of Physics and James Franck Institute University of Chicago Chicago, Illinois A. R. P. Rau Department of Physics and Astronomy louisiana State University
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 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 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 informationSelect/Special Topics in Atomic Physics Prof. P.C. Deshmukh Department Of Physics Indian Institute of Technology, Madras
Select/Special Topics in Atomic Physics Prof. P.C. Deshmukh Department Of Physics Indian Institute of Technology, Madras Lecture - 37 Stark - Zeeman Spectroscopy Well, let us continue our discussion on
More informationSelect/ Special Topics in Atomic Physics Prof. P. C. Deshmukh Department of Physics Indian Institute of Technology, Madras
Select/ Special Topics in Atomic Physics Prof. P. C. Deshmukh Department of Physics Indian Institute of Technology, Madras Lecture No. # 06 Angular Momentum in Quantum Mechanics Greetings, we will begin
More informationPhysics 215 Quantum Mechanics 1 Assignment 1
Physics 5 Quantum Mechanics Assignment Logan A. Morrison January 9, 06 Problem Prove via the dual correspondence definition that the hermitian conjugate of α β is β α. By definition, the hermitian conjugate
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 informationANGULAR MOMENTUM. We have seen that when the system is rotated through an angle about an axis α the unitary operator producing the change is
ANGULAR MOMENTUM We have seen that when the system is rotated through an angle about an ais α the unitary operator producing the change is U i J e Where J is a pseudo vector obeying the commutation relations
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 informationQuantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid
Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Announcement This handout includes 9 problems. The first 5 are the problem set due. The last 4 cover material from the final few lectures
More informationCLEBSCH-GORDAN COEFFICIENTS - EXAMPLES
CLEBSCH-GORDAN COEFFICIENTS - EXAMPLES Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog. Shankar, R. (994), Principles of Quantum Mechanics, Plenum
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 informationQuantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 14 Exercises on Quantum Expectation Values (Refer Slide Time: 00:07) In the last couple
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 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 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 informationVector Spaces for Quantum Mechanics J. P. Leahy January 30, 2012
PHYS 20602 Handout 1 Vector Spaces for Quantum Mechanics J. P. Leahy January 30, 2012 Handout Contents Examples Classes Examples for Lectures 1 to 4 (with hints at end) Definitions of groups and vector
More informationParticle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision)
Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e - e + - + e - e + - + e - e + - + e - e + - Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 Non-Relativistic QM (Revision)
More informationB. PHENOMENOLOGICAL NUCLEAR MODELS
B. PHENOMENOLOGICAL NUCLEAR MODELS B.0. Basic concepts of nuclear physics B.0. Binding energy B.03. Liquid drop model B.04. Spherical operators B.05. Bohr-Mottelson model B.06. Intrinsic system of coordinates
More informationClebsch-Gordan Coefficients
Phy489 Lecture 7 Clebsch-Gordan Coefficients 2 j j j2 m m m 2 j= j j2 j + j j m > j m > = C jm > m = m + m 2 2 2 Two systems with spin j and j 2 and z components m and m 2 can combine to give a system
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 informationPhysics 221A Fall 2017 Notes 20 Parity
Copyright c 2017 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 20 Parity 1. Introduction We have now completed our study of proper rotations in quantum mechanics, one of the important space-time
More informationADDITION OF ANGULAR MOMENTUM
ADDITION OF ANGULAR MOMENTUM QM 1 I Total angular momentum Recall that quantum mechanical particles can possess both orbital, L, and spin, S, angular momentum The state of such a particle may be written
More informationIf you like us, please share us on social media. The latest UCD Hyperlibrary newsletter is now complete, check it out.
