December 4, U(x) = U 0 cos 4 πx 8
|
|
- Lionel King
- 6 years ago
- Views:
Transcription
1 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 Rm 11 Office hours: TR 3 pm- pm Plese help your instructor by doing your work netly Every (lgebric) finl result must be sup December, 013 P1 [0 points ] Consider 1D electron system subject to wek periodic potentil [ U(x) = U 0 cos πx 3 ], 8 Find nd plot the dispersions of energy bnds in the first Brillouin zone Plot the energies in units of π /m, the wve numbers in units of 1/, nd ssume tht U 0 = 01 in these units Solution Using the trigonometric identity cos x = (1 + cos(x))/, the potentil cn be re-written s ( 1 πx U(x) = U 0 cos + 1 ) ( πx 1 cos = U 0 8 eiπx/ + 1 ) 16 eπix/ + cc There re two non-zero hrmonics with mplitudes u 1 = U 0 / nd u = U 0 /16 with wvenumbers q = π/ nd q = π/ Scttering t hrmonic with q = π/ lifts denegercy of the freeelectron sttes with energies ε 1 = k /m nd ε = (k π/) /m The degenercy points re k = π/ The energy levels ner these points re given by ε = ε 1 + ε (ε 1 ε ) + ( ) U0 Likewise, scttering t hrmonic with q = π/ lifts denegercy of the free-electron sttes with energies ε 1 = k /m nd ε 3 = (k π/) /m The degenercy points re k = π/ The energy levels ner these points re given by ε = ε 1 + ε 3 (ε 1 ε 3 ) + ( ) U0 16 1
2 P [0 points ] Consider D electrons subject to wek periodic potentil ( U(x, y) = U 0 cos πx + cos πy ) [8 points ] Find the energy bnds ner the edges of the first Brillouin zone (Hint: the edges of the Brillouin zone re the Brgg plnes where the eigensttes re doubly degenerte) Plot the bnds nd isoenergetic surfces Solution The non-zero hrmonics of the potentil hve wvenumbers (b, q y = 0) nd (q x = 0, q y = b), where b = π/, with mplitudes U 0 / Consider one of the edges, eg, k x = π/; by symmetry, the energy spectrum will be the sme t other edges The two free-electron sttes with energies ε 0 k x,k y = ( kx + ky) /m nd ε 0 kx π/,k y = ( (k x π/) + ky) /m re degenerte t this edge The energy levels ner the edge re given by ε = ε0 k x,k y + ε 0 k x π/,k y ) ) (ε 0kx,ky ε 0kx π/,ky + ( ) U0 b) [1 points ] Repet the nlysis of prt ) for regions ner the corners of the first Brillouin zone, where there re four degenerte eigensttes Solution The bsis must now include four degenerte sttes with energies ε 0 k x,k y, ε 0 k x b,k y, ε 0 k x b,k y q y, nd ε 0 k x b,k y b Using the fct the only non-zero hrmonics of the periodic potentil re U qx=b,0 = U qx=0,q y=b = U 0 /, we obtin equtions for the weighing fctors of the four degenerte sttes [cf Eq (919) in AM]: (ε ε 0 k x,k y )c kx,k y = U (c k x b,k y + c kx,k y b) (ε ε 0 k x b,k y )c kx b,k y = U (c k x,k y + c kx b,k y b) (ε ε 0 k x,k y b)c kx,k y b = U (c k x,k y + c kx b,k y b) (ε ε 0 k x b,k y b)c kx b,k y b = U (c k x b,k y + c kx,k y b) (1) Inverting the mtrix nd mesuring the wvenumbers in units of 1/ nd energies in units of /m, we obtin for the four energy levels where ε 1 = E 0 + R + ε = E 0 R + ε 3 = E 0 + R ε = E 0 R E 0 = kx + ky π(k x + k y ) + π R = 1 [ U0 + 3π (kx + ky π(k x + k y ) + π ) (U0 (π k x ) )(U0 + 16π (π k y ) ) P3 [0 points ] The Kronig-Penney model llows one to consider both the nerly-free-electron nd tight-binding limits Previously, we focused on the ltter Now, consider the (repulsive) Dirc-Kronig-Penney model, which, s we showed erlier, gives the following implicit eqution for the energy bnds: cos k = cos q + u sin q q, () ] 1/
3 where q = me/ nd u = mu 0 / > 0 Assuming tht u 1, find the solution of Eq () ner the Brgg points k π/, π/ (Reminder: the sign mens tht k is close yet not equl to the wvenumber of the Brgg point) Compre your result with tht of the nerly-free-electron model Solution The mximum vlue of the rhs of Eq () (= 1 + u) occurs t q = 0 The minimum vlue occurs ner q = π To see this, substitute q = π + δ nd expnd the rhs to second order in δ: sin q R cos q + u = 1 + δ q uδ π + R/ δ = 0 gives δ = u/π, nd the minimum vlue of R is R min = 1 u /π < 1 Since cos k 1, Eq () hs no soluton in the region round the minimum of R, where R min R 1 This region corresponds to gp in the energy spectrum To find the energy levels ner the gp, substitute k = π + p into Eq () nd expnd cos k in p s cos(π + p) = 1 + p / The resulting eqution δ π