Lecture #1. Review. Postulates of quantum mechanics (1-3) Postulate 1
|
|
- Elfrieda Owen
- 5 years ago
- Views:
Transcription
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, if a system is in the state, the average of any physical observable relevant he this system in time is Only normalizable wave functions represent physical states. The set of all squareintegrable functions, on a specified interval, constitutes a Hilbert space. Wave functions live in Hilbert space. Exercise 1 Problem 1.5 Consider the wave function where A, λ, and ω are positive real constants. (a) Normalize (b) Determine the expectation values of x and x 2.
2 Solution L1.P2
3 L1.P3
4 L1.P4 Postulate 2 To any self-consistently and well-defined observable Q, such as linear momentum, energy, angular momentum, or a number of particles, there correspond an operator such that measurement of Q yields values (call these measured values q) which are eigenvalues of Q. That is, the values q are those for which the equation has a solution. The function is called the eigenfunction of corresponding to the eigenvalue q. Exercise 7 Consider the operator angle in polar coordinates, and the functions are subject to (1)Is hermitian? (2) Find its eigenvalues and eigenfunctions. (3) What is the spectrum of Q? Is the spectrum degenerate?
5 Solution L1.P5 (3) The spectrum is doubly degenerate, for a given n there are two eigenfunctions except special case n=0, which is not degenerate.
6 Postulate 3 L1.P6 Measurement of the observable Q that yields the value q leaves the system in the state, where is the eigenfunction of Q that corresponds to the eigenvalue q. Generalized statistical interpretation: If your measure observable Q on a particle in a state you will get one of the eigenvalues of the hermitian operator Q. If the spectrum of Q is discrete, the probability of getting the eigenvalue associated with orthonormalized eigenfunction is It the spectrum is continuous, with real eigenvalues q(z) and associated Dirac-orthonormalized eigenfunctions, the probability of getting a result in the range dz is The wave function "collapses" to the corresponding eigenstate upon measurement. The uncertainty principle:
7 L1.P7 Schrödinger equation: summary The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent Schrödinger equation If potential V is spherically symmetric, i.e. only depends on distance to the origin r, then the separable solutions are where are solutions of radial equation with normalization condition The spherical harmonics are associated Legendre functions
8 L1.P8 Summary for radial equation: Angular momentum Eigenfunctions of and are labeled by m and l: For a given value of l, there are 2l+1 values of m: Elementary particles carry intrinsic angular momentum S in addition to L. Spin of elementary particles has nothing to do with rotation, does not depend on coordinates and, and is purely a quantum mechanical phenomena.
9 L1.P9 Class exercise #12 The electron in a hydrogen atom occupies the combined spin an position state: (a) If you measure the orbital angular momentum squared your get and what is the probability of each?, what values might (b) Same for z component of the orbital angular momentum. (c) Same for the spin angular momentum squared. (d) Same for z component of the spin angular momentum.
10 L1.P10 Class exercise #12 The electron in a hydrogen atom occupies the combined spin an position state: Note that in both cases (a) If you measure the orbital angular momentum squared your get and what is the probability of each?, what values might You get with probability P=1 (100%). (b) Same for z component of the orbital angular momentum. Possible values of m : (c) Same for the spin angular momentum squared. (d) Same for z component of the spin angular momentum.
11
The 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 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 informationPotential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form
Lecture 6 Page 1 Atoms L6.P1 Review of hydrogen atom Heavy proton (put at the origin), charge e and much lighter electron, charge -e. Potential energy, from Coulomb's law Potential is spherically symmetric.
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 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 informationAngular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z.
