# Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Size: px
Start display at page:

Transcription

1 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, where we mde three pproximtions Ignore the lttice bckground Ignore the interctions between electrons Ignore the interctions between phonons nd electrons In this chpter, we will tke the lttice bckground into considertion nd see wht will hppen Energy Bnds nd Bnd Gp Phenomenon Consider 1D solid. In the bsence of lttice bckground, the kinetic energy of one electron cn tke ny positive vlues (ϵ = p 2 /2 m > ). In the presence of lttice bckground, the kinetic energy (s function of the momentum) breks into pieces. Ech piece is known s n energy bnd. Between two energy bnds, there my be forbidden region, which the energy of n electron cn never enter. This forbidden region is clled the bnd gp. Fig. 1. The dispersion reltion for () free electron gs nd (b) electrons moving in crystl.

2 5 Phys463.nb Fig. 2. The probbility density for the ψ + nd ψ - wves Why re there two possible energy t k = π /? Here we consdier the gp tht ppers t the momentum k = ±π/. At these two momentum points, there re two liner-independent plne wves 1 ei π/ x nd 1 e-i π/ x. We cn form two stnding wves using these two plne wves ψ + (x) = ei π/ x -i π/ x + e 2 ψ - (x) = ei π/ x -i π/ x - e 2 i The probbility density is = 2/ cos π x = 2/ sin π x (7.1) (7.2) ρ ± = 2 ψ ± (x) 2 cos2 π x for + = 2 sin2 π x for - (7.3) In the bsence of lttices, these two stnding wves hve exctly the sme energy ϵ = ħ2 k 2 ħ 2 ± π 2 m = 2 2 m = ħ2 π2 2 m 2 So, we hve ϵ + = ϵ - = ħ 2 π 2 /2 m 2 However, in the presence of the lttice, the ψ + wve hs lower energy thn ψ -, becuse the electron hs higher probbly to pper ner n ion for ψ +. This is the reson why t momentum k = π/ we hve two possible energies (point A nd B in figure 1). Similr bnd gps will rise t k = n π/ for ny integer n. (7.4)

3 Phys463.nb 51 The momentum region -n π < k < -(n - 1) π nd (n - 1) π < k < n π is clled the nth Brillouin zones (This is the sme Brillouin zones s we lerned in the reciprocl lttice). In side the of these Brillouin zones, the energy is smooth function nd this smooth function is clled the nth bnd. At ech boundry of the Brillouin zones, the energy curve shows jump nd thus n energy gp opens up Mgnitude of the bnd gp The size of the gp cn be estimted s the following E g = U(x) ρ + (x) d x - U(x) ρ - (x) d x (7.5) where U(x) is the potentil energy from the ions. Becuse the lttice is periodic, U(x) is periodic function U(x + ) = U(x) where is the lttice constnt. In ddition, with out loss of generlity, we cn ssume tht U(x) is n even function U (x) = -U(x). For such n even periodic function, we cn write it s Fourier series. nd U(x) = U 2 + U n cos 2 π n n U n = 2 U(x) cos 2 π n x d x The energy gp t k = ±π/ is E g = x U(x)[ρ + (x) - ρ - (x)] d x = 2 π U(x) cos 2 x - sin2 The energy gp t k = ±n π/ is E g = π x d x = 2 U(x) cos 2 π (7.6) (7.7) x d x = U 1 (7.8) U(x)[ρ + (x) - ρ - (x)] d x = 2 U(x) cos 2 n π x - sin2 n π x d x = 2 U(x) cos 2 π n x d x = U n (7.9) 7.2. Bloch wves The effect of the lttice potentil: the Hmiltonin nd the Schrodinger eqution The Hmiltonin H = p2 ħ2 + U(x) = - 2 m 2 m x 2 +U(x) (7.1) The Schrodinger eqution - ħ2 2 m x 2 ψ(x) + U(x) ψ(x) = ϵ ψ(x) (7.11) This eqution is in generl hrd (or impossible) to solve nlyticlly. However, F. Bloch mnged to prove very importnt theorem, which sttes tht the solution to this eqution must tke the following form: ψ k (x) = u k (x) e i k x Here, u(x) is periodic function u k (x + ) = u k (x), which hs the sme period s the lttice. This conclusion is true in ny dimensions. For exmple, in 3D, we hve ψ r = u k r exp i k r where u r is 3D periodic function u k r = u k r + T where T is ny lttice vector. This type of wves re known s the Bloch wve. They hve some nice properties The plne wves re specil type of Bloch wves with the function u(x) = constnt. (7.12)

