# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

4 Pages

### FreeParticleWaves

Course: PHYS 491, Fall 2009
School: New Mexico
Rating:

Word Count: 1029

#### Document Preview

particle Free waves So, we have confirmed the idea that matter behaves like waves, but what is the nature of these waves, and how do we describe them. Let's begin with a free particle, that is, a particle moving in one dimension with no force acting on it, therefore with no potential changes. We start by writing a function that will describe the dynamics of the wave, what we will henceforth refer to as the...

Register Now

#### Unformatted Document Excerpt

Coursehero >> New Mexico >> New Mexico >> PHYS 491

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
particle Free waves So, we have confirmed the idea that matter behaves like waves, but what is the nature of these waves, and how do we describe them. Let's begin with a free particle, that is, a particle moving in one dimension with no force acting on it, therefore with no potential changes. We start by writing a function that will describe the dynamics of the wave, what we will henceforth refer to as the wavefunction: ( x , t ) = sin (kx - t ) This function describes an oscillating wave with wavelength = 2/k and frequency = /2. Let us try to extract the dynamics of the particle from this function. There are zeros of this function whenever (kx-t)=n or whenever xn=n/k+t/k. The velocity of these zeros is w = d/dt(xn) = /k = 2/(2/) = as we expect for the relationship between the velocity, wavelength and frequency. But, let's now go back to the de Broglie relationship between the wavelength and momentum, and the Einstien relationship between the energy and the frequency: p = h/ and E = h. Then = (E/h)(h/p) = E/p = ()mv2/mv = v. This gives us the puzzling result that the group velocity (the velocity of the group of waves) is of the particle velocity!? So, how can we reasonable describe the particle with such a wavefunction? Well, there is another problem with the wavefunction that we initially chose, that is that there is no defined spatial extent. There are an infinite number of zeros, or, if we choose, an infinite number of maxima. So it would be a stretch to call this a particle wavefunction something that intuitively we would expect to describe a localized particle moving with velocity v. We make one step towards creating a wave packet. Let's look at the sum of two wavefunctions, one with a slightly different wavelength and frequency: ( x, t ) = 1 ( x, t ) + 2 ( x, t ) 1 ( x, t ) = sin (kx - t ) 1 ( x, t ) = sin ((k + dk )x - ( + d )t ) using sinA +sinB = 2cos[(A-B)/2]sin[(A+B)/2], then dk d ( x, t ) = 2 cos x- 2 2 dk d = 2 cos x- 2 2 2k + dk 2 + d t sin x - 2 2 t sin (kx - t ) t where the last step is a simplification from the fact that d<<2 and dk<<2k. Looking at this wavefunction, we can see that it is the same sin function, moderated by a changing amplitude (cos function). From the figure below we see now larger "packets of waves" with wavelengths 1/dk moving with velocity g = d/dk. So, now we have two distinct velocities for the wavefunction: 1) the wave velocity w = /k, and 2) the group velocity g = d/dk. If we now return to the de Broglie relationships: = 2E/h and k = 2/ = 2p/h, then d = 2dE/h and dk = 2dp/h. This gives us for the group velocity, g = d/dk = dE/dp. Using E = mv2/2 and p = mv, dE = mvdv and dp =mdv or, dE/dp = mvdv/mdv = v. OK, so now we have that the group velocity for the wavefunction is the same as the particle velocity, but what about localization? We still have an infinite of number these "groups" moving at v. The next step we make is to sum more waves together. The figure below shows seven different waves with different k values (k/2) = 9, 10 ... 15. The amplitude and phase of each wave has been chosen such that all the waves are in phase at x = 0 and that the sum of all the waves = k (shown at the bottom) gives a packet k of particular width and shape. Notice that there is still a repeated pattern at the right side of the plot of the sum. If we were to extend this process using the exact same k-range (i.e. k = 9-15) but with a greater number of k values (say, k = 9, 9.5, 10, 10.5, ... 14.5, 15) one would find that the width of the packet centered around x = 0 would remain the same, but the distance between the repeated patterns would increase, and the amplitude of the oscillations between the repeated patterns would diminish. We could continue this process of subdividing the k-space and adding waves and we will find that as the number of waves in this interval goes to infinity, the number of packets that are left goes to one. Instead of a sum of waves, we could now write this as an integral: ( x, t ) = - (k ) sin(kx - t )dk where (k) now represents the relative amplitude of the waves in the integral. OK, so now we have a wavefunction that is localized (is non-zero over a limited x-range), travels with a group velocity equal to that of a moving particle, v. But we have arrived at this wave description at a price. In order to make the packet spatial dimension increasingly small, we must increase the range of the k-values that we include. Hence, if we want a localized wave packet around some very small x-value a (in the figure below), then we must in...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

New Mexico - PHYS - 491
Uncertainty in Free Particle Waves Now let us return to the free particle waves and our loose understanding of the relation between knowledge of the position, and knowledge of the momentum, i.e., that the narrower we make the wave packet, the more k-
New Mexico - PHYS - 491
Unbound States OK, up until now, we have dealt solely with states that are bound inside a potential well. [Well, except for our treatment of the free particle and we want to keep that in mind here.] We want to now consider what happens if the partic
New Mexico - PHYS - 491
New Mexico - PHYS - 491
Classical and Quantum TrajectoriesLets take a closer look at the behavior of a quantum system given a potential, and the energy of the particle. Shown below is some part of a potential, V(x), and, on the same scale, the particles energy E. The regi
Idaho State - B - 442
Coevolution of Plant-Animal InteractionsBIOS 442/542 Spring 2009Origins of Coevolution Mode (1958) - Coined Term - Plant Pathology Ehrlich and Raven (1964) - Popularized Idea - Butterflies and Host Plants Janzen (1980 and earlier) - Questioned
Idaho State - B - 358
TRANSLATION Tuesday, April 4, 2006 1. QUIZ a. Review Solutions 2. The Genetic Code a. Take Home Points i. Triplet Code (Codon) is degenerate ii. Non-overlapping iii. Code is ordered iv. Start &amp; Stop Codons v. Code is universal (nearly). b. Triplet C
New Mexico - MGT - 331
C:\Program Files\Oracle\JDeveloper9iR3\jdk\bin\javaw.exe -ojvm -classpath C:\Temp\Exam2a\classes;C:\Program Files\Oracle\JDeveloper9iR3\jdev\lib\jdev-rt.jar mypackage1.DumpDB * Dumping Vehicle Table 123456,Toyota,Camry,49.99,true 234567,Toyota
New Mexico - MGT - 331
Sample Output from TestFilm - Homework #4 -MGT 331 - Spring 2003calling Film(&quot;Ben Hur&quot;,1959)mypackage1.Film.title=Ben Hurmypackage1.Film.year=1959mypackage1.Film.length=212mypackage1.Film.mPAA_Rating=mypackage1.Film.studio=MGMmypackage1.Fil
Idaho State - MATH - 360
Review Problems for Midterm 2, Math 360, Fall 20021. What is the best statement that you can make about the existence and uniqueness of the solution of the following initial value problem? y + (1/x2 )y 3y = ex y(3) = 2, y (3) = 72. An 8 lb weight
New Mexico - MATH - 215
Syllabus University of New Mexico Valencia Campus Math 215: Mathematics for Elementary and Middle School Teachers III Fall 2008 Instructor: Julie DePree, Ph.D. Phone: 925-8607 Office Hours: T 10:00-10:30, T&amp;TH 11:45-3:00 or by appointment Office: Aca
New Mexico - IT - 152
THE UNIVERSITY OF NEW MEXICOLOS ALAMOS COURSE SYLLABUS IT152: Advanced Cascading Style SheetsIT152: Advanced Cascading Style Sheets UNM-LA Room 504 M-W 10:00AM 11:15AM Fall 2008 Robert (Bob) L Judd Home: 505-662-4940 Work: 505-667-0690 (at LANL) Em
New Mexico - IT - 244
THE UNIVERSITY OF NEW MEXICOLOS ALAMOS COURSE SYLLABUS IT244LT: Information Architecture for Complex Web SitesIT244LT: Information Architecture for Complex Web Sites UNM-LA Room 504 M-W 12:00PM 1:15PM Fall 2007 Robert (Bob) L Judd Home: 505-662-494
New Mexico - IT - 244
Visual chaosFrom waking up to the morning news, to visiting countless Web sites all day at work, to driving to dinner through a sea of ashing lights and larger than life advertisements, our days are full of visual chaos demanding attention.Actual
New Mexico - IT - 243
THE UNIVERSITY OF NEW MEXICOLOS ALAMOS COURSE SYLLABUS IT243: Introduction to PHPIT243: Introduction to PHP UNM-LA Room 504 M-W 8:00AM 9:15AM Spring 2008 Robert (Bob) L Judd Home: 505-662-4940 Work: 505-667-0690 (at LANL) Email: it243@att.net Web:h
New Mexico - IT - 245
THE UNIVERSITY OF NEW MEXICOLOS ALAMOS COURSE SYLLABUS IT245: Introduction to MySQLIT245: Introduction to MySQL UNM-LA Room 504 M-W 8:00AM 9:15AM Fall 2008 Robert (Bob) L Judd Home: 505-662-4940 Work: 505-667-0690 (at LANL) Email: it245@att.net Web
New Mexico - IT - 244
Well.APPLE DOESN'T HAVE SOMEspecial place where their marketing secrets are kept, unless of course you count their charismatic CEO's brain. The ve secrets I offer here are careful deductions, empirical results and the product of my career as an eng
New Mexico - IT - 244
THE UNIVERSITY OF NEW MEXICOLOS ALAMOS COURSE SYLLABUS IT244LT: Information Architecture for Complex Web SitesIT244LT: Information Architecture for Complex Web Sites UNM-LA Room 504 Tues &amp; Thurs 2:00PM 3:15PM January 17 to May 11, 2005 Robert L (Bo
New Mexico - MGT - 337
SYSTEM DESCRIPTIONSIBM Intellistation Pro workstations are systems designed for business production environments concerned with reliability and performance. At first glance they appear to be ordinary mass market personal computers, but the technolo
New Mexico - MGT - 337
IBM Product customizationPage 1 of 8Customize IntelliStation Z ProIntelliStation Z Pro Model Configured price Lease price SuccessLease for Small Business* Monitor sold separately unless otherwise stated.IBM recommends Microsoft Windows XP Profe
Idaho State - UGRAD - 98
270ISU Campus Map271ISU Campus Map Key10 ADMINISTRATION BUILDING (ADMIN) . G5 81 BANNOCK NURSING HOME (BNHC) . D2 80 BANNOCK REGIONAL MEDICAL CENTER (BRMC) .. D3 55 BARTZ FIELD . E8 66 BECKLEY NURSING BUILDING (NURS) . C3 5 BUSINESS ADMINISTRA
Idaho State - GRAD - 99
112x Graduate StudiesCollege of Pharmacy x 113 Individual requirements for each student must be approved by the students major professor within the emphasis area. All classified graduate students must register for the appropriate graduate seminar
Idaho State - UGRAD - 99
185College of PharmacyBarbara G. Wells, Pharm.D., Dean Virginia J. Galizia, Ph.D., Associate Dean ceutical Education, 311 West Superior Street, Suite 512, Chicago, IL 60610. Professional Standards Students enrolled in the programs of the College o
Idaho State - UGRAD - 99
113College of BusinessWilliam E. Stratton, Ph.D., Dean Kenneth A. Smith, Ph.D., Associate Dean George A. Johnson, Ph.D., MBA Director Darrell Scott, MBA, Assistant Dean Department of Accounting Chair and Professor: K. Smith Professors: Boes, Plewa
Idaho State - UGRAD - 98
177College of PharmacyBarbara G. Wells, Pharm.D., Dean Virginia J. Galizia, Ph.D., Associate Dean Accreditation The Doctor of Pharmacy program is accredited by the American Council on Pharmaceutical Education, 311 West Superior Street, Suite 512,
Idaho State - UGRAD - 98
151College of EngineeringJay Kunze, Ph.D., Dean D. T. Neill, Ph.D., Associate Dean for Graduate Studies R. E. Stuffle, Ph.D., Associate Dean for Undergraduate Studies Professors: Campo, Kunze, Naidu, Neill, Robinson, Smedley, G. Stuffle Associate
Idaho State - GRAD - 98
64 x Graduate StudiesCollege of Business x 65 business: Accounting, Economics, Finance, and Marketing. The core also examines behavioral, information system, and industry analysis issues that cut across the functional boundaries and provide a basic
Idaho State - GRAD - 99
62 x Graduate StudiesCollege of Business x 63 The first-year core courses develop a broad competence in the functional fields of business: Accounting, Economics, Finance, and Marketing. The core also examines behavioral, information system, and ind
Idaho State - GRAD - 99
86x Graduate StudiesCollege of Engineeringx 87College of EngineeringJay F. Kunze, Dean Professors Naidu, Neill, Robinson, Smedley, Stuffle Associate Professors Leung, Rao, Sadid, Sato, Wabrek Assistant Professors Bennion, Blotter, McWhirter
Idaho State - GRAD - 02
2002-2003 Academic Calendar(for registration information, obtain current Class Schedule from Registrar) To Register, Add/Drop, Change Sections, or Withdraw, use Academic Web Registration located on MyISU portal:Fall Semester 2002August 23 .. DISE
Idaho State - UGRAD - 99
155College of EngineeringJay Kunze, Ph.D., Dean R. E. Stuffle, Ph.D., Associate Dean for Undergraduate Studies Professors: Campo, Kunze, Naidu, Neill, Robinson, G. Stuffle Associate Professors: Ellis, Rao, Sadid, Sato, Wabrek Assistant Professors:
Idaho State - UGRAD - 99
161College of Health ProfessionsLinda C. Hatzenbuehler, Ph.D., Dean Delane C. Kritsky, Ph.D., Associate Dean The College of Health Professions offers programs of professional education leading to the Associate of Science degree in Radiographic Sci
Idaho State - UGRAD - 99
Idaho State - UGRAD - 98
Idaho State - UGRAD - 98
119College of EducationLarry B. Harris, Ph.D., Dean The College of Education prepares students who are interested in the profession of teaching and who are seeking a comprehensive understanding of the complex processes associated with learning and
Idaho State - UGRAD - 99
123Degrees OfferedCollege of EducationLarry B. Harris, Ph.D., Dean Stephanie Salzman, Ph.D., Associate Dean T.C. Mattocks, Ed.D., Assistant Dean The College of Education prepares students who are interested in the profession of teaching and who
New Mexico - PHYS - 262
UNM Physics 262, Problem Set 7, Fall 2006Instructor: Dr. Landahl Issued: October 11, 2006 Due: October 18, 2006Do all of the exercises and problems listed below.Hand in your problem set in therolling cart hand-in box, either before class or a
New Mexico - PHYS - 452
UNM Physics 452/581: Introduction to Quantum InformationLecture 1: Course overview; bits &amp; pbitsAndrew J. Landahl, alandahl@unm.edu (DRAFT: Thursday 30 August, 2007) Welcome to UNM Physics 452/581, Introduction to Quantum Information. In class, I
Idaho State - ADSS - 2006
CETON Project(Cooled Electron Target-Optimized for Neutron-production)Team Members: Tim Beller Ryan LeCounte Brice HowardApril 3, 2006SolidWorks DrawingsSolidWorks Cross SectionMachiningThe aluminum and copper were machined at UNLV by Kev
New Mexico - CS - 150
CS-150 Spring 2009 Lab Schedule Monday 11:00-11:50 Tuesday Section: 6 (38) Moe (lead) Mohammad WednesdayUniversity of New Mexico Thursday Section: 7 (26) Moe (lead) Mohammad Friday1:00 - 1:50Section: 1 (40) Melissa (lead) Matthew Section: 8 (25
New Mexico - CS - 261
CS 261 - Homework 2Section 2.12.(a) 2 + 7 &gt; 11 (b) 2 is not an even integer or 8 is not an odd integer4.(a) 3 + 1 &lt; 5 and 7 = 3 6 3 + 1 &lt; 5 or 7 = 3 6 (b) I am rich and I am happy. I am rich or I am happy.8.(a) False (b) False (c) True (d
New Mexico - CS - 261
Section 1.16.{x | x is a positive even integer less than twelve}8.{x | x = y3 and y is in {1, 2, 3, 4, 5}16.All True.26.(a) {}, {3}, {7}, {3, 7} (b) |A| = 2 (c) |P(A)| = 4Section 1.26.(a) {1, 6, 8} (b) {5, 9} (c) {1, 2, 3, 4} (e) {3, 5
New Mexico - CS - 150
Prerequisite Computer Skills for Success in CS-150Prerequisite computer skills for CS-150 are not as important as the prerequisite mathematics skill. This is because, with some extra effort, the prerequisite computer skills can be learned in the fir
New Mexico - CS - 149
Adams,First Name,Role,Cross-Listing Information,Lab 1,Lab 2,Q-1,Lab 3,Q-2,Lab 4,Exam-1,Lab 5,Q-3,Q-4,Lab 6,Q-5,Q-6,Exam-2,Lab 7,Lab 8,Q-7Allen,Dabria ,Student,CS-150L-007 (Fall 2008),55,85,4,77,5,88,86,91,10,9,90,15,10,79,94, ,24Almaraz,Dacey ,Stu
New Mexico - M - 100
EXTRA PRACTICE 1 Addition Of Real Numbers Use after Section 1.3 Name_ _ Examples: Add. 4 + 7 = 11 4+ 7 = 3 Add. 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 39. 5 + ( 6) = _ 9 + ( 4 ) = _ 14 + ( 13) = _ 3 + 10 = _ 14 + (
New Mexico - PPPS - 2007
PPPS-2007 SOCIAL FUNCTIONSSunday 6/17 7:00 AM 9:30 -10:00 AM Noon -1:00 PM 3:00 -3:30 PM Monday 6/18 Continental Breakfast Convention Center Lobby Coffee Break Lunch (on your own) Afternoon Break (NE Exhibit Hall) Tuesday 6/19 Continental Breakfast
New Mexico - PPPS - 2007
IEEE ICOPS 2007 Minicourse June 22 23, 2007Diagnostics for High Density Plasmas and Pulsed Power SystemsDay 1 7:30 8:15 8:30 8:30 9:15 9:15 10:15 10:15 10:30 10:30 11:45 11:45 1:15 1:15 2:15 2:15 3:15 3:15 3:30 3:30 4:30 4:45 6:00 Day 2
New Mexico - PPPS - 2007
2007 International Conference on Pulsed Power and Plasma Science(34th IEEE ICOPS + 16th IEEE Pulsed Power Conference)June 1722, 2007 Convention Center Albuquerque, New Mexico USAPAC 2007 (IEEE Particle Accelerator Conference) will follow, June 2
New Mexico - PPPS - 2007
PPPS-2007 SATELLITE MEETINGS, FORUMS, MINICOURSE, WORKSHOPSSunday 6/17 7:00 AM 8:00 AM 9:00 AM 10:00 AM 11:00 AM NOON 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM 7:00 PM 8:00 PM Judging Student Papers Pavilion I-II, Hyatt Regency Dave Abe Monday
New Mexico - PPPS - 2007
ScheduleIEEE PPPS-2007 Albuquerque CourseSyd Field &amp; Alex Singer Hyatt Regency Albuquerque - Pavilion IV-V Continental Breakfast 7:30-8:30 Outside the room 8:30AM 9:30AM Syd FieldIntroduction - The Overview Distinctions What is a Screenplay?
New Mexico - PPPS - 2007
Call for PapersIEEE Transactions on Plasma ScienceThe Special Issue onInvited and Plenary Talks of PPPS 2007The IEEE Transactions on Plasma Science is proud to announce a Special Issue dedicated to the invited and plenary talks given at the Puls
New Mexico - PPPS - 2007
Call for PapersIEEE Transactions on Plasma ScienceThe Special Issue onPulsed Power Science and TechnologyThe IEEE Transactions on Plasma Science is proud to announce another Special Issue on Pulsed Power Science and Technology, to be published
New Mexico - TSOCS - 2004
Accepted by Journal of Chemical PhysicsDNA-Psoralen Interaction: A Single Molecule ExperimentM. S. Rocha, N. B. Viana, and O. N. Mesquita Departamento de Fsica, ICEX, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, CEP 301
New Mexico - TSOCS - 2004
Global Persistence in a cooperative game model. Roberto da Silva Alexandre Baraviera Silvio Dahmen Ana BazzanTSOCS21 Motivation I In the context of simple random walks in regular lattices in Zd it is known that d = 1 and 2 a random walker alw
New Mexico - TSOCS - 2004
Guillermo AbramsonCentro Atmico Bariloche and CONICET, Bariloche, Argentina. Consortium of the Americas for Interdisciplinary Science, University of New Mexico, Albuquerque, USA.with L. Giuggioli and V.M. KenkreTSOCS 2004 Porto AlegreThe diff
New Mexico - TSOCS - 2004
Need, Greed and Noise: Competing Strategies in a Trading ModelR. Donangelo, 1 A. Hansen, 2 K. Sneppen, and S. R. Souza 1The Niels Bohr Institute, Blegdamsvej 17, Copenhagen , DenmarkarXiv:cond-mat/0408531 v1 25 Aug 2004Abstract We study an econ
New Mexico - TSOCS - 2004
Disorder and correlation in 1D complex systemsGerardo Martnez, UFRGS, Porto Alegre Hugo N. Nazareno, ICCMP, UnB, BrasliaIntroduction Non-linear response and time-dependent phenomena are an open frontier in the physics of strongly electron corr
New Mexico - TSOCS - 2004
Wealth exchanges in a dynamical social networkJos Roberto IglesiasInstituto de Fsica, Universidade Federal do Rio Grande do Sul, Porto Alegre, BrazilThe Economic gang of 7J.R. Iglesias, S. Pianegonda Porto Alegre, BrazilG. Abramson S. C. de B
New Mexico - TSOCS - 2004
Workshop on TRANSPORT AND SELF-ORGANIZATION IN COMPLEX SYSTEMSA model for the closure of stomata in a leafCarmen P. C. Prado (IFUSP) Marcus Cima Ferraz (IFUSP, student) Gustavo Maia Souza (UNOESTE, SP)TSOCS2004 Porto Alegre, Brasil, agosto 2004
New Mexico - TSOCS - 2004
Magnetism and pattern formationP. Molho Laboratoire Louis Nel, CNRS, Grenoble, FranceW. Herreman, V. Heresanu, R. Ballou, L. Del Rey Laboratoire Louis Nel, CNRS Grenoble S. Bodea IRPHE, Universit de Marseille S. Neveu LI2C, Universit Pierre et Mar
New Mexico - TSOCS - 2004
Out of equilibrium dynamics in systems with modulated patternsDaniel A. StarioloUFRGS - BRASIL stariolo@if.ufrgs.br Colaborators: Sergio Cannas, Francisco Tamarit, Pablo Glaiser (UNC, Argentina) Mateus Michelon, Lucas Nicolao (UFRGS, Brazil). p.
New Mexico - TSOCS - 2004
Deterministic walks for animal foragingO. Miramontes (1), D. Boyer (1), G. Ramos (3), G. Cocho (1), JL Mateos (1), H. Larralde (2) &amp; H. Ramos (1) (1) Instituto de Fisica - Mexico City (2) Centro de Ciencias Fisicas-Cuernavaca Universidad Nacional Au