4.1 Time evolution of superpositions. Slides: Video Introduction to time evolution of superpositions
|
|
- Rudolph Hubert Peters
- 6 years ago
- Views:
Transcription
1 4.1 Time evolution of superpositions Slides: Video Introduction to time evolution of superpositions
2 Time evolution of superpositions Quantum mechanics for scientists and engineers David Miller
3
4 4.1 Time evolution of superpositions Slides: Video Superposition for the particle in a box Text reference: Quantum Mechanics for Scientists and Engineers Section 3.6 ( Simple linear superposition in an infinite potential well )
5 Time evolution of superpositions Superposition for a particle in a box Quantum mechanics for scientists and engineers David Miller
6 Superposition for a particle in a box Suppose we have an infinitely deep potential well a particle in a box with the particle in a linear superposition for example, with equal parts of the first and second states of the well 1 E1 E2 2 t, expi tsin expi tsin L L L
7 Superposition for a particle in a box Note for each eigenfunction in the superposition it is multiplied by the appropriate complex exponential time-varying function En exp i t This superposition is also normalied 1 E1 E2 2 t, expi tsin expi tsin L L L
8 Superposition for a particle in a box From this superposition 1 E1 E2 2 t, expi tsin expi tsin L L L t, 2 we can multiply it by its complex conjugate to get the probability density sin sin E E 2cos t sin sin L L L L L
9 Superposition for a particle in a box E1 E2 2 exp i tsin exp i tsin L L E1 E2 2 expi tsin expi tsin L L E2E1 E2E1 sin sin sin sin exp i t exp i t L L L L sin sin E E 2cos t sin sin L L L L multiplied by its complex conjugate
10 Superposition for a particle in a box t, 2 Note this probability density sin sin E E 2cos t sin sin L L L L L has a part that is oscillating in time at an angular frequency 21 E2 E1 / 3 E1 / Note also that the absolute energy origin does not matter here for this measurable quantity only the energy difference E E matters 2 1
11 Particle in a box As a reminder here are the first few particlein-a-box energy levels and their associated wavefunctions plotted with the orange dashed lines as horiontal axes Energy n 3 n 2 n 1 E 3 E 2 E 1
12 Superposition The n 1 spatial eigenfunction 1 is plotted here with the bottom of the box as its horiontal axis n 1 Wavefunction 1
13 Superposition For the probability density 2 1 note the different shape Multiplying by the time dependent factor gives E1 1, texp i t 1 The probability densities are the same, t n 1 Probability density 2 1 t 2 1,
14 Superposition Similarly The n 2 spatial eigenfunction 2 is plotted here with the bottom of the box as its horiontal axis n 2 Wavefunction 2
15 Superposition The probability density 2 2 is a positive function Multiplying by the time dependent factor gives E2 2, texp i t 2 The probability densities are the same, t n 2 Probability density 2 2 t 2 2,
16 Superposition An equal superposition of the two oscillates at the angular frequency E E / 3 E / , t, t, t E2 E1 t 1 2 2cos Probability density
17 Superposition An equal superposition of the two oscillates at the angular frequency E E / 3 E / , t, t, t E2 E1 t 1 2 2cos Probability density, t, t 2 1 2
18
19 4.1 Time evolution of superpositions Slides: Video Superposition for the harmonic oscillator Text reference: Quantum Mechanics for Scientists and Engineers Section 3.6 ( Harmonic oscillator example )
20 Time evolution of superpositions Superposition for the harmonic oscillator Quantum mechanics for scientists and engineers David Miller
21 Superpositions and oscillation Quite generally if we make a linear combination of two energy eigenstates with energies E a and E b the resulting probability distribution will oscillate at the (angular) frequency E E / ab a b
22 Superpositions and oscillation So, if we have a superposition wavefunction, exp E E a b r ab t c a i t a cb exp i t b r r then the probability distribution will be 2 2 r, t c r c r ab a a b b Ea Eb t 2 c rc r cos where arg ab caa r c b b r a a b b ab
23 Harmonic oscillator As a reminder here are the first two harmonic energy levels and their associated wavefunctions plotted with the orange dashed lines as horiontal axes Energy 3 /2 /2
24 Superposition The n 0 spatial eigenfunction 0 is plotted here with the bottom of the parabolic well as its horiontal axis n 0 Wavefunction 0
25 Superposition For the probability density 0 2 note the narrower shape Multiplying 0 by the time dependent factor gives E0 0, texp i t 0 The probability densities are the same, t n 0 Probability density 2 0
26 Superposition The n 1 spatial eigenfunction 1 is plotted here with the bottom of the parabolic well as its horiontal axis n 1 Wavefunction 1
27 Superposition For the probability density 2 1 note it is positive Multiplying by the time dependent factor gives E1 1, texp i t 1 The probability densities are the same, t n 1 Probability density 2 1
28 Superposition An equal superposition of the two oscillates at the angular frequency E E, t, t, t 2cos t / Probability density
29 Superposition An equal superposition of the two oscillates at the angular frequency E E, t, t, t 2cos t / Probability density, t, t 2 0 1
30
31 4.1 Time evolution of superpositions Slides: Video The coherent state Text reference: Quantum Mechanics for Scientists and Engineers Section 3.6 ( Coherent state )
32 Time evolution of superpositions The coherent state Quantum mechanics for scientists and engineers David Miller
33 The coherent state The coherent state for a harmonic oscillator of frequency is 1 N, tcnnexpin t n n0 2 where n N expn cnn n! and the n are the harmonic oscillator eigenstates
34 The coherent state Incidentally, note that for the expansion coefficients c Nn n 2 N expn cnn n! This is the Poisson distribution from statistics with mean N and standard deviation N We will make no direct use of this here but in the end it explains, e.g., the Poissonian distribution of photons in a laser beam
35 Coherent state Coherent state oscillations with n0 N, t 1 cnn exp in t n 2 c Nn N n expn n!
36 Coherent state N, t 2 N 1 Coherent state oscillations with N, t n0 1 cnn exp in t n 2 c Nn N n expn n!
37 Coherent state N, t 2 N 3 Coherent state oscillations with N, t n0 1 cnn exp in t n 2 c Nn N n expn n!
38 Coherent state N, t 2 N 10 Coherent state oscillations with N, t n0 1 cnn exp in t n 2 c Nn N n expn n!
39 Coherent state N, t 2 N 100 Coherent state oscillations with N, t n0 1 cnn exp in t n 2 c Nn N n expn n!
40 Finite well superposition Make an equal superposition of the first three states of a finite potential well as in our previous example Because the energies are not rationally related the superposition never repeats Vo 8 E
41 Finite well superposition Make an equal superposition of the first three states of a finite potential well as in our previous example Because the energies are not rationally related the superposition never repeats e.g., in the probability density in time Vo 8 E
42
Quantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechanics for Scientists and Engineers David Miller Time evolution of superpositions Time evolution of superpositions Superposition for a particle in a box Superposition for a particle in a box
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 informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechanics for Scientists and Engineers David Miller Particles in potential wells The finite potential well Insert video here (split screen) Finite potential well Lesson 7 Particles in potential
More information5.1 Uncertainty principle and particle current
5.1 Uncertainty principle and particle current Slides: Video 5.1.1 Momentum, position, and the uncertainty principle Text reference: Quantum Mechanics for Scientists and Engineers Sections 3.1 3.13 Uncertainty
More informationECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline:
ECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline: Time-Dependent Schrödinger Equation Solutions to thetime-dependent Schrödinger Equation Expansion of Energy Eigenstates Things you
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechanics for Scientists and Engineers David Miller The particle in a box The particle in a box Linearity and normalization Linearity and Schrödinger s equation We see that Schrödinger s equation
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 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 informationECE 487 Lecture 5 : Foundations of Quantum Mechanics IV Class Outline:
ECE 487 Lecture 5 : Foundations of Quantum Mechanics IV Class Outline: Linearly Varying Potential Triangular Potential Well Time-Dependent Schrödinger Equation Things you should know when you leave Key
More information2.2 Schrödinger s wave equation
2.2 Schrödinger s wave equation Slides: Video 2.2.1 Schrödinger wave equation introduction Text reference: Quantum Mechanics for Scientists and Engineers Section Chapter 2 introduction Schrödinger s wave
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 informationAngular momentum. Angular momentum operators. Quantum mechanics for scientists and engineers
7.1 Angular momentum Slides: Video 7.1.1 Angular momentum operators Text reference: Quantum Mechanics for Scientists and Engineers Chapter 9 introduction and Section 9.1 (first part) Angular momentum Angular
More informationQuantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras. Lecture - 16 The Quantum Beam Splitter
Quantum Mechanics - I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 16 The Quantum Beam Splitter (Refer Slide Time: 00:07) In an earlier lecture, I had
More information1 Time-Dependent Two-State Systems: Rabi Oscillations
Advanced kinetics Solution 7 April, 16 1 Time-Dependent Two-State Systems: Rabi Oscillations a In order to show how Ĥintt affects a bound state system in first-order time-dependent perturbation theory
More informationCreation and Destruction Operators and Coherent States
Creation and Destruction Operators and Coherent States WKB Method for Ground State Wave Function state harmonic oscillator wave function, We first rewrite the ground < x 0 >= ( π h )1/4 exp( x2 a 2 h )
More information6.2 Unitary and Hermitian operators
6.2 Unitary and Hermitian operators Slides: Video 6.2.1 Using unitary operators Text reference: Quantum Mechanics for Scientists and Engineers Section 4.10 (starting from Changing the representation of
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechanics for Scientists and Engineers David Miller Wavepackets Wavepackets Group velocity Group velocity Consider two waves at different frequencies 1 and 2 and suppose that the wave velocity
More informationQuantum Chemistry Exam 2 Solutions
Chemistry 46 Fall 17 Dr. Jean M. Standard November 8, 17 Name KEY Quantum Chemistry Exam Solutions 1.) ( points) Answer the following questions by selecting the correct answer from the choices provided.
More informationLecture 12: Particle in 1D boxes & Simple Harmonic Oscillator
Lecture 12: Particle in 1D boxes & Simple Harmonic Oscillator U(x) E Dx y(x) x Dx Lecture 12, p 1 Properties of Bound States Several trends exhibited by the particle-in-box states are generic to bound
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 informationFor example, in one dimension if we had two particles in a one-dimensional infinite potential well described by the following two wave functions.
Identical particles In classical physics one can label particles in such a way as to leave the dynamics unaltered or follow the trajectory of the particles say by making a movie with a fast camera. Thus
More informationSo far, we considered quantum static, as all our potentials did not depend on time. Therefore, our time dependence was trivial and always the same:
Lecture 20 Page 1 Lecture #20 L20.P1 Time-dependent perturbation theory So far, we considered quantum static, as all our potentials did not depend on time. Therefore, our time dependence was trivial and
More informationB2.III Revision notes: quantum physics
B.III Revision notes: quantum physics Dr D.M.Lucas, TT 0 These notes give a summary of most of the Quantum part of this course, to complement Prof. Ewart s notes on Atomic Structure, and Prof. Hooker s
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 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 informationLecture-XXVI. Time-Independent Schrodinger Equation
Lecture-XXVI Time-Independent Schrodinger Equation Time Independent Schrodinger Equation: The time-dependent Schrodinger equation: Assume that V is independent of time t. In that case the Schrodinger equation
More informationDepartment of Physics and Astronomy University of Georgia
Department of Physics and Astronomy University of Georgia August 2007 Written Comprehensive Exam Day 1 This is a closed-book, closed-note exam. You may use a calculator, but only for arithmetic functions
More informationCavity QED with Rydberg Atoms Serge Haroche, Collège de France & Ecole Normale Supérieure, Paris
Cavity QED with Rydberg Atoms Serge Haroche, Collège de France & Ecole Normale Supérieure, Paris A three lecture course Goal of lectures Manipulating states of simple quantum systems has become an important
More informationBASICS OF QUANTUM MECHANICS. Reading: QM Course packet Ch 5
BASICS OF QUANTUM MECHANICS 1 Reading: QM Course packet Ch 5 Interesting things happen when electrons are confined to small regions of space (few nm). For one thing, they can behave as if they are in an
More information( ) = 9φ 1, ( ) = 4φ 2.
Chemistry 46 Dr Jean M Standard Homework Problem Set 6 Solutions The Hermitian operator A ˆ is associated with the physical observable A Two of the eigenfunctions of A ˆ are and These eigenfunctions are
More informationCoherent states, beam splitters and photons
Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.
More informationChemistry 432 Problem Set 4 Spring 2018 Solutions
Chemistry 4 Problem Set 4 Spring 18 Solutions 1. V I II III a b c A one-dimensional particle of mass m is confined to move under the influence of the potential x a V V (x) = a < x b b x c elsewhere and
More informationPhysics 742 Graduate Quantum Mechanics 2 Solutions to Second Exam, Spring 2017
Physics 74 Graduate Quantum Mechanics Solutions to Second Exam Spring 17 The points for each question are marked. Each question is worth points. Some possibly useful formulas appear at the end of the test.
More informationReading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom.
Chemistry 356 017: Problem set No. 6; Reading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom. The H atom involves spherical coordinates and angular momentum, which leads to the shapes
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 6.2 6.3 6.4 6.5 6.6 6.7 The Schrödinger Wave Equation Expectation Values Infinite Square-Well Potential Finite Square-Well Potential Three-Dimensional Infinite-Potential
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 informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More information16. GAUGE THEORY AND THE CREATION OF PHOTONS
6. GAUGE THEORY AD THE CREATIO OF PHOTOS In the previous chapter the existence of a gauge theory allowed the electromagnetic field to be described in an invariant manner. Although the existence of this
More informationShort Course in Quantum Information Lecture 2
Short Course in Quantum Information Lecture Formal Structure of Quantum Mechanics Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html Syllabus Lecture
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechanics for Scientists and Engineers David Miller Background mathematics 5 Sum, factorial and product notations Summation notation If we want to add a set of numbers a 1, a 2, a 3, and a 4, we
More informationEE 223 Applied Quantum Mechanics 2 Winter 2016
EE 223 Applied Quantum Mechanics 2 Winter 2016 Syllabus and Textbook references Version as of 12/29/15 subject to revisions and changes All the in-class sessions, paper problem sets and assignments, and
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More informationPhysics 217 Problem Set 1 Due: Friday, Aug 29th, 2008
Problem Set 1 Due: Friday, Aug 29th, 2008 Course page: http://www.physics.wustl.edu/~alford/p217/ Review of complex numbers. See appendix K of the textbook. 1. Consider complex numbers z = 1.5 + 0.5i and
More informationElectron in a Box. A wave packet in a square well (an electron in a box) changing with time.
Electron in a Box A wave packet in a square well (an electron in a box) changing with time. Last Time: Light Wave model: Interference pattern is in terms of wave intensity Photon model: Interference in
More informationPhysics 221A Fall 2017 Notes 27 The Variational Method
Copyright c 2018 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 27 The Variational Method 1. Introduction Very few realistic problems in quantum mechanics are exactly solvable, so approximation methods
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 informationE = 1 c. where ρ is the charge density. The last equality means we can solve for φ in the usual way using Coulomb s Law:
Photons In this section, we want to quantize the vector potential. We will work in the Coulomb gauge, where A = 0. In gaussian units, the fields are defined as follows, E = 1 c A t φ, B = A, where A and
More informationThe Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case:
The Schrodinger Equation and Postulates Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about
More informationGrading. Class attendance: (1 point/class) x 9 classes = 9 points maximum Homework: (10 points/hw) x 3 HW = 30 points maximum
Grading Class attendance: (1 point/class) x 9 classes = 9 points maximum Homework: (10 points/hw) x 3 HW = 30 points maximum Maximum total = 39 points Pass if total >= 20 points Fail if total
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 informationBuilding Blocks for Quantum Computing Part IV. Design and Construction of the Trapped Ion Quantum Computer (TIQC)
Building Blocks for Quantum Computing Part IV Design and Construction of the Trapped Ion Quantum Computer (TIQC) CSC801 Seminar on Quantum Computing Spring 2018 1 Goal Is To Understand The Principles And
More information2 Quantization of the Electromagnetic Field
2 Quantization of the Electromagnetic Field 2.1 Basics Starting point of the quantization of the electromagnetic field are Maxwell s equations in the vacuum (source free): where B = µ 0 H, D = ε 0 E, µ
More informationQualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!
Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics
More information7.3 The hydrogen atom
7.3 The hydrogen atom Slides: Video 7.3.1 Multiple particle wavefunctions Text reference: Quantum Mechanics for Scientists and Engineers Chapter 10 introduction and Section 10.1 The hydrogen atom Multiple
More informationFOURIER ANALYSIS. (a) Fourier Series
(a) Fourier Series FOURIER ANAYSIS (b) Fourier Transforms Useful books: 1. Advanced Mathematics for Engineers and Scientists, Schaum s Outline Series, M. R. Spiegel - The course text. We follow their notation
More information09a. Collapse. Recall: There are two ways a quantum state can change: 1. In absence of measurement, states change via Schrödinger dynamics:
09a. Collapse Recall: There are two ways a quantum state can change:. In absence of measurement, states change via Schrödinger dynamics: ψ(t )!!! ψ(t 2 ) Schrödinger evolution 2. In presence of measurement,
More informationInternational Journal of Scientific and Engineering Research, Volume 7, Issue 9,September-2016 ISSN
1574 International Journal of Scientific and Engineering Research, Volume 7, Issue 9,September-216 Energy and Wave-function correction for a quantum system after a small perturbation Samuel Mulugeta Bantikum
More informationOften, it is useful to know the number of states per unit energy, called the density of states (or DOS), D. In any number of dimensions, we can use
Densit of States The hamiltonian for a electron in an isotropic electronic band of a material depends on the band-edge energ of E and an effective mass m ˆ pˆ H E m In an number of dimensions, the dispersion
More informationQuantum Wells The Shooting Method (application to the simple harmonic oscillator and other problems of a particle in a potential)
quantumwellsho.nb Quantum Wells The Shooting Method (application to the simple harmonic oscillator and other problems of a particle in a potential) Introduction In the previous handout we found the eigenvalues
More informationPreliminary Examination - Day 2 August 16, 2013
UNL - Department of Physics and Astronomy Preliminary Examination - Day August 16, 13 This test covers the topics of Quantum Mechanics (Topic 1) and Thermodynamics and Statistical Mechanics (Topic ). Each
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 information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More information2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements
1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical
More informationFemtochemistry. Mark D. Ellison Department of Chemistry Wittenberg University Springfield, OH
Femtochemistry by Mark D. Ellison Department of Chemistry Wittenberg University Springfield, OH 45501 mellison@wittenberg.edu Copyright Mark D. Ellison, 2002. All rights reserved. You are welcome to use
More information4 Power Series Solutions: Frobenius Method
4 Power Series Solutions: Frobenius Method Now the ODE adventure takes us to series solutions for ODEs, a technique A & W, that is often viable, valuable and informative. These can be readily applied Sec.
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 informationPHY4905: Intro to Solid State Physics. Tight-binding Model
PHY495: Intro to Solid State Physics Tight-binding Model D. L. Maslov Department of Physics, University of Florida REMINDER: TUNNELING Quantum particles can penetrate into regions where classical motion
More informationPH425 Spins Homework 5 Due 4 pm. particles is prepared in the state: + + i 3 13
PH45 Spins Homework 5 Due 10/5/18 @ 4 pm REQUIRED: 1. A beam of spin- 1 particles is prepared in the state: ψ + + i 1 1 (a) What are the possible results of a measurement of the spin component S z, and
More informationThe Two Level Atom. E e. E g. { } + r. H A { e e # g g. cos"t{ e g + g e } " = q e r g
E e = h" 0 The Two Level Atom h" e h" h" 0 E g = " h# 0 g H A = h" 0 { e e # g g } r " = q e r g { } + r $ E r cos"t{ e g + g e } The Two Level Atom E e = µ bb 0 h" h" " r B = B 0ˆ z r B = B " cos#t x
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 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 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 informationNotes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14
Notes on excitation of an atom or molecule by an electromagnetic wave field. F. Lanni / 11feb'12 / rev9sept'14 Because the wavelength of light (400-700nm) is much greater than the diameter of an atom (0.07-0.35
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 informationWaves, the Wave Equation, and Phase Velocity. We ll start with optics. The one-dimensional wave equation. What is a wave? Optional optics texts: f(x)
We ll start with optics Optional optics texts: Waves, the Wave Equation, and Phase Velocity What is a wave? f(x) f(x-) f(x-) f(x-3) Eugene Hecht, Optics, 4th ed. J.F. James, A Student's Guide to Fourier
More informationQuantum Mechanics: The Hydrogen Atom
Quantum Mechanics: The Hydrogen Atom 4th April 9 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen
More informationd)p () = A = Z x x p b Particle in a Box 3. electron is in a -dimensional well with innitely high sides and width An Which of following statements mus
4 Spring 99 Problem Set Optional Problems Physics April, 999 Handout a) Show that (x; t) =Ae i(kx,!t) satises wave equation for a string: (x; t) @ = v @ (x; t) @t @x Show that same wave function (x; t)
More informationPhysics of Semiconductors (Problems for report)
Physics of Semiconductors (Problems for report) Shingo Katsumoto Institute for Solid State Physics, University of Tokyo July, 0 Choose two from the following eight problems and solve them. I. Fundamentals
More informationProblem Set 6: Workbook on Operators, and Dirac Notation Solution
Moern Physics: Home work 5 Due ate: 0 March. 014 Problem Set 6: Workbook on Operators, an Dirac Notation Solution 1. nswer 1: a The cat is being escribe by the state, ψ >= ea > If we try to observe it
More information= X = X ( ~) } ( ) ( ) On the other hand, when the Hamiltonian acts on ( ) one finds that
6. A general normalized solution to Schrödinger s equation of motion for a particle moving in a time-independent potential is of the form ( ) = P } where the and () are, respectively, eigenvalues and normalized
More informationQuantum Theory of Light and Matter
Quantum Theory of Light and Matter Field quantization Paul Eastham February 23, 2012 Quantization in an electromagnetic cavity Quantum theory of an electromagnetic cavity e.g. planar conducting cavity,
More informationCHEM 301: Homework assignment #5
CHEM 30: Homework assignment #5 Solutions. A point mass rotates in a circle with l =. Calculate the magnitude of its angular momentum and all possible projections of the angular momentum on the z-axis.
More informationExternal (differential) quantum efficiency Number of additional photons emitted / number of additional electrons injected
Semiconductor Lasers Comparison with LEDs The light emitted by a laser is generally more directional, more intense and has a narrower frequency distribution than light from an LED. The external efficiency
More informationIntegrated optical circuits for classical and quantum light. Part 2: Integrated quantum optics. Alexander Szameit
Integrated optical circuits for classical and quantum light Part 2: Integrated quantum optics Alexander Szameit Alexander Szameit alexander.szameit@uni-jena.de +49(0)3641 947985 +49(0)3641 947991 Outline
More information1 Expansion in orthogonal functions
Notes "orthogonal" January 9 Expansion in orthogonal functions To obtain a more useful form of the Green s function, we ll want to expand in orthogonal functions that are (relatively) easy to integrate.
More informationChemistry 532 Practice Final Exam Fall 2012 Solutions
Chemistry 53 Practice Final Exam Fall Solutions x e ax dx π a 3/ ; π sin 3 xdx 4 3 π cos nx dx π; sin θ cos θ + K x n e ax dx n! a n+ ; r r r r ˆL h r ˆL z h i φ ˆL x i hsin φ + cot θ cos φ θ φ ) ˆLy i
More informationLINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. General framework
Differential Equations Grinshpan LINEAR EQUATIONS OF HIGHER ORDER. EXAMPLES. We consider linear ODE of order n: General framework (1) x (n) (t) + P n 1 (t)x (n 1) (t) + + P 1 (t)x (t) + P 0 (t)x(t) = 0
More informationQuantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie
Quantum Mechanics II Lecture (www.sp.phy.cam.ac.u/~dar/pdf) David Ritchie Michaelmas. So far we have found solutions to Section 4:Transitions Ĥ ψ Eψ Solutions stationary states time dependence with time
More information4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam
Lecture Notes for Quantum Physics II & III 8.05 & 8.059 Academic Year 1996/1997 4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam c D. Stelitano 1996 As an example of a two-state system
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationAppendix A. The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System
Appendix A The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System Real quantum mechanical systems have the tendency to become mathematically
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 informationStrongly correlated systems in atomic and condensed matter physics. Lecture notes for Physics 284 by Eugene Demler Harvard University
Strongly correlated systems in atomic and condensed matter physics Lecture notes for Physics 284 by Eugene Demler Harvard University September 18, 2014 2 Chapter 5 Atoms in optical lattices Optical lattices
More informationTime-Independent Perturbation Theory
4 Phys46.nb Time-Independent Perturbation Theory.. Overview... General question Assuming that we have a Hamiltonian, H = H + λ H (.) where λ is a very small real number. The eigenstates of the Hamiltonian
More 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 informationPhysics for Scientists & Engineers 2
Electromagnetic Oscillations Physics for Scientists & Engineers Spring Semester 005 Lecture 8! We have been working with circuits that have a constant current a current that increases to a constant current
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
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 informationPHYS-454 The position and momentum representations
PHYS-454 The position and momentum representations 1 Τhe continuous spectrum-a n So far we have seen problems where the involved operators have a discrete spectrum of eigenfunctions and eigenvalues.! n
More informationDifferential calculus. Background mathematics review
Differential calculus Background mathematics review David Miller Differential calculus First derivative Background mathematics review David Miller First derivative For some function y The (first) derivative
More information