Sign In Forgot Password Register username username password password Sign In If you like us, please share us on social media. The latest UCD Hyperlibrary newsletter is now complete, check it out. ChemWiki
More informationParticle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation
Dr. A. Mitov Particle Physics 45 Particle Physics Dr. Alexander Mitov µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 46 Non-Relativistic
More informationQuantum Physics II (8.05) Fall 2002 Outline
Quantum Physics II (8.05) Fall 2002 Outline 1. General structure of quantum mechanics. 8.04 was based primarily on wave mechanics. We review that foundation with the intent to build a more formal basis
More informationCHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM. (From Cohen-Tannoudji, Chapter XII)
CHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM (From Cohen-Tannoudji, Chapter XII) We will now incorporate a weak relativistic effects as perturbation
More information,, rectilinear,, spherical,, cylindrical. (6.1)
Lecture 6 Review of Vectors Physics in more than one dimension (See Chapter 3 in Boas, but we try to take a more general approach and in a slightly different order) Recall that in the previous two lectures
More informationThis is the important completeness relation,
Observable quantities are represented by linear, hermitian operators! Compatible observables correspond to commuting operators! In addition to what was written in eqn 2.1.1, the vector corresponding to
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 If two operators A, B commute then they have same set of eigenkets.
Lecture 14 Matrix representing of Operators While representing operators in terms of matrices, we use the basis kets to compute the matrix elements of the operator as shown below < Φ 1 x Φ 1 >< Φ 1 x Φ
More informationRepresentations of angular momentum
Representations of angular momentum Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 September 26, 2008 Sourendu Gupta (TIFR Graduate School) Representations of angular momentum QM I 1 / 15 Outline
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 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 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 informationThe Hydrogen Atom. Dr. Sabry El-Taher 1. e 4. U U r
The Hydrogen Atom Atom is a 3D object, and the electron motion is three-dimensional. We ll start with the simplest case - The hydrogen atom. An electron and a proton (nucleus) are bound by the central-symmetric
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 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 informationComputing Generalized Racah and Clebsch-Gordan Coefficients for U(N) groups
Computing Generalized Racah and Clebsch-Gordan Coefficients for U(N) groups Stephen V. Gliske May 9, 006 Abstract After careful introduction and discussion of the concepts involved, procedures are developed
More informationParticle Physics. Michaelmas Term 2009 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims
Particle Physics Michaelmas Term 2009 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2009 205 Introduction/Aims Symmetries play a central role in particle physics;
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 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 informationMolecular Bonding. Molecular Schrödinger equation. r - nuclei s - electrons. M j = mass of j th nucleus m 0 = mass of electron
Molecular onding Molecular Schrödinger equation r - nuclei s - electrons 1 1 W V r s j i j1 M j m i1 M j = mass of j th nucleus m = mass of electron j i Laplace operator for nuclei Laplace operator for
More informationΑΜ Α and ΒΜ Β angular momentum basis states to form coupled ΑΒCΜ C basis states RWF Lecture #4. The Wigner-Eckart Theorem
MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 5.74 RWF Lecture #4 4 The Wigner-Ecart Theorem It is always possible to evaluate the angular
More 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 informationLecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More informationPhysics 221A Fall 2005 Homework 11 Due Thursday, November 17, 2005
Physics 221A Fall 2005 Homework 11 Due Thursday, November 17, 2005 Reading Assignment: Sakurai pp. 234 242, 248 271, Notes 15. 1. Show that Eqs. (15.64) follow from the definition (15.61) of an irreducible
More informationarxiv:quant-ph/ v1 20 Apr 1995
Combinatorial Computation of Clebsch-Gordan Coefficients Klaus Schertler and Markus H. Thoma Institut für Theoretische Physik, Universität Giessen, 3539 Giessen, Germany (February, 008 The addition of
More informationGroup Theory and Its Applications in Physics
T. Inui Y Tanabe Y. Onodera Group Theory and Its Applications in Physics With 72 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Contents Sections marked with
More informationCONTENTS. vii. CHAPTER 2 Operators 15
CHAPTER 1 Why Quantum Mechanics? 1 1.1 Newtonian Mechanics and Classical Electromagnetism 1 (a) Newtonian Mechanics 1 (b) Electromagnetism 2 1.2 Black Body Radiation 3 1.3 The Heat Capacity of Solids and
More informationPhysics 557 Lecture 5
Physics 557 Lecture 5 Group heory: Since symmetries and the use of group theory is so much a part of recent progress in particle physics we will take a small detour to introduce the basic structure (as
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 informationProb (solution by Michael Fisher) 1
Prob 975 (solution by Michael Fisher) We begin by expressing the initial state in a basis of the spherical harmonics, which will allow us to apply the operators ˆL and ˆL z θ, φ φ() = 4π sin θ sin φ =
More informationQuantum Theory of Many-Particle Systems, Phys. 540
Quantum Theory of Many-Particle Systems, Phys. 540 IPM? Atoms? Nuclei: more now Other questions about last class? Assignment for next week Wednesday ---> Comments? Nuclear shell structure Ground-state
More informationIsospin. K.K. Gan L5: Isospin and Parity 1
Isospin Isospin is a continuous symmetry invented by Heisenberg: Explain the observation that the strong interaction does not distinguish between neutron and proton. Example: the mass difference between
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 information= a. a = Let us now study what is a? c ( a A a )
7636S ADVANCED QUANTUM MECHANICS Solutions 1 Spring 010 1 Warm up a Show that the eigenvalues of a Hermitian operator A are real and that the eigenkets of A corresponding to dierent eigenvalues are orthogonal
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 informationThe rotation group and quantum mechanics 1 D. E. Soper 2 University of Oregon 30 January 2012
The rotation group and quantum mechanics 1 D. E. Soper 2 University of Oregon 30 January 2012 I offer here some background for Chapter 3 of J. J. Sakurai, Modern Quantum Mechanics. 1 The rotation group
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 information3. Quantum Mechanics in 3D
3. Quantum Mechanics in 3D 3.1 Introduction Last time, we derived the time dependent Schrödinger equation, starting from three basic postulates: 1) The time evolution of a state can be expressed as a unitary
More informationComparing and Improving Quark Models for the Triply Bottom Baryon Spectrum
Comparing and Improving Quark Models for the Triply Bottom Baryon Spectrum A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science degree in Physics from the
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 informationTensor Operators and the Wigner Eckart Theorem
November 11, 2009 Tensor Operators and the Wigner Eckart Theorem Vector operator The ket α transforms under rotation to α D(R) α. The expectation value of a vector operator in the rotated system is related
More informationPhysics 221B Fall 2018 Notes 28 Identical Particles
Copyright c 2018 by Robert G. Littlejohn Physics 221B Fall 2018 Notes 28 Identical Particles 1. Introduction Understanding the quantum mechanics of systems of identical particles requires a new postulate,
More informationMultielectron Atoms.
Multielectron Atoms. Chem 639. Spectroscopy. Spring 00 S.Smirnov Atomic Units Mass m e 1 9.109 10 31 kg Charge e 1.60 10 19 C Angular momentum 1 1.055 10 34 J s Permittivity 4 0 1 1.113 10 10 C J 1 m 1
More informationSpin Dynamics Basics of Nuclear Magnetic Resonance. Malcolm H. Levitt
Spin Dynamics Basics of Nuclear Magnetic Resonance Second edition Malcolm H. Levitt The University of Southampton, UK John Wiley &. Sons, Ltd Preface xxi Preface to the First Edition xxiii Introduction
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 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 informationPY 351 Modern Physics Short assignment 4, Nov. 9, 2018, to be returned in class on Nov. 15.
PY 351 Modern Physics Short assignment 4, Nov. 9, 2018, to be returned in class on Nov. 15. You may write your answers on this sheet or on a separate paper. Remember to write your name on top. Please note:
More informationPhysics 221A Fall 1996 Notes 21 Hyperfine Structure in Hydrogen and Alkali Atoms
Physics 221A Fall 1996 Notes 21 Hyperfine Structure in Hydrogen and Alkali Atoms Hyperfine effects in atomic physics are due to the interaction of the atomic electrons with the electric and magnetic multipole
More information