Angular momentum is an important concept in quantum theory, necessary for analyzing motion in 3D as well as intrinsic properties such as spin Classically the orbital angular momentum with respect to a
More 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 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 informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation
Lecture 27st Page 1 Lecture 27 L27.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 informationDegeneracy & in particular to Hydrogen atom
Degeneracy & in particular to Hydrogen atom In quantum mechanics, an energy level is said to be degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely,
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 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 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 informationAtomic Systems (PART I)
Atomic Systems (PART I) Lecturer: Location: Recommended Text: Dr. D.J. Miller Room 535, Kelvin Building d.miller@physics.gla.ac.uk Joseph Black C407 (except 15/1/10 which is in Kelvin 312) Physics of Atoms
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 informationOne-electron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation:
One-electron Atom The atomic orbitals of hydrogen-like atoms are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's
More information1 Measurement and expectation values
C/CS/Phys 191 Measurement and expectation values, Intro to Spin 2/15/05 Spring 2005 Lecture 9 1 Measurement and expectation values Last time we discussed how useful it is to work in the basis of energy
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 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 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 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 informationQuantum Mechanics Exercises and solutions
Quantum Mechanics Exercises and solutions P.J. Mulders Department of Physics and Astronomy, Faculty of Sciences, Vrije Universiteit Amsterdam De Boelelaan 181, 181 HV Amsterdam, the Netherlands email:
More information2m r2 (~r )+V (~r ) (~r )=E (~r )
Review of the Hydrogen Atom The Schrodinger equation (for 1D, 2D, or 3D) can be expressed as: ~ 2 2m r2 (~r, t )+V (~r ) (~r, t )=i~ @ @t The Laplacian is the divergence of the gradient: r 2 =r r The time-independent
More 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 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 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 informationUGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM PH 05 PHYSICAL SCIENCE TEST SERIES # 1. Quantum, Statistical & Thermal Physics
UGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM BOOKLET CODE SUBJECT CODE PH 05 PHYSICAL SCIENCE TEST SERIES # Quantum, Statistical & Thermal Physics Timing: 3: H M.M: 00 Instructions. This test
More 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 informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More 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 informationSolution Set of Homework # 6 Monday, December 12, Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second Volume
Department of Physics Quantum II, 570 Temple University Instructor: Z.-E. Meziani Solution Set of Homework # 6 Monday, December, 06 Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second
More 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 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 informationC/CS/Phys C191 Particle-in-a-box, Spin 10/02/08 Fall 2008 Lecture 11
C/CS/Phys C191 Particle-in-a-box, Spin 10/0/08 Fall 008 Lecture 11 Last time we saw that the time dependent Schr. eqn. can be decomposed into two equations, one in time (t) and one in space (x): space
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 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 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 informationLECTURES ON QUANTUM MECHANICS
LECTURES ON QUANTUM MECHANICS GORDON BAYM Unitsersity of Illinois A II I' Advanced Bock Progrant A Member of the Perseus Books Group CONTENTS Preface v Chapter 1 Photon Polarization 1 Transformation of
More informationChemistry 881 Lecture Topics Fall 2001
Chemistry 881 Lecture Topics Fall 2001 Texts PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon MATHEMATICS for PHYSICAL CHEMISTRY, Mortimer i. Mathematics Review (M, Chapters 1,2,3 & 4; M&S,
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 informationQUANTUM MECHANICS SECOND EDITION G. ARULDHAS
QUANTUM MECHANICS SECOND EDITION G. ARULDHAS Formerly, Professor and Head of Physics and Dean, Faculty of Science University of Kerala New Delhi-110001 2009 QUANTUM MECHANICS, 2nd Ed. G. Aruldhas 2009
More informationeff (r) which contains the influence of angular momentum. On the left is
1 Fig. 13.1. The radial eigenfunctions R nl (r) of bound states in a square-well potential for three angular-momentum values, l = 0, 1, 2, are shown as continuous lines in the left column. The form V (r)
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 informationParadigms in Physics: Quantum Mechanics
Paradigms in Physics: Quantum Mechanics David H. McIntyre Corinne A. Manogue Janet Tate Oregon State University 23 November 2010 Copyright 2010 by David H. McIntyre, Corinne A. Manogue, Janet Tate CONTENTS
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 informationThe energy of the emitted light (photons) is given by the difference in energy between the initial and final states of hydrogen atom.
Lecture 20-21 Page 1 Lectures 20-21 Transitions between hydrogen stationary states The energy of the emitted light (photons) is given by the difference in energy between the initial and final states of
More informationLecture 14 The Free Electron Gas: Density of States
Lecture 4 The Free Electron Gas: Density of States Today:. Spin.. Fermionic nature of electrons. 3. Understanding the properties of metals: the free electron model and the role of Pauli s exclusion principle.
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 informationPlane wave solutions of the Dirac equation
Lecture #3 Spherical spinors Hydrogen-like systems again (Relativistic version) irac energy levels Chapter, pages 48-53, Lectures on Atomic Physics Chapter 5, pages 696-76, Bransden & Joachain,, Quantum
More 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 informationChapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found.
Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. In applying quantum mechanics to 'real' chemical problems, one is usually faced with a Schrödinger
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules CHEM3006P or similar background knowledge is required for this course. This course has two parts: Part 1: Quantum Chemistry techniques for simulations of molecular
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 20, March 8, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer Lecture 20, March 8, 2006 Solved Homework We determined that the two coefficients in our two-gaussian
More information(2 pts) a. What is the time-dependent Schrödinger Equation for a one-dimensional particle in the potential, V (x)?
Part I: Quantum Mechanics: Principles & Models 1. General Concepts: (2 pts) a. What is the time-dependent Schrödinger Equation for a one-dimensional particle in the potential, V (x)? (4 pts) b. How does
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 informationPhysics 401: Quantum Mechanics I Chapter 4
Physics 401: Quantum Mechanics I Chapter 4 Are you here today? A. Yes B. No C. After than midterm? 3-D Schroedinger Equation The ground state energy of the particle in a 3D box is ( 1 2 +1 2 +1 2 ) π2
More 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 informationGroup representation theory and quantum physics
Group representation theory and quantum physics Olivier Pfister April 29, 2003 Abstract This is a basic tutorial on the use of group representation theory in quantum physics, in particular for such systems
More informationChapter 6. Electronic Structure of Atoms. Lecture Presentation. John D. Bookstaver St. Charles Community College Cottleville, MO
Lecture Presentation Chapter 6 John D. Bookstaver St. Charles Community College Cottleville, MO Waves To understand the electronic structure of atoms, one must understand the nature of electromagnetic
More informationAtomic Structure and Atomic Spectra
Atomic Structure and Atomic Spectra Atomic Structure: Hydrogenic Atom Reading: Atkins, Ch. 10 (7 판 Ch. 13) The principles of quantum mechanics internal structure of atoms 1. Hydrogenic atom: one electron
More information1 Commutators (10 pts)
Final Exam Solutions 37A Fall 0 I. Siddiqi / E. Dodds Commutators 0 pts) ) Consider the operator  = Ĵx Ĵ y + ĴyĴx where J i represents the total angular momentum in the ith direction. a) Express both
More 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 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 informationONE AND MANY ELECTRON ATOMS Chapter 15
See Week 8 lecture notes. This is exactly the same as the Hamiltonian for nonrigid rotation. In Week 8 lecture notes it was shown that this is the operator for Lˆ 2, the square of the angular momentum.
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 informationLecture 5 (Sep. 20, 2017)
Lecture 5 8.321 Quantum Theory I, Fall 2017 22 Lecture 5 (Sep. 20, 2017) 5.1 The Position Operator In the last class, we talked about operators with a continuous spectrum. A prime eample is the position
More informationPOSTULATES OF QUANTUM MECHANICS
POSTULATES OF QUANTUM MECHANICS Quantum-mechanical states - In the coordinate representation, the state of a quantum-mechanical system is described by the wave function ψ(q, t) = ψ(q 1,..., q f, t) (in
More informationQUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer
Franz Schwabl QUANTUM MECHANICS Translated by Ronald Kates Second Revised Edition With 122Figures, 16Tables, Numerous Worked Examples, and 126 Problems ff Springer Contents 1. Historical and Experimental
More information(n, l, m l ) 3/2/2016. Quantum Numbers (QN) Plots of Energy Level. Roadmap for Exploring Hydrogen Atom
PHYS 34 Modern Physics Atom III: Angular Momentum and Spin Roadmap for Exploring Hydrogen Atom Today Contents: a) Orbital Angular Momentum and Magnetic Dipole Moment b) Electric Dipole Moment c) Stern
More informationPostulates of Quantum Mechanics
EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 Exercise 1: Stern-Gerlach experiment Postulates of Quantum Mechanics AStern-Gerlach(SG)deviceisabletoseparateparticlesaccordingtotheirspinalonga
More informationI. RADIAL PROBABILITY DISTRIBUTIONS (RPD) FOR S-ORBITALS
5. Lecture Summary #7 Readings for today: Section.0 (.9 in rd ed) Electron Spin, Section. (.0 in rd ed) The Electronic Structure of Hydrogen. Read for Lecture #8: Section. (. in rd ed) Orbital Energies
More informationAnnouncements. Lecture 20 Chapter. 7 QM in 3-dims & Hydrogen Atom. The Radial Part of Schrodinger Equation for Hydrogen Atom
Announcements! HW7 : Chap.7 18, 20, 23, 32, 37, 38, 45, 47, 53, 57, 60! Physics Colloquium: Development in Electron Nuclear Dynamics Theory on Thursday @ 3:40pm! Quiz 2 (average: 9), Quiz 3: 4/19 *** Course
More informationChm 331 Fall 2015, Exercise Set 4 NMR Review Problems
Chm 331 Fall 015, Exercise Set 4 NMR Review Problems Mr. Linck Version.0. Compiled December 1, 015 at 11:04:44 4.1 Diagonal Matrix Elements for the nmr H 0 Find the diagonal matrix elements for H 0 (the
More information5.111 Lecture Summary #6
5.111 Lecture Summary #6 Readings for today: Section 1.9 (1.8 in 3 rd ed) Atomic Orbitals. Read for Lecture #7: Section 1.10 (1.9 in 3 rd ed) Electron Spin, Section 1.11 (1.10 in 3 rd ed) The Electronic