uδ p = 0 hs two roots δ = u ( u ) p π + π Returning to the originl vribles, we obtin for the energy levels where the lst line is vlid to first order in u (k E = π 1 m + u ) ( u ) π π 1 + π (k π 1 m + u ) ( u ) π π 1 + π, (3) Recll the result of the NFE model (with digonl mtrix elements U 11 nd U equl to zero) E = ε k + ε k π/ + U 1 (εk ε k π/ ) + U 1, () where ε k = k /m The difference of the free-electron energies cn be re-written s ε k ε k π/ = π ( ) k m π 1, while in their sum we cn replce k by π/: ε k + ε k π/ = π m Now it is obvious tht the NFE result is equivlent to Eq () P [15 points ] Find the dispersions of energy bnds for fcc nd bcc lttices tking into ccount hopping between the nerest nd next-to-nerest neighbors Solution:
4 The tight binding dispersion reltion is: E k = E 0 t Rnn e i k R nn t () cse of bcc 8 nn: (1, +1, +1), (1, +1, 1), (1, 1, +1), (1, 1, 1) 6 nnn: (1, 0, 0), (0, 1, 0), (0, 0, 1) R nnn e i k Rnnn (5) R bcc nn e i bcc k R nn = e (1/)i(+k x+k y+k z) + e (1/)i( kx+ky+kz) +e (1/)i(+kx+ky kz) + e (1/)i( kx+ky kz) +e (1/)i(+kx ky+kz) + e (1/)i( kx ky+kz) +e (1/)i(+kx ky kz) + e (1/)i( kx ky kz) = cos[(/)(k x + k y + k z )] + cos[(/)(k x k y + k z )] + cos[(/)(k x + k y k z )] + cos[(/)(k x k y k z )] = 8 cos(k x /) cos(k y /) cos(k y /) (6) R bcc nnn e i bcc k R nnn = e ik x + e ikx + e iky + e iky + e ikz + e ikz = cos(k x ) + cos(k y ) + cos(k z ) (7) (b) cse of fcc 1 nn: (+1, 0, 1), (0, +1, 1), ( 1, 0, 1), (0, 1, 1), (+1, +1, 0), (+1, 1, ), ( 1, +1, 0), ( 1, 1, 0) ; 6 nnn: (1, 0, 0), (0, 1, 0), (0, 0, 1); (sme s bcc) R fcc nn e i fcc k R nn = cos[(/)(kx + k y )] + cos[(/)(k x k y )] + cos[(/)(k x + k z )] + cos[(/)(k x k z )] + cos[(/)(k y + k z )] + cos[(/)(k y k z )] = [cos(k x /) cos(k y /) + cos(k y /) cos(k z /) + cos(k z /) cos(k x /)] (8) R fcc nnn e i fcc k R nnn = R bcc nnn e i bcc k R nnn (9) P5 [5 points ] Bilyer grphene consists of two grphene lyers stcked on top of ech other in Bernl mnner, ie, such tht only one of the two inequivlent lttice sites, eg, A, hs the nerest neighbor long the
5 verticl (see Fig 1) The tight-binding Hmiltonin in the nerest-neighbor-hopping pproximtion reds Ĥ = γ 0 â m,rˆb m,r γ 1 â 1,Râ,R + Hc (10) m,r,r R,R where â m nd ˆb m with m = 1, re the electron nnihiltion opertors for sites A nd B in the bottom (1) nd top () plnes, correspondingly, sum over R nd R includes only the nerest neighbors, nd prmeters γ 0 nd γ 1 re rel (FYI, γ 0 3 ev nd γ 1 0 ev) The plnes re seprted by distnce c ( 3Å) Find the energy spectrum for rbitrry k in the Brillouin zone Plot the isonergetic contours Anlyze the spectrum ner the K nd K points Solution The unit cell of grphene bilyer consists of four toms The electron stte is -spinor 1 ψ = b 1, b where 1 b re the mplitudes of finding n electron on site A 1 B In bsis of ψ sttes, the Hmiltonin cn be written s mtrix 0 S γ 1 0 Ĥ = S γ S, (11) 0 0 S 0 where S is the mtrix element connecting A nd B sites in the sme plne The top left nd bottom right blocks re the sme s for monolyer Digonlizing Hmiltonin in Eq (11), we obtin for the energy levels ε = γ 1 + S γ 1 + γ 1 S 3 S (1) Recll tht S = (3/)γ 0 k v F k ner the K nd K points, where k is mesured from either of these two points In the low-energy limit S γ 1, two out of the four brnches in Eq (1) pproch finite vlue (equl to γ 1 ), wheres the other two vnish Expnding these lst two dispersions in Tylor series, we find ε = S /γ 1 = ( v F /γ 1 )k k /m, where m γ 1 /v F Therefore, the low energy excittions in bilyer grphene re mssive electrons nd holes Degenercy of the spectrum t K nd K points is not lifted by inter-lyer coupling but the functionl form of the dispersion round the degenercy point is modified from liner to qudrtic
6 Figure 1: Crystl structure of bilyer grphene, from A H Cstro Neto, F Guine, N M R Peres, K S Novoselov, A K Geim, Rev Mod Phys 81, 109 (009)
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
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 informationPhysics 525, Condensed Matter Homework 3 Due Tuesday, 16 th October 2006
Physics 55 Condensed Mtter Homework Due Tuesdy 6 th October 6 Jcob Lewis Bourjily Problem : Electron in Wek Sinusoidl Potentil Consider n electron moving in one-dimensionl periodic potentil Ur = cosπr/.
More informationDepartment of Electrical and Computer Engineering, Cornell University. ECE 4070: Physics of Semiconductors and Nanostructures.
Deprtment of Electricl nd Computer Engineering, Cornell University ECE 4070: Physics of Semiconductors nd Nnostructures Spring 2014 Exm 2 ` April 17, 2014 INSTRUCTIONS: Every problem must be done in the
More informationfiziks 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
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 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 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 informationCrystalline Structures The Basics
Crystlline Structures The sics Crystl structure of mteril is wy in which toms, ions, molecules re sptilly rrnged in 3-D spce. Crystl structure = lttice (unit cell geometry) + bsis (tom, ion, or molecule
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 informationR. 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
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 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 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 informationThis final is a three hour open book, open notes exam. Do all four problems.
Physics 55 Fll 27 Finl Exm Solutions This finl is three hour open book, open notes exm. Do ll four problems. [25 pts] 1. A point electric dipole with dipole moment p is locted in vcuum pointing wy from
More informationg 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
More informationPHY 140A: Solid State Physics. Solution to Midterm #1
PHY 140A: Solid Stte Physics Solution to Midterm #1 TA: Xun Ji 1 October 24, 2006 1 Emil: jixun@physics.ucl.edu Problem #1 (20pt)Clculte the pcking frction of the body-centered cubic lttice. Solution:
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 informationJackson 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
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 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 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 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 informationLecture 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
More informationKai 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
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 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 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 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 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 informationElectromagnetism Answers to Problem Set 10 Spring 2006
Electromgnetism 76 Answers to Problem Set 1 Spring 6 1. Jckson Prob. 5.15: Shielded Bifilr Circuit: Two wires crrying oppositely directed currents re surrounded by cylindricl shell of inner rdius, outer
More informationChapter 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
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 informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationPhysics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011
Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you
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 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 informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
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 informationElectron Correlation Methods
Electron Correltion Methods HF method: electron-electron interction is replced by n verge interction E HF c E 0 E HF E 0 exct ground stte energy E HF HF energy for given bsis set HF Ec 0 - represents mesure
More informationLecture Outline. Dispersion Relation Electromagnetic Wave Polarization 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3c
Course Instructor Dr. Rymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mil: rcrumpf@utep.edu EE 4347 Applied Electromgnetics Topic 3c Wve Dispersion & Polriztion Wve Dispersion These notes & Polriztion
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More informationProblem 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 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 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
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 informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
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 informationQUB XRD Course. The crystalline state. The Crystalline State
QUB XRD Course Introduction to Crystllogrphy 1 The crystlline stte Mtter Gseous Stte Solid stte Liquid Stte Amorphous (disordered) Crystlline (ordered) 2 The Crystlline Stte A crystl is constructed by
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 informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationThe 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
More informationinteratomic distance
Dissocition energy of Iodine molecule using constnt devition spectrometer Tbish Qureshi September 2003 Aim: To verify the Hrtmnn Dispersion Formul nd to determine the dissocition energy of I 2 molecule
More informationDiscrete Least-squares Approximations
Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
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 informationTHE 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
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationthan 1. It means in particular that the function is decreasing and approaching the x-
6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the
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 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 informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More information63. 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 =
More informationTheoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4
WiSe 1 8.1.1 Prof. Dr. A.-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Mtthis Sb m Lehrstuhl für Theoretische Physik I Deprtment für Physik Friedrich-Alexnder-Universität Erlngen-Nürnberg Theoretische
More informationExam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B
PHY 249, Fll 216 Exm 1 Solutions nswer 1 is correct for ll problems. 1. Two uniformly chrged spheres, nd B, re plced t lrge distnce from ