Ammonia molecule, from Chapter 9 of the Feynman Lectures, Vol 3. Example of a 2-state system, with a small energy difference between the symmetric
|
|
- Julie Boyd
- 6 years ago
- Views:
Transcription
1 Ammonia molecule, from Chapter 9 of the Feynman Lectures, Vol 3. Eample of a -state system, with a small energy difference between the symmetric and antisymmetric combinations of states and. This energy difference is eploited by the Ammonia Maser.
2 Transmission and Reflection Coefficients for Barrier Penetration m 3 3 π a. V C m V a.75 The graph depends only on this combination of parameters. T( ) 4 sinh C ( ) ( ) MathCad will evaluate the hyperbolic functions properly when the argument becomes imaginary (when the energy is above the barrier height). R( ) T( ) In these epressions, is the ratio E/V Transmission and Reflection Coefficients TE ( ) RE ( ) E Energy/Potential Note that the transmission is % when the width of the barrier is an integer multiple / wavelength, such that the wave reflected from the second step interferes destructively with that reflected from the first, completely cancelling the reflected wave. There is a small gap at E/V= in the plot, due to the fact that MathCad cannot calculate numerically the limit there correctly, but you can see by eye that the curve appears to be continuous and smooth there. In fact, the limit is well defined.
3 Barrier Penetration (tunneling) m 3 π Units: time in ps, distance in nm, mass in kg a. V E.5V Barrier width and potential, and the energy k me 3.48 m V E κ κ a.6 Wave number outside of the barrier region Decay coefficient within the barrier region V( ) if( if ( a )) Potential shape, for plotting Wave eigenfunction for a plane wave incident from the left with unit amplitude: e ika κ k sinh( κa) F( kκ ) κ k Bkκ ( ) k κ sinh( κa) i κkcosh( κa) cosh( κa) i sinh( κa) κk C( kκ ) ( κ ik ) ( κ ik ) B( kκ ) κ Dk ( κ ) ( κ ik ) ( κ ik ) B( kκ ) κ ψ( kκ ) if e ik Bkκ ( ) e ik if afk ( κ ) e ik Ckκ ( ) e κ P ( ) ψ k κ ψ k κ.9 R Bk κ.7 T Fk κ R T Probability density Check that they add up to unity. Reflection coefficient Transmission coefficient ( ) e κ Dkκ 3 (not normalized) P ( ) V ( ) E V....
4 Now, use the eigenfunctions calculated above to simulate a wave packet incident on the barrier. The behavior is rather different in this case, because there is a significant range of wave numbers included. In fact, the upper end of the range included below etends above the well, so the transmission is not entirely from tunneling. On the other hand, the lower end of the range has an eponentially lower tunneling probability. a σ a Initial position and gaussian width of the wave packet mv Wave number for an energy right at the top of the potential barrier σ k 5 k σ 3.48 k v t m v Width and central value in k space (momentum space) Packet speed, and elapsed time when it hits the barrier. ϕ( k) πσ k 4 e kk 4σ k i kk e This makes a gaussian distribution of wave numbers designed to make a gaussian-shaped wave packet centered around at time and moving to the right with speed v. Now make a wave function by summing (integrating) eigenfunctions for all values of k within 4-sigma of the peak. The integral is done numerically, which is rather time consuming but within the capability of a modern PC. κ( k) mv k ω( k) k ω k m Ψ( t) k 4σ k ϕ( k) ψ( k κ ( k) ) e iω( k) t dk π k 4σ k ρ( t) Ψ( t)
5 t FRAMEt 4 ρ( t ) V ( ) 6a E 6aV 5 t ρ t V ( ) 6a E 6aV
6 Passing over a Potential Barrier (E>V) m 3 π Units: time in ps, distance in nm, mass in kg a.8 E V.7E Barrier width and potential, and the energy me k k 6.96 m E V k k a Wave number outside of the barrier region V( ) if( if ( a )) Potential shape, for plotting Wave number in the barrier region Wave eigenfunction for a plane wave incident from the left with unit amplitude: F kk C kk if ψ k k e ika k k Bkk cosk a sink i k k a k k k kbkk k Dkk k k k k sin k a sin k a k k k kbkk k i kk cos k a e ik Bkk e ik if afkk e ik ik Ckk ik e Dkk e P ( ) ψ k k ψ k k.69 R Bk k.73 T Fk k Reflection coefficient Transmission coefficient R T Check that they add up to unity.
7 6 (not normalized) P ( ) V ( ) 4.. Now, use the eigenfunctions calculated above to simulate a wave packet incident on the barrier. The behavior is rather different in this case, because there is a significant range of wave numbers included, including some corresponding to energies below the top of the barrier. a a σ 5 mv σ k 3.5 k σ 6.96 Initial position and gaussian width of the wave packet. In this case, the packet is smaller than the barrier thickness, so we epect to see two distinct reflections, rather than a superposition of two reflections. Wave number for an energy right at the top of the potential barrier Width and central value in k space (momentum space) k v t m.53 7 v Packet speed, and elapsed time when it hits the barrier. ϕ( k) πσ k 4 e kk 4σ k i kk e This makes a gaussian distribution of wave numbers designed to make a gaussian-shaped wave packet centered around at time and moving to the right with speed v. ω( k) k m κ( k) k mv κ k ω k Frequency and the wave number in the barrier region. Ψ( t) k 4σ k ϕ( k) ψ( k κ ( k) ) e iω( k) t dk π k 4σ k The wave function. The limits are not set to infinity, in order to save some calculation time.
8 N i N ( 8a ) 4a i i N Evaluate the wave function at a fied grid of N values for plotting. If MathCad is told to plot the function itself, instead of the array of function values, then it tries to evaluate the function too many times and takes forever. t FRAMEt 4 Time at which to evaluate the wave function, set up for making an animation. ρ Ψ t v 5 V 5E e i i i i i V Arrays of values to plot 3 ρ v e t..
9 t t.53 7 A later time at which to evaluate the wave function ρ Ψ t v 5 V 5E e i i i i i V Arrays of values to plot 3 ρ v e t
Scattering in One Dimension
Chapter 4 The door beckoned, so he pushed through it. Even the street was better lit. He didn t know how, he just knew the patron seated at the corner table was his man. Spectacles, unkept hair, old sweater,
More informationPhysics 505 Homework No. 4 Solutions S4-1
Physics 505 Homework No 4 s S4- From Prelims, January 2, 2007 Electron with effective mass An electron is moving in one dimension in a potential V (x) = 0 for x > 0 and V (x) = V 0 > 0 for x < 0 The region
More informationIntroduction to Quantum Mechanics
Introduction to Quantum Mechanics INEL 5209 - Solid State Devices - Spring 2012 Manuel Toledo January 23, 2012 Manuel Toledo Intro to QM 1/ 26 Outline 1 Review Time dependent Schrödinger equation Momentum
More informationLecture 15: Time-Dependent QM & Tunneling Review and Examples, Ammonia Maser
ecture 15: Time-Dependent QM & Tunneling Review and Examples, Ammonia Maser ψ(x,t=0) 2 U(x) 0 x ψ(x,t 0 ) 2 x U 0 0 E x 0 x ecture 15, p.1 Special (Optional) ecture Quantum Information One of the most
More informationModern Physics. Unit 3: Operators, Tunneling and Wave Packets Lecture 3.3: The Momentum Operator
Modern Physics Unit 3: Operators, Tunneling and Wave Packets Lecture 3.3: The Momentum Operator Ron Reifenberger Professor of Physics Purdue University 1 There are many operators in QM H Ψ= EΨ, or ˆop
More informationLecture 5. Potentials
Lecture 5 Potentials 51 52 LECTURE 5. POTENTIALS 5.1 Potentials In this lecture we will solve Schrödinger s equation for some simple one-dimensional potentials, and discuss the physical interpretation
More information3-2 Classical E&M Light Waves In classical electromagnetic theory we express the electric and magnetic fields in a complex form. eg.
Chapter-3 Wave Particle Duality 3- Classical EM Light Waves In classical electromagnetic theory we epress the electric and magnetic fields in a comple form. eg. Plane Wave E! B Poynting Vector! S = 1 µ
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 informationSection 4: Harmonic Oscillator and Free Particles Solutions
Physics 143a: Quantum Mechanics I Section 4: Harmonic Oscillator and Free Particles Solutions Spring 015, Harvard Here is a summary of the most important points from the recent lectures, relevant for either
More informationPeter Hertel. University of Osnabrück, Germany. Lecture presented at APS, Nankai University, China.