4 52 Phys463.nb The Bloch wve is NOT n eigenstte of the momentum opertor. In other words, it doesn t hve well-defined momentum (unless u(x) = constnt). However, the prmeter k in the Bloch wve behves very similrly to the momentum. We cll ħ k the lttice momentum. Lttice momentum is not the momentum, but it is conserved in lttice The proof of the Bloch theorem Define the trnsltion opertor T ψ(r) = ψ(r + ) (7.13) where is the lttice constnt. This opertor shift our quntum system in rel spce by the lttice constnt. Under trnsltion by, quntum opertor O chnges s O T O T -1. Therefore, for the Hmiltonin T H(x) T -1 = H(x + ) (7.14) Becuse nd H(x) = - ħ2 H(x + ) = - ħ2 2 m x 2 +U(x) (7.15) 2 m x 2 +U(x + ) = - ħ2 we know immeditely tht H (x) = H(x + ). And thus 2 m x 2 +U(x) (7.16) T H(x) T -1 = H(x + ) = H(x) (7.17) In other words, T H(x) = H(x) T (7.18) so [T, H] = (7.19) If two opertors commute, we know tht we cn find common eigensttes of these two opertors. For T nd H, this mens tht we cn find set of (complete orthonorml) bsis ψ k (x), such tht H ψ k (x) = ϵ ψ k (x) nd T ψ k (x) = e i k ψ k (x) (7.2) Here, we used the fct tht H is n Hermitin opertor, so tht its eigenvlues ϵ must be rel. For T, becuse it is unitry opertor, its eigenvlue must be complex with bsolute vlue 1. Define u k (x) = e -i k x ψ k (x) (7.21) It is esy to prove tht u k (x) is periodic function u k (x) = u k (x + ). u k (x + ) = e -i k (x+) ψ k (x + ) = e -i k (x+) T ψ k (x) = e -i k (x+) e i k ψ k (x) = e -i k x ψ k (x) = u k (x) (7.22) Therefore, ψ k (x) = u k (x) e i k x (7.23) where u k (x) is periodic function Properties of Bloch wves nd crystl momentum Eigen wve functions for single prticle moving in periodic potentil: Bloch wves ψ n k (x) = u n k(x) e i k x (7.24) Here k is known s the crystl momentum. The corresponding eigenenergy is ϵ n,k, which stisfies

5 Phys463.nb 53 H ψ n,k (x) = ϵ n,k ψ n,k (x) (7.25) vlue of k? The crystl moment re only well-defined modulo G, where G is reciprocl vector. Therefore, we cn limit the crystl momentum to the first Brillouin zone without loss of informtion (e.g. for 1D system, -π/ k < π/). Proof : Suppose we hve Bloch wve with crystl momentum k ψ n k (x) = u n k (x) e i k x (7.26) Define k ' = k + G ψ n k (x) = u n k (x) e i k x = u n k'-g (x) e i (k'-g ) x = u n k'-g (x) e -i G x e i k' x (7.27) Define u n k' (x) = u n, k'-g (x) e -i G x, ψ n k (x) = u n k'-g (x) e -i G x e i k' x = u n k' (x) e i k' x (7.28) It is esy to check tht u n,k' (x) = u n,k' (x + T) where T is ny lttice vector. u n,k' (x + T) = u n, k'-g (x + T) e -i G (x+t) = u n, k'-g (x) e -i G x = u n,k' (x) (7.29) Therefore, ψ n k (x) = u n k' (x) e i k' x is lso Bloch wve function with crystl momentum k ' ψ n k (x) = u n k' (x) e i k' x = ψ n,k' (x) (7.3) Conclusion: s long s k nd k ' differs by reciprocl lttice vector G, we cn write Bloch wvefunction with crystl momentum k s Bloch wvefunction with crystl momentum k '. In other words, crystl momentum is only well defined modulo G Fig. 3. Dispersion reltion ϵ n (k) shown in extended zone, reduced zone nd periodic zone. (Figure 4 in text book pge 225).

6 54 Phys463.nb Reduced Brillouin zone Since the crystl momentum is only well defined modulo G, we cn limit the momentum to the first Brillouin zone (e.g. for 1D system, -π/ k < π/ ) without loss of informtion. We cn plot the eigen-energy ϵ n,k s function of k in the first Brillouin zone. This wy of plotting ϵ n,k is known s the reduced Brillouin zone (See figure b bove) The bnd index n is (from bottom to top) n = 1, 2, 3, Periodic Brillouin zone There is nother wy of plotting ϵ n,k. We cn strt from the reduced Brillouin zone nd then we plot ϵ n,k s periodic function of k, repeting the sme curve in the second, third,... Brillouin zone (See figure c bove) Folding, Reduced Brillouin zone nd extended Brillouin zone for free prticles without lttices We know tht plne wves is specil cse of Bloch wves (where the periodic potentil is V = ). Therefore, we cn present the dispersion of free prticle ϵ = ħ 2 k 2 /2 m in the sme wy (in the reduced BZ nd periodic BZ). The cn be chieve using the following procedures: strt from the dispersion ϵ(k) = ħ 2 k 2 /2 m, then move the curve in the regions (2 n - 1) π/ < k < 2 n π/ to -π/ < k < nd move the curve in the regions 2 n π/ < k < (2 n + 1) π/ to < k < π/. So we get the figure below. Fig. 4. Reduced zone for free prticle. (textbook p 177 Fig. 8) This construction is known s Brillouin zone folding. Similrly, the reverse procedure is known s unfolding. We cn unfold the reduced Brillouin zone to get the extended Brillouin zone, going bck to the cse - < k < + nd get one single curve with k 2 /2 m for free prticle.

### Quantum 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

### The 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

### Massachusetts 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

### Chapter 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

### Name 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

### Practice 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

### Chapter 3 The Schrödinger Equation and a Particle in a Box

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

### Do 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

### C. 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

### dx 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

### Continuous 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

### 221B 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

### Quantum 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

### Lecture 8. Band theory con.nued

Lecture 8 Bnd theory con.nued Recp: Solved Schrodinger qu.on for free electrons, for electrons bound in poten.l box, nd bound by proton. Discrete energy levels rouse. The Schrodinger qu.on pplied to periodic

### 221A 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

### Summary: 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

### PHY4605 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

### Homework 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:

### ODE: Existence and Uniqueness of a Solution

Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

### Quantum 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

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

### Infinite Geometric Series

Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to

### ( ) 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

### Quantum Analogs Chapter 4 Student Manual

Quntum Anlogs Chpter 4 Student Mnul Modeling One Dimensionl Solid Professor Rene Mtzdorf Universitet Kssel Stud. Mn. Rev 2.0 12/09 4. Modeling one-dimensionl solid There re two different wys to explin

### MATH34032: 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

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

### Classical 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,

### Point 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:

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

### Problem 3: Band Structure of YBa 2 Cu 3 O 7

HW 5 SSP 601-2017. here is very relistic clcultion which uses the concepts of lttice, reciprocl spce, Brillouin zone nd tight-binding pproximtion. Go over the solution nd fill up every step nd every detil

### Aike 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

### 2.57/2.570 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,

### 7.2 The Definite Integral

7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

### Theoretical foundations of Gaussian quadrature

Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

### Physics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2011

Physics 74 Grdute Quntum Mechnics Solutions to Finl Exm, Fll 0 You my use () clss notes, () former homeworks nd solutions (vilble online), (3) online routines, such s Clebsch, provided by me, or (4) ny

### Solutions 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

### PH12b 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,

### Physics 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

### 20 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

### g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f

1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where

### Math 360: A primitive integral and elementary functions

Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

### We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.

Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn

### Jack 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,

### Problems for HW X. C. Gwinn. November 30, 2009

Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object

### R. I. Badran Solid State Physics

I Bdrn Solid Stte Physics Crystl vibrtions nd the clssicl theory: The ssmption will be mde to consider tht the men eqilibrim position of ech ion is t Brvis lttice site The ions oscillte bot this men position

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

### The 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

### MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

### State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

### 1 1D heat and wave equations on a finite interval

1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

### Problem 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

### ARITHMETIC 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

### 5.04 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

### NOTES 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

### Physics 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

### STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2

### Chapter 14. Matrix Representations of Linear Transformations

Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

### Numerical integration

2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

### Kinematic Waves. These are waves which result from the conservation equation. t + I = 0. (2)

Introduction Kinemtic Wves These re wves which result from the conservtion eqution E t + I = 0 (1) where E represents sclr density field nd I, its outer flux. The one-dimensionl form of (1) is E t + I

### Taylor Polynomial Inequalities

Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil

### The Dirac distribution

A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution

### Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

### 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,

### Best Approximation in the 2-norm

Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

### Kai Sun. University of Michigan, Ann Arbor

Ki Sun University of Michign, Ann Arbor How to see toms in solid? For conductors, we cn utilize scnning tunneling microscope (STM) to see toms (Nobel Prize in Physics in 1986) Limittions: (1) conductors

### Notes on length and conformal metrics

Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued

### Ph2b 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

### Numerical Integration

Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the

### AQA 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

### 4.4 Areas, Integrals and Antiderivatives

. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

### Math Lecture 23

Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of

### Recitation 3: More Applications of the Derivative

Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

### 13: Diffusion in 2 Energy Groups

3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

### Chapter 3 Polynomials

Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling

### Convergence of Fourier Series and Fejer s Theorem. Lee Ricketson

Convergence of Fourier Series nd Fejer s Theorem Lee Ricketson My, 006 Abstrct This pper will ddress the Fourier Series of functions with rbitrry period. We will derive forms of the Dirichlet nd Fejer

### SUPPLEMENTARY INFORMATION

DOI:.38/NMAT343 Hybrid Elstic olids Yun Li, Ying Wu, Ping heng, Zho-Qing Zhng* Deprtment of Physics, Hong Kong University of cience nd Technology Cler Wter By, Kowloon, Hong Kong, Chin E-mil: phzzhng@ust.hk

### Chapter 28. Fourier Series An Eigenvalue Problem.

Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why

### Separation of Variables in Linear PDE

Seprtion of Vribles in Liner PDE Now we pply the theory of Hilbert spces to liner differentil equtions with prtil derivtives (PDE). We strt with prticulr exmple, the one-dimensionl (1D) wve eqution 2 u

### MASSACHUSETTS 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

### Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

### THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

### Math 61CM - Solutions to homework 9

Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ

### December 4, U(x) = U 0 cos 4 πx 8

PHZ66: Fll 013 Problem set # 5: Nerly-free-electron nd tight-binding models: Solutions due Wednesdy, 11/13 t the time of the clss Instructor: D L Mslov mslov@physufledu 39-0513 Rm 11 Office hours: TR 3

### CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

### Math 5440 Problem Set 3 Solutions

Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 213 1: (Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping

### Lecture 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

### Stuff You Need to Know From Calculus

Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you

### fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics

Solid Stte Physics JEST-0 Q. bem of X-rys is incident on BCC crystl. If the difference between the incident nd scttered wvevectors is K nxˆkyˆlzˆ where xˆ, yˆ, zˆ re the unit vectors of the ssocited cubic

### Lecture 3. Limits of Functions and Continuity

Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

### Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.

Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous

### Chapter 3 Solving Nonlinear Equations

Chpter 3 Solving Nonliner Equtions 3.1 Introduction The nonliner function of unknown vrible x is in the form of where n could be non-integer. Root is the numericl vlue of x tht stisfies f ( x) 0. Grphiclly,

### Math 5440 Problem Set 3 Solutions

Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 25 1: Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping

### Chapter 3. Vector Spaces

3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct