Chapter 3 The Schrödinger Equation and a Particle in a Box
|
|
- August McKenzie
- 5 years ago
- Views:
Transcription
1 Chpter 3 The Schrödinger Eqution nd Prticle in Bo Bckground: We re finlly ble to introduce the Schrödinger eqution nd the first quntum mechnicl model prticle in bo. This eqution is the bsis of quntum mechnics nd s we will lern in chpter 4 is postulte rther thn derivble epression. The Schrödinger eqution is the eqution for the wvefunction of prticle - we begin with the clssicl D wve eqution u u with ut (, ) ( )cos( t) & ( ) is the sptil mplitude v t u u ( )cos( t) ( )cos( t) v t v t cos( t) ( ) ( ) cos( t) v t ( ) ( ) cos( t) v cos( t) d ( ) ( ) 0 with v & v v d v d v d 4 ( ) ( ) ( ) ( ) 0 d v d - If we use de Broglie s reltion h nd E = K.E. + P.E. p p h E V( ) p mev( ) m mev( ) - Using h nd substituting bck into our eqution: d 4 d ( ) ( ) ( ) m EV( ) ( ) d d h d mev( ) ( ) ( ) 0 d d mev( ) ( ) ( ) 0 d d ( ) EV( ) ( ) 0 md d ( ) E( ) V( ) ( ) 0 md d ( ) V( ) ( ) E( ) md The lst eqution is the time-independent Schrödinger eqution.
2 Clssicl-Mechnicl quntities re represented by liner opertors in quntum mechnics - In generl, liner opertor  A ht is defined s A[ ˆ cf( ) c ˆ ˆ f( )] ca f( ) ca f( ) where c nd c re constnts - Emples: liner : d d d d  =, cf( ) cf( ) c f( ) c f( ) d d d d non liner : cf cf c f cc f f c f etr term  =, ( ) ( ) ( ) ( ) ( ) ( ) The Schrödinger eqution cn be formulted s n eigenvlue problem - definition of n eigenvlue eqution: A ˆ where = constnt - Emples: -- liner momentum opertor pˆ i ik pˆ ( ) where ( ) e ik ik ik i e iike i ke ik ke k ( ) constnt -- direction opertor ˆ ik ˆ ( ) e constnt ( ) ( ) is not eigenfunction of opertor ˆ - Bck to Schrödinger d ˆ V( ) ( ) ( ) or ( ) ( ) E H E md where Hˆ is clled the Hmiltonin opertor E is the energy of our wvefunction ( ) pˆ d d d KE.. i i nd P. E. = V( ) m m d d m d Wve functions hve probbilistic interprettion - s we will lter point out wvefunctions re often thought of s vectors - the intensity of the wve is the squre of the mplitude or mgnitude of the Wvefunction, I ( ) ( ) ( ) - tody, ( ) ( d ) is the probbility of finding prticle locted btwn nd + d
3 The Energy of prticle in bo is quntized - this will become the model tht UV-vis spectroscopy is bsed on - D cse: prticle is confined to move long one coordinte, -- boundry conditions: ψ(0) = 0 nd ψ() = physiclly we re sying tht the potentil t the ends of our bo is 0 V( ) the prticle is thereby restricted to be locted btwn 0 & or 0 -- How does this effect the Schrödinger eqution? Given our conditions for the potentil, V() d ( ) V( ) ( ) E ( ) md becomes d ( ) E ( ) md d me Or, ( ) ( ) 0 d -- Now, let s look t ψ() --- from Chpter tht possible eigen solution to our Schrödinger eqution is d ( ) e ( ) e ( ) d --- plugging this bck into the eqution: d me me ( ) ( ) 0 ( ) ( ) 0 d me me me or ( ) 0 so i me ik ik -- if we let k then ( ) ce ce Or using Euler s equtions: ( ) Acos( k) Bsin( k) -- Now let s pply the boundry conditions, ψ(0) = 0 nd ψ() = 0 (0) 0 A 0 ( ) 0 B sin( k) 0 so sin( k) 0 So, k n where n,,3, Plugging k bck in: k n gives me me me 8 me k n n n n h h nh n En nd ( ) sin where,,3, n B n 8m -- Gret so wht does ll this men? --- first, energy is quntized given our integer n dependence
4 --- n is quntum number such tht n describes the stte of our system ---- for ny n we will hve n En nd ψn for tht stte ---- lso the smller the n the closer our system is to ground stte (g.s.) --- see figure 3. from below so --- We cn determine how much energy is required to promote n e- from stte to the net e.g. to go from n = to n = 4, 6h 4h h E E4 E 8m 8m 8m Since electronic trnsitions often occur in the visible, then let λm = 500 nm which mens = m 34 h Js 9 E4.890 J 8m kg m Wve functions must be normlized - remember we re doing probbilistic pproch here s such the probbility of finding prticle inside our D bo must be no greter thn unity - A normlized wvefunction, ψ(), will dhere to ( ) ( d ) - In our cse we re tlking bout prticle which is confined to the limits of our bo n So, to normlize our wvefunction, n( ) Bsin, the limits of integrtion Are 0 nd : n n( ) n( ) d B sin d B or B 0 the normlized D prticle in bo wvefunction is: ( ) sin n n n n
5 The verge momentum of prticle in bo is zero - Averge/Men -- We cn determine the verge vlue of ny opertor, system ˆ O ( ) Oˆ( ) d this is postulte from Ch-4 Ô, which is pplied to our -- the verge or men position of prticle in constrined to our bo is: n n ( ) ˆ ( ) d sin sin d for ll n This is the midpoint of the bo. - Vrince -- Like the men, the vrince of ny opertor pplied to the system is defined s -- Since we hve, we just need : n n sin sin d 3 0 n n 3 n 4 n 3 - Stndrd Devition, -- s we know the stndrd devition is the n -- for our model, n 3 - We know wht the verge position, let s see bout momentum n d n i n n p sin i sin d sin cos d 0 d 0 0 Which mens tht the prticle is moving in either directions eqully n d n n n p sin sin d 3 d 0 Revisiting Heisenberg nd his uncertinty n n - p p p 0 -- from this we see tht p / which mens the more we try to loclize (or mke the bo smller) the more devition we hve in p -- free prticle hs no bo limiting it, in this cse we will hve no uncertinty in the momentum or
6 n - n 3 -- from this we see tht which mens the smller the bo the smller our devition in position -- for free prticle the uncertinty in position is infinity - once gin, if we hve certinty in position we hve uncertinty in momentum nd vice-vers mthemticlly: p n since n, 3 n n n n 3 3 > we cn sy tht p Unto rel bo 3D time - boundry conditions: 0 (0, yz, ) ( yz,, ) for ll y & z 0 0 yb V( ) (, y, z) (,0, z) (, b, z) for ll & z 0 z c (, y,0) (, y, c) for ll & y elsewhere Once gin the prticle is restricted in ll three directions - the Schrödinger eqution: (, yz, ) E( yz,, ) m y z Lplcin or -- using seprtion of vribles gin we cn derive the solution for the wvefunction nd the energy description see the tet. Appliction to prticle in bo conjugted polymers - For emple C6H8: -- ech crbon tom hs 3 sp orbitls nd -- it turns out tht trnsitions for these conjugted systems re UV-Vis ctive
7 --- the verge C C bond length is.4å this trnsltes to 5.4 Å = 7 Å for the length of our bo --- the first ecited stte is from E3 E4, so Js 9 E J kg 70 m -- this trnsltes to wvelength of: 34 8 hc J s 30 m s m or 69nm E which is in the UV rnge -- let s tlk bout the sttes 9 J -- HOMO (highest occupied moleculr orbitl)/lumo (lowest unoccupied MO) --- for the ground stte the HOMO is ψ3 nd the LUMO is ψ4 --- for the first ecited stte HOMO is ψ4 nd the LUMO is ψ5 -- the gp between the nd bnd is clled the bnd gp --- the smller the bnd gp the more conducting the system is --- -bnd is lso referred to s the vlence bnd --- bnd is lso referred to s the conduction bnd Dirc Nottion br nd ket - this is shorthnd form of sptil integrtion which is commonly used by chemists & physists - d n n n n n n -- n is the br nd is the conjugte of n -- n is the ket n n - Kronecker Delt, nn 0 n n -- this works for normlized wvefunctions
8 -- the n n implies tht ll the sttes re orthogonl to ech other -- these two stipultions combined led to set of wvefunctions which re orthonorml, normlized nd orthogonl wvefunctions
Problem Set 3 Solutions
Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,
More informationdx x x = 1 and + dx α x x α x = + dx α ˆx x x α = α ˆx α as required, in the last equality we used completeness relation +
Physics 5 Assignment #5 Solutions Due My 5, 009. -Dim Wvefunctions Wvefunctions ψ α nd φp p α re the wvefunctions of some stte α in position-spce nd momentum-spce, or position representtion nd momentum
More informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More informationHomework Problem Set 1 Solutions
Chemistry 460 Dr. Jen M. Stnr Homework Problem Set 1 Solutions 1. Determine the outcomes of operting the following opertors on the functions liste. In these functions, is constnt..) opertor: / ; function:
More informationQuantum Mechanics Qualifying Exam - August 2016 Notes and Instructions
Quntum Mechnics Qulifying Exm - August 016 Notes nd Instructions There re 6 problems. Attempt them ll s prtil credit will be given. Write on only one side of the pper for your solutions. Write your lis
More informationProblem Set 2 Solutions
Chemistry 362 Dr. Jen M. Stnr Problem Set 2 Solutions 1. Determine the outcomes of operting the following opertors on the functions liste. In these functions, is constnt.).) opertor: /x ; function: x e
More informationDo the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?
1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the
More information( ) 2. ( ) is the Fourier transform of! ( x). ( ) ( ) ( ) = Ae i kx"#t ( ) = 1 2" ( )"( x,t) PC 3101 Quantum Mechanics Section 1
1. 1D Schrödinger Eqution G chpters 3-4. 1.1 the Free Prticle V 0 "( x,t) i = 2 t 2m x,t = Ae i kxt "( x,t) x 2 where = k 2 2m. Normliztion must hppen: 2 x,t = 1 Here, however: " A 2 dx " " As this integrl
More informationMassachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 6
Msschusetts Institute of Technology Quntum Mechnics I (8.) Spring 5 Solutions to Problem Set 6 By Kit Mtn. Prctice with delt functions ( points) The Dirc delt function my be defined s such tht () (b) 3
More informationPhysics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15
Physics H - Introductory Quntum Physics I Homework #8 - Solutions Fll 4 Due 5:1 PM, Mondy 4/11/15 [55 points totl] Journl questions. Briefly shre your thoughts on the following questions: Of the mteril
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10
University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationChapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1
Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4
More informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More informationContinuous Quantum Systems
Chpter 8 Continuous Quntum Systems 8.1 The wvefunction So fr, we hve been tlking bout finite dimensionl Hilbert spces: if our system hs k qubits, then our Hilbert spce hs n dimensions, nd is equivlent
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Deprtment 8.044 Sttisticl Physics I Spring Term 03 Problem : Doping Semiconductor Solutions to Problem Set # ) Mentlly integrte the function p(x) given in
More informationPh2b Quiz - 1. Instructions
Ph2b Winter 217-18 Quiz - 1 Due Dte: Mondy, Jn 29, 218 t 4pm Ph2b Quiz - 1 Instructions 1. Your solutions re due by Mondy, Jnury 29th, 218 t 4pm in the quiz box outside 21 E. Bridge. 2. Lte quizzes will
More informationAike ikx Bike ikx. = 2k. solving for. A = k iκ
LULEÅ UNIVERSITY OF TECHNOLOGY Division of Physics Solution to written exm in Quntum Physics F0047T Exmintion dte: 06-03-5 The solutions re just suggestions. They my contin severl lterntive routes.. Sme/similr
More informationPhysics 137A - Quantum Mechanics - Spring 2018 Midterm 1. Mathematical Formulas
Copyright c 8 by Austin J. Hedemn Physics 7A - Quntum Mechnics - Spring 8 Midterm Mondy, Februry 6, 6:-8: PM You hve two hours, thirty minutes for the exm. All nswers should be written in blue book. You
More informationQuantum Physics I (8.04) Spring 2016 Assignment 8
Quntum Physics I (8.04) Spring 206 Assignment 8 MIT Physics Deprtment Due Fridy, April 22, 206 April 3, 206 2:00 noon Problem Set 8 Reding: Griffiths, pges 73-76, 8-82 (on scttering sttes). Ohnin, Chpter
More informationPractice Problems Solution
Prctice Problems Solution Problem Consier D Simple Hrmonic Oscilltor escribe by the Hmiltonin Ĥ ˆp m + mwˆx Recll the rte of chnge of the expecttion of quntum mechnicl opertor t A ī A, H] + h A t. Let
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationNOTES ON HILBERT SPACE
NOTES ON HILBERT SPACE 1 DEFINITION: by Prof C-I Tn Deprtment of Physics Brown University A Hilbert spce is n inner product spce which, s metric spce, is complete We will not present n exhustive mthemticl
More information5.04 Principles of Inorganic Chemistry II
MIT OpenCourseWre http://ocw.mit.edu 5.04 Principles of Inorgnic Chemistry II Fll 2008 For informtion bout citing these mterils or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.04, Principles of
More informationPhysics 215 Quantum Mechanics 1 Assignment 2
Physics 15 Quntum Mechnics 1 Assignment Logn A. Morrison Jnury, 16 Problem 1 Clculte p nd p on the Gussin wve pcket α whose wve function is x α = 1 ikx x 1/4 d 1 Solution Recll tht where ψx = x ψ. Additionlly,
More informationu t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx
Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationClassical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011
Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More information221B Lecture Notes WKB Method
Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,
More informationDescribe in words how you interpret this quantity. Precisely what information do you get from x?
WAVE FUNCTIONS AND PROBABILITY 1 I: Thinking out the wve function In quntum mechnics, the term wve function usully refers to solution to the Schrödinger eqution, Ψ(x, t) i = 2 2 Ψ(x, t) + V (x)ψ(x, t),
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationChapter 9 Many Electron Atoms
Chem 356: Introductory Quntum Mechnics Chpter 9 Mny Electron Atoms... 11 MnyElectron Atoms... 11 A: HrtreeFock: Minimize the Energy of Single Slter Determinnt.... 16 HrtreeFock Itertion Scheme... 17 Chpter
More informationQUANTUM CHEMISTRY. Hückel Molecular orbital Theory Application PART I PAPER:2, PHYSICAL CHEMISTRY-I
Subject PHYSICAL Pper No nd Title TOPIC Sub-Topic (if ny) Module No., PHYSICAL -II QUANTUM Hückel Moleculr orbitl Theory CHE_P_M3 PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory TABLE OF CONTENTS.
More informationPhysics Graduate Prelim exam
Physics Grdute Prelim exm Fll 2008 Instructions: This exm hs 3 sections: Mechnics, EM nd Quntum. There re 3 problems in ech section You re required to solve 2 from ech section. Show ll work. This exm is
More information221A Lecture Notes WKB Method
A Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using ψ x, t = e
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More information2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm
2.57/2.570 Midterm Exm No. 1 Mrch 31, 2010 11:00 m -12:30 pm Instructions: (1) 2.57 students: try ll problems (2) 2.570 students: Problem 1 plus one of two long problems. You cn lso do both long problems,
More informationSECTION 9-4 Translation of Axes
9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationThe solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr
Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the e-vectors nd e-vlues
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2009
University of Wshington Deprtment of Chemistry Chemistry Winter Qurter 9 Homework Assignment ; Due t pm on //9 6., 6., 6., 8., 8. 6. The wve function in question is: ψ u cu ( ψs ψsb * cu ( ψs ψsb cu (
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationPhysics 220. Exam #1. April 21, 2017
Physics Exm # April, 7 Nme Plese red nd follow these instructions crefully: Red ll problems crefully before ttempting to solve them. Your work must be legible, nd the orgniztion cler. You must show ll
More informationOptimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.
Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd
More informationPh 219b/CS 219b. Exercises Due: Wednesday 9 March 2016
1 Ph 219b/CS 219b Eercises Due: Wednesdy 9 Mrch 2016 3.1 Positivity of quntum reltive entropy ) Show tht ln 1 for ll positive rel, with equlity iff = 1. b) The (clssicl) reltive entropy of probbility distribution
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More information1 2-D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2-D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +
More informationQuantum Physics III (8.06) Spring 2005 Solution Set 5
Quntum Physics III (8.06 Spring 005 Solution Set 5 Mrch 8, 004. The frctionl quntum Hll effect (5 points As we increse the flux going through the solenoid, we increse the mgnetic field, nd thus the vector
More informationSolutions to Problems in Merzbacher, Quantum Mechanics, Third Edition. Chapter 7
Solutions to Problems in Merzbcher, Quntum Mechnics, Third Edition Homer Reid April 5, 200 Chpter 7 Before strting on these problems I found it useful to review how the WKB pproimtion works in the first
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationVector potential quantization and the photon wave-particle representation
Vector potentil quntiztion nd the photon wve-prticle representtion Constntin Meis, Pierre-Richrd Dhoo To cite this version: Constntin Meis, Pierre-Richrd Dhoo. Vector potentil quntiztion nd the photon
More informationPHY4605 Introduction to Quantum Mechanics II Spring 2005 Final exam SOLUTIONS April 22, 2005
. Short Answer. PHY4605 Introduction to Quntum Mechnics II Spring 005 Finl exm SOLUTIONS April, 005 () Write the expression ψ ψ = s n explicit integrl eqution in three dimensions, ssuming tht ψ represents
More informationOrthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
More information3 Mathematics of the Poisson Equation
3 Mthemtics of the Poisson Eqution 3. Green functions nd the Poisson eqution () The Dirichlet Green function stisfies the Poisson eqution with delt-function chrge 2 G D (r, r o ) = δ 3 (r r o ) (3.) nd
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationC. Bulutay Topics on Semiconductor Physics. In This Lecture: Electronic Bandstructure: General Info
C. Buluty Topics on Semiconductor Physics In This Lecture: Electronic Bndstructure: Generl Info C. Buluty Topics on Semiconductor Physics Electronic Bndstructure Acronyms FPLAPW: Full-potentil linerized
More informationLinear Differential Equations Physics 129a Solutions to Problems Frank Porter Revision F. Porter
Liner Differentil Equtions Physics 19 Solutions to Problems 051018 Frnk Porter Revision 11106 F. Porter 1 Exercises 1. Consider the generl liner second order homogeneous differentil eqution in one dimemsion:
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationLecture 13 - Linking E, ϕ, and ρ
Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More informationlim P(t a,b) = Differentiate (1) and use the definition of the probability current, j = i (
PHYS851 Quntum Mechnics I, Fll 2009 HOMEWORK ASSIGNMENT 7 1. The continuity eqution: The probbility tht prticle of mss m lies on the intervl [,b] t time t is Pt,b b x ψx,t 2 1 Differentite 1 n use the
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner
More informationPH12b 2010 Solutions HW#3
PH 00 Solutions HW#3. The Hmiltonin of this two level system is where E g < E e The experimentlist sis is H E g jgi hgj + E e jei hej j+i p (jgi + jei) j i p (jgi jei) ) At t 0 the stte is j (0)i j+i,
More information1 APPLICATIONS OF SCHRÖDINGER S EQUATION AND BAND THEORY
1 APPLICATIONS OF SCHRÖDINGER S EQUATION AND BAND THEORY 1.1 INTRODUCTION We hve lredy noted tht Schrödinger ws influenced by the mtter wve postulte of de Broglie. In order to describe the behviour of
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
More informationpotentials A z, F z TE z Modes We use the e j z z =0 we can simply say that the x dependence of E y (1)
3e. Introduction Lecture 3e Rectngulr wveguide So fr in rectngulr coordintes we hve delt with plne wves propgting in simple nd inhomogeneous medi. The power density of plne wve extends over ll spce. Therefore
More informationk and v = v 1 j + u 3 i + v 2
ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationM344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
More informationENGI 9420 Lecture Notes 7 - Fourier Series Page 7.01
ENGI 940 ecture Notes 7 - Fourier Series Pge 7.0 7. Fourier Series nd Fourier Trnsforms Fourier series hve multiple purposes, including the provision of series solutions to some liner prtil differentil
More informationHeat flux and total heat
Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce
More information21.6 Green Functions for First Order Equations
21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to
More informationFor the percentage of full time students at RCC the symbols would be:
Mth 17/171 Chpter 7- ypothesis Testing with One Smple This chpter is s simple s the previous one, except it is more interesting In this chpter we will test clims concerning the sme prmeters tht we worked
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for One-Dimensionl Eqution The reen s function provides complete solution to boundry
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More information5 Probability densities
5 Probbility densities 5. Continuous rndom vribles 5. The norml distribution 5.3 The norml pproimtion to the binomil distribution 5.5 The uniorm distribution 5. Joint distribution discrete nd continuous
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationSTURM-LIOUVILLE THEORY, VARIATIONAL APPROACH
STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH XIAO-BIAO LIN. Qudrtic functionl nd the Euler-Jcobi Eqution The purpose of this note is to study the Sturm-Liouville problem. We use the vritionl problem s
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationModule 2: Rate Law & Stoichiomtery (Chapter 3, Fogler)
CHE 309: Chemicl Rection Engineering Lecture-8 Module 2: Rte Lw & Stoichiomtery (Chpter 3, Fogler) Topics to be covered in tody s lecture Thermodynmics nd Kinetics Rection rtes for reversible rections
More informationVariational Techniques for Sturm-Liouville Eigenvalue Problems
Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationPoint Lattices: Bravais Lattices
Physics for Solid Stte Applictions Februry 18, 2004 Lecture 7: Periodic Structures (cont.) Outline Review 2D & 3D Periodic Crystl Structures: Mthemtics X-Ry Diffrction: Observing Reciprocl Spce Point Lttices:
More informationChapter 2. Vectors. 2.1 Vectors Scalars and Vectors
Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More information