crystal The Hopping University of Osnabrück, Germany Lecture presented at APS, Nankai University, China http://www.home.uni-osnabrueck.de/phertel Spring 2011 States and amplitudes crystal The system has
More informationOpinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6. Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- 6.6 Simple Harmonic
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 informationPHYS Concept Tests Fall 2009
PHYS 30 Concept Tests Fall 009 In classical mechanics, given the state (i.e. position and velocity) of a particle at a certain time instant, the state of the particle at a later time A) cannot be determined
More informationQuantum Mechanical Tunneling
Chemistry 460 all 07 Dr Jean M Standard September 8, 07 Quantum Mechanical Tunneling Definition of Tunneling Tunneling is defined to be penetration of the wavefunction into a classically forbidden region
More informationPHYS 3313 Section 001 Lecture # 22
PHYS 3313 Section 001 Lecture # 22 Dr. Barry Spurlock Simple Harmonic Oscillator Barriers and Tunneling Alpha Particle Decay Schrodinger Equation on Hydrogen Atom Solutions for Schrodinger Equation for
More informationProbability, Expectation Values, and Uncertainties
Chapter 5 Probability, Epectation Values, and Uncertainties As indicated earlier, one of the remarkable features of the physical world is that randomness is incarnate, irreducible. This is mirrored in
More informationThere is light at the end of the tunnel. -- proverb. The light at the end of the tunnel is just the light of an oncoming train. --R.
A vast time bubble has been projected into the future to the precise moment of the end of the universe. This is, of course, impossible. --D. Adams, The Hitchhiker s Guide to the Galaxy There is light at
More informationPHYSICS 4750 Physics of Modern Materials Chapter 5: The Band Theory of Solids
PHYSICS 4750 Physics of Modern Materials Chapter 5: The Band Theory of Solids 1. Introduction We have seen that when the electrons in two hydrogen atoms interact, their energy levels will split, i.e.,
More informationOn Resonant Tunnelling in the Biased Double Delta-Barrier
Vol. 116 (2009) ACTA PHYSICA POLONICA A No. 6 On Resonant Tunnelling in the Biased Double Delta-Barrier I. Yanetka Department of Physics, Faculty of Civil Engineering, Slovak University of Technology Radlinského
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 informationThe Simple Harmonic Oscillator
The Simple Harmonic Oscillator Michael Fowler, University of Virginia Einstein s Solution of the Specific Heat Puzzle The simple harmonic oscillator, a nonrelativistic particle in a potential ½C, is a
More informationRelevant self-assessment exercises: [LIST SELF-ASSESSMENT EXERCISES HERE]
Chapter 5 Fourier Analysis of Finite Difference Methods In this lecture, we determine the stability of PDE discretizations using Fourier analysis. First, we begin with Fourier analysis of PDE s, and then
More informationLecture 10: The Schrödinger Equation Lecture 10, p 1
Lecture 10: The Schrödinger Equation Lecture 10, p 1 Overview Probability distributions Schrödinger s Equation Particle in a Bo Matter waves in an infinite square well Quantized energy levels y() U= n=1
More informationPhysics 43 Chapter 41 Homework #11 Key
Physics 43 Chapter 4 Homework # Key π sin. A particle in an infinitely deep square well has a wave function given by ( ) for and zero otherwise. Determine the epectation value of. Determine the probability
More informationMATH 1010E University Mathematics Lecture Notes (week 8) Martin Li
MATH 1010E University Mathematics Lecture Notes (week 8) Martin Li 1 L Hospital s Rule Another useful application of mean value theorems is L Hospital s Rule. It helps us to evaluate its of indeterminate
More informationContents. 1 Solutions to Chapter 1 Exercises 3. 2 Solutions to Chapter 2 Exercises Solutions to Chapter 3 Exercises 57
Contents 1 Solutions to Chapter 1 Exercises 3 Solutions to Chapter Exercises 43 3 Solutions to Chapter 3 Exercises 57 4 Solutions to Chapter 4 Exercises 77 5 Solutions to Chapter 5 Exercises 89 6 Solutions
More informationSTEP Support Programme. Pure STEP 3 Solutions
STEP Support Programme Pure STEP 3 Solutions S3 Q6 Preparation Completing the square on gives + + y, so the centre is at, and the radius is. First draw a sketch of y 4 3. This has roots at and, and you
More informationCHM 671. Homework set # 4. 2) Do problems 2.3, 2.4, 2.8, 2.9, 2.10, 2.12, 2.15 and 2.19 in the book.
CHM 67 Homework set # 4 Due: Thursday, September 28 th ) Read Chapter 2 in the 4 th edition Atkins & Friedman's Molecular Quantum Mechanics book. 2) Do problems 2.3, 2.4, 2.8, 2.9, 2.0, 2.2, 2.5 and 2.9
More informationLecture 10: The Schrödinger Equation. Lecture 10, p 2
Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that
More informationChapter 4 (Lecture 6-7) Schrodinger equation for some simple systems Table: List of various one dimensional potentials System Physical correspondence
V, E, Chapter (Lecture 6-7) Schrodinger equation for some simple systems Table: List of various one dimensional potentials System Physical correspondence Potential Total Energies and Probability density
More informationThe Schrödinger Equation in One Dimension
The Schrödinger Equation in One Dimension Introduction We have defined a comple wave function Ψ(, t) for a particle and interpreted it such that Ψ ( r, t d gives the probability that the particle is at
More informationLecture 10: The Schrödinger Equation. Lecture 10, p 2
Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that
More informationLecture 7. 1 Wavepackets and Uncertainty 1. 2 Wavepacket Shape Changes 4. 3 Time evolution of a free wave packet 6. 1 Φ(k)e ikx dk. (1.
Lecture 7 B. Zwiebach February 8, 06 Contents Wavepackets and Uncertainty Wavepacket Shape Changes 4 3 Time evolution of a free wave packet 6 Wavepackets and Uncertainty A wavepacket is a superposition
More informationApplied Nuclear Physics Homework #2
22.101 Applied Nuclear Physics Homework #2 Author: Lulu Li Professor: Bilge Yildiz, Paola Cappellaro, Ju Li, Sidney Yip Oct. 7, 2011 2 1. Answers: Refer to p16-17 on Krane, or 2.35 in Griffith. (a) x
More informationWave nature of particles
Wave nature of particles We have thus far developed a model of atomic structure based on the particle nature of matter: Atoms have a dense nucleus of positive charge with electrons orbiting the nucleus
More informationQM I Exercise Sheet 2
QM I Exercise Sheet 2 D. Müller, Y. Ulrich http://www.physik.uzh.ch/de/lehre/phy33/hs207.html HS 7 Prof. A. Signer Issued: 3.0.207 Due: 0./2.0.207 Exercise : Finite Square Well (5 Pts.) Consider a particle
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 information8.04 Quantum Physics Lecture XV One-dimensional potentials: potential step Figure I: Potential step of height V 0. The particle is incident from the left with energy E. We analyze a time independent situation
More informationLecture 21 Matter acts like waves!
Particles Act Like Waves! De Broglie s Matter Waves λ = h / p Schrodinger s Equation Announcements Schedule: Today: de Broglie and matter waves, Schrodinger s Equation March Ch. 16, Lightman Ch. 4 Net
More informationTime Evolution in Diffusion and Quantum Mechanics. Paul Hughes & Daniel Martin
Time Evolution in Diffusion and Quantum Mechanics Paul Hughes & Daniel Martin April 29, 2005 Abstract The Diffusion and Time dependent Schrödinger equations were solved using both a Fourier based method
More informationTraversal time in periodically loaded waveguides
Z. Phys. B 100, 595 599 (1996) Traversal time in periodically loaded waveguides E. Cuevas, V. Gasparian, M. Ortun o, J. Ruiz Departamento de Física, Universidad de Murcia, E-30080 Murcia, Spain Department
More informationXI. INTRODUCTION TO QUANTUM MECHANICS. C. Cohen-Tannoudji et al., Quantum Mechanics I, Wiley. Outline: Electromagnetic waves and photons
XI. INTRODUCTION TO QUANTUM MECHANICS C. Cohen-Tannoudji et al., Quantum Mechanics I, Wiley. Outline: Electromagnetic waves and photons Material particles and matter waves Quantum description of a particle:
More informationQuantum Physics Lecture 9
Quantum Physics Lecture 9 Potential barriers and tunnelling Examples E < U o Scanning Tunelling Microscope E > U o Ramsauer-Townsend Effect Angular Momentum - Orbital - Spin Pauli exclusion principle potential
More informationQuantum Mechanical Tunneling
The square barrier: Quantum Mechanical Tunneling Behaviour of a classical ball rolling towards a hill (potential barrier): If the ball has energy E less than the potential energy barrier (U=mgy), then
More informationPHYS 3220 PhET Quantum Tunneling Tutorial
PHYS 3220 PhET Quantum Tunneling Tutorial Part I: Mathematical Introduction Recall that the Schrödinger Equation is i Ψ(x,t) t = ĤΨ(x, t). Usually this is solved by first assuming that Ψ(x, t) = ψ(x)φ(t),
More informationFinal Exam: Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall.
Final Exam: Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Chapter 38 Quantum Mechanics Units of Chapter 38 38-1 Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the
More informationLøsningsforslag Eksamen 18. desember 2003 TFY4250 Atom- og molekylfysikk og FY2045 Innføring i kvantemekanikk
Eksamen TFY450 18. desember 003 - løsningsforslag 1 Oppgave 1 Løsningsforslag Eksamen 18. desember 003 TFY450 Atom- og molekylfysikk og FY045 Innføring i kvantemekanikk a. With Ĥ = ˆK + V = h + V (x),
More informationQMI PRELIM Problem 1. All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work.
QMI PRELIM 013 All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work. Problem 1 L = r p, p = i h ( ) (a) Show that L z = i h y x ; (cyclic
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationSOLUTIONS for Homework #2. 1. The given wave function can be normalized to the total probability equal to 1, ψ(x) = Ne λ x.
SOLUTIONS for Homework #. The given wave function can be normalized to the total probability equal to, ψ(x) = Ne λ x. () To get we choose dx ψ(x) = N dx e λx =, () 0 N = λ. (3) This state corresponds to
More informationSolutions to Problem Sheet for Week 6
THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Solutions to Problem Sheet for Week 6 MATH90: Differential Calculus (Advanced) Semester, 07 Web Page: sydney.edu.au/science/maths/u/ug/jm/math90/
More informationExplanations of quantum animations Sohrab Ismail-Beigi April 22, 2009
Explanations of quantum animations Sohrab Ismail-Beigi April 22, 2009 I ve produced a set of animations showing the time evolution of various wave functions in various potentials according to the Schrödinger
More informationModule 40: Tunneling Lecture 40: Step potentials
Module 40: Tunneling Lecture 40: Step potentials V E I II III 0 x a Figure 40.1: A particle of energy E is incident on a step potential of hight V > E as shown in Figure 40.1. The step potential extends
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 informationSuperposition of electromagnetic waves
Superposition of electromagnetic waves February 9, So far we have looked at properties of monochromatic plane waves. A more complete picture is found by looking at superpositions of many frequencies. Many
More information1. The infinite square well
PHY3011 Wells and Barriers page 1 of 17 1. The infinite square well First we will revise the infinite square well which you did at level 2. Instead of the well extending from 0 to a, in all of the following
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 informationModel Problems 09 - Ch.14 - Engel/ Particle in box - all texts. Consider E-M wave 1st wave: E 0 e i(kx ωt) = E 0 [cos (kx - ωt) i sin (kx - ωt)]
VI 15 Model Problems 09 - Ch.14 - Engel/ Particle in box - all texts Consider E-M wave 1st wave: E 0 e i(kx ωt) = E 0 [cos (kx - ωt) i sin (kx - ωt)] magnitude: k = π/λ ω = πc/λ =πν ν = c/λ moves in space
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 informationName Solutions to Test 3 November 7, 2018
Name Solutions to Test November 7 8 This test consists of three parts. Please note that in parts II and III you can skip one question of those offered. Some possibly useful formulas can be found below.
More informationc. Better work with components of slowness vector s (or wave + k z 2 = k 2 = (ωs) 2 = ω 2 /c 2. k=(kx,k z )
.50 Introduction to seismology /3/05 sophie michelet Today s class: ) Eikonal equation (basis of ray theory) ) Boundary conditions (Stein s book.3.0) 3) Snell s law Some remarks on what we discussed last
More informationMore on waves + uncertainty principle
More on waves + uncertainty principle ** No class Fri. Oct. 18! The 2 nd midterm will be Nov. 2 in same place as last midterm, namely Humanities at 7:30pm. Welcome on Columbus day Christopher Columbus
More informationQUANTUM MECHANICS Intro to Basic Features
PCES 4.21 QUANTUM MECHANICS Intro to Basic Features 1. QUANTUM INTERFERENCE & QUANTUM PATHS Rather than explain the rules of quantum mechanics as they were devised, we first look at a more modern formulation
More information1 Orders of Magnitude
Quantum Mechanics M.T. 00 J.F. Wheater Problems These problems cover all the material we will be studying in the lectures this term. The Synopsis tells you which problems are associated with which lectures.
More informationlecture 7: Trigonometric Interpolation
lecture : Trigonometric Interpolation 9 Trigonometric interpolation for periodic functions Thus far all our interpolation schemes have been based on polynomials However, if the function f is periodic,
More informationTunneling via a barrier faster than light
Tunneling via a barrier faster than light Submitted by: Evgeniy Kogan Numerous theories contradict to each other in their predictions for the tunneling time. 1 The Wigner time delay Consider particle which
More informationif trap wave like violin string tied down at end standing wave
VI 15 Model Problems 9.5 Atkins / Particle in box all texts onsider E-M wave 1st wave: E 0 e i(kx ωt) = E 0 [cos (kx - ωt) i sin (kx - ωt)] magnitude: k = π/λ ω = πc/λ =πν ν = c/λ moves in space and time
More informationFinal Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall.
Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Summary of Chapter 38 In Quantum Mechanics particles are represented by wave functions Ψ. The absolute square of the wave function Ψ 2
More informationx(t+ δt) - x(t) = slope δt t+δt
Techniques of Physics Worksheet 2 Classical Vibrations and Waves Introduction You will have encountered many different examples of wave phenomena in your courses and should be familiar with most of the
More informationECE 6341 Spring 2016 HW 2
ECE 6341 Spring 216 HW 2 Assigned problems: 1-6 9-11 13-15 1) Assume that a TEN models a layered structure where the direction (the direction perpendicular to the layers) is the direction that the transmission
More informationModern physics. 4. Barriers and wells. Lectures in Physics, summer
Modern physics 4. Barriers and wells Lectures in Physics, summer 016 1 Outline 4.1. Particle motion in the presence of a potential barrier 4.. Wave functions in the presence of a potential barrier 4.3.
More informationWave Properties of Particles Louis debroglie:
Wave Properties of Particles Louis debroglie: If light is both a wave and a particle, why not electrons? In 194 Louis de Broglie suggested in his doctoral dissertation that there is a wave connected with
More informationScattering in one dimension
Scattering in one dimension Oleg Tchernyshyov Department of Physics and Astronomy, Johns Hopkins University I INTRODUCTION This writeup accompanies a numerical simulation of particle scattering in one
More informationRelativity Problem Set 9 - Solutions
Relativity Problem Set 9 - Solutions Prof. J. Gerton October 3, 011 Problem 1 (10 pts.) The quantum harmonic oscillator (a) The Schroedinger equation for the ground state of the 1D QHO is ) ( m x + mω
More informationDavid J. Starling Penn State Hazleton PHYS 214
All the fifty years of conscious brooding have brought me no closer to answer the question, What are light quanta? Of course today every rascal thinks he knows the answer, but he is deluding himself. -Albert
More informationReview of Fundamental Equations Supplementary notes on Section 1.2 and 1.3
Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:
More informationChapter (5) Matter Waves
Chapter (5) Matter Waves De Broglie wavelength Wave groups Consider a one- dimensional wave propagating in the positive x- direction with a phase speed v p. Where v p is the speed of a point of constant
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 informationChapter 38. Photons and Matter Waves
Chapter 38 Photons and Matter Waves The sub-atomic world behaves very differently from the world of our ordinary experiences. Quantum physics deals with this strange world and has successfully answered
More information8.04 Spring 2013 April 09, 2013 Problem 1. (15 points) Mathematical Preliminaries: Facts about Unitary Operators. Uφ u = uφ u
Problem Set 7 Solutions 8.4 Spring 13 April 9, 13 Problem 1. (15 points) Mathematical Preliminaries: Facts about Unitary Operators (a) (3 points) Suppose φ u is an eigenfunction of U with eigenvalue u,
More informationElectromagnetic Waves Across Interfaces
Lecture 1: Foundations of Optics Outline 1 Electromagnetic Waves 2 Material Properties 3 Electromagnetic Waves Across Interfaces 4 Fresnel Equations 5 Brewster Angle 6 Total Internal Reflection Christoph
More informationPhysics 342 Lecture 30. Solids. Lecture 30. Physics 342 Quantum Mechanics I
Physics 342 Lecture 30 Solids Lecture 30 Physics 342 Quantum Mechanics I Friday, April 18th, 2008 We can consider simple models of solids these highlight some special techniques. 30.1 An Electron in a
More informationVibrating Strings and Heat Flow
Vibrating Strings and Heat Flow Consider an infinite vibrating string Assume that the -ais is the equilibrium position of the string and that the tension in the string at rest in equilibrium is τ Let u(,
More informationComputation of the scattering amplitude in the spheroidal coordinates
Computation of the scattering amplitude in the spheroidal coordinates Takuya MINE Kyoto Institute of Technology 12 October 2015 Lab Seminar at Kochi University of Technology Takuya MINE (KIT) Spheroidal
More informationQUANTUM PHYSICS II. Challenging MCQ questions by The Physics Cafe. Compiled and selected by The Physics Cafe
QUANTUM PHYSICS II Challenging MCQ questions by The Physics Cafe Compiled and selected by The Physics Cafe 1 Suppose Fuzzy, a quantum-mechanical duck of mass 2.00 kg, lives in a world in which h, the Planck
More informationProbability and Normalization
Probability and Normalization Although we don t know exactly where the particle might be inside the box, we know that it has to be in the box. This means that, ψ ( x) dx = 1 (normalization condition) L
More informationPath Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Michael Fowler 10/4/07 Huygen s Picture of Wave Propagation If a point source of light is switched on, the wavefront is an expanding sphere centered at the source. Huygens
More information2 z = ±b and the bottom is the plane y = h. The free surface is located at y = (x; z; t), which is unknown. We assume irrotational flow and incompress
1 I-campus project School-wide Program on Fluid Mechanics Modules on Waves in fluids T. R. Akylas & C. C. Mei CHAPTER SIX FORCED DISPERSIVE WAVES ALONG A NARROW CHANNEL Linear surface gravity w aves propagating
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationA 2 sin 2 (n x/l) dx = 1 A 2 (L/2) = 1
VI 15 Model Problems 014 - Particle in box - all texts, plus Tunneling, barriers, free particle Atkins(p.89-300),ouse h.3 onsider E-M wave first (complex function, learn e ix form) E 0 e i(kx t) = E 0
More informationGuided waves - Lecture 11
Guided waves - Lecture 11 1 Wave equations in a rectangular wave guide Suppose EM waves are contained within the cavity of a long conducting pipe. To simplify the geometry, consider a pipe of rectangular
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles? 5.6 Uncertainty Principle 5.7 Probability,
More information4E : The Quantum Universe. Lecture 9, April 13 Vivek Sharma
4E : The Quantum Universe Lecture 9, April 13 Vivek Sharma modphys@hepmail.ucsd.edu Just What is Waving in Matter Waves? For waves in an ocean, it s the water that waves For sound waves, it s the molecules
More informationFigure 4.4: b < f. Figure 4.5: f < F b < f. k 1 = F b, k 2 = F + b. ( ) < F b < f k 1 = f, k 2 = F + b. where:, (see Figure 4.6).
Figure 4.4: b < f ii.) f < F b < f where: k = F b, k = F + b, (see Figure 4.5). Figure 4.5: f < F b < f. k = F b, k = F + b. ( ) < F b < f k = f, k = F + b iii.) f + b where:, (see Figure 4.6). 05 ( )
More informationTotal Internal Reflection & Metal Mirrors
Phys 531 Lecture 7 15 September 2005 Total Internal Reflection & Metal Mirrors Last time, derived Fresnel relations Give amplitude of reflected, transmitted waves at boundary Focused on simple boundaries:
More informationModern Physics notes Paul Fendley Lecture 3
Modern Physics notes Paul Fendley fendley@virginia.edu Lecture 3 Electron Wavelength Probability Amplitude Which slit? Photons Born, IV.4 Feynman, 1.6-7, 2.1 Fowler, Rays and Particles The wavelength of
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.
PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms
More informationIntroduction to the Discrete Fourier Transform
Introduction to the Discrete ourier Transform Lucas J. van Vliet www.ph.tn.tudelft.nl/~lucas TNW: aculty of Applied Sciences IST: Imaging Science and technology PH: Linear Shift Invariant System A discrete
More informationLecture 6 Scattering theory Partial Wave Analysis. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 6 Scattering theory Partial Wave Analysis SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 The Born approximation for the differential cross section is valid if the interaction
More informationPhysics 505 Homework No. 3 Solutions S3-1
Physics 55 Homework No. 3 s S3-1 1. More on Bloch Functions. We showed in lecture that the wave function for the time independent Schroedinger equation with a periodic potential could e written as a Bloch
More information