More informationPhysics 342 Lecture 17. Midterm I Recap. Lecture 17. Physics 342 Quantum Mechanics I
Physics 342 Lecture 17 Midterm I Recap Lecture 17 Physics 342 Quantum Mechanics I Monday, March 1th, 28 17.1 Introduction In the context of the first midterm, there are a few points I d like to make about
More informationAngular momentum & spin
Angular momentum & spin January 8, 2002 1 Angular momentum Angular momentum appears as a very important aspect of almost any quantum mechanical system, so we need to briefly review some basic properties
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationIdentical Particles in Quantum Mechanics
Identical Particles in Quantum Mechanics Chapter 20 P. J. Grandinetti Chem. 4300 Nov 17, 2017 P. J. Grandinetti (Chem. 4300) Identical Particles in Quantum Mechanics Nov 17, 2017 1 / 20 Wolfgang Pauli
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Chemistry 3502/4502 Final Exam Part I May 14, 2005 1. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle (e) The
More informationPHY492: Nuclear & Particle Physics. Lecture 5 Angular momentum Nucleon magnetic moments Nuclear models
PHY492: Nuclear & Particle Physics Lecture 5 Angular momentum Nucleon magnetic moments Nuclear models eigenfunctions & eigenvalues: Classical: L = r p; Spherical Harmonics: Orbital angular momentum Orbital
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Advocacy chit Chemistry 350/450 Final Exam Part I May 4, 005. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle
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 informationTime Independent Perturbation Theory Contd.
Time Independent Perturbation Theory Contd. A summary of the machinery for the Perturbation theory: H = H o + H p ; H 0 n >= E n n >; H Ψ n >= E n Ψ n > E n = E n + E n ; E n = < n H p n > + < m H p n
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 informationFYS-6306 QUANTUM THEORY OF MOLECULES AND NANOSTRUCTURES
i FYS-6306 QUANTUM THEORY OF MOLECULES AND NANOSTRUCTURES Credit units: 6 ECTS Lectures: 48 h Tapio Rantala, prof. Tue 10 12 SC203 SG219 8 10 SG312 FirstName.LastName@tut.fi http://www.tut.fi/~trantala/opetus/
More informationChem 110 Practice Midterm 2014
Name Chem 110 Practice Midterm 2014 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Which of the following statements is true? 1) A) Two electrons
More informationTopics for the Qualifying Examination
Topics for the Qualifying Examination Quantum Mechanics I and II 1. Quantum kinematics and dynamics 1.1 Postulates of Quantum Mechanics. 1.2 Configuration space vs. Hilbert space, wave function vs. state
More informationLine spectrum (contd.) Bohr s Planetary Atom
Line spectrum (contd.) Hydrogen shows lines in the visible region of the spectrum (red, blue-green, blue and violet). The wavelengths of these lines can be calculated by an equation proposed by J. J. Balmer:
More informationChapter 6: Electronic Structure of Atoms
Chapter 6: Electronic Structure of Atoms Learning Outcomes: Calculate the wavelength of electromagnetic radiation given its frequency or its frequency given its wavelength. Order the common kinds of radiation
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 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 information8.1 The hydrogen atom solutions
8.1 The hydrogen atom solutions Slides: Video 8.1.1 Separating for the radial equation Text reference: Quantum Mechanics for Scientists and Engineers Section 10.4 (up to Solution of the hydrogen radial
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 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 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 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 informationChemistry 483 Lecture Topics Fall 2009
Chemistry 483 Lecture Topics Fall 2009 Text PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon A. Background (M&S,Chapter 1) Blackbody Radiation Photoelectric effect DeBroglie Wavelength Atomic
More informationLegendre Polynomials and Angular Momentum
University of Connecticut DigitalCommons@UConn Chemistry Education Materials Department of Chemistry August 006 Legendre Polynomials and Angular Momentum Carl W. David University of Connecticut, Carl.David@uconn.edu
More informationQuantum Physics II (8.05) Fall 2002 Assignment 11
Quantum Physics II (8.05) Fall 00 Assignment 11 Readings Most of the reading needed for this problem set was already given on Problem Set 9. The new readings are: Phase shifts are discussed in Cohen-Tannoudji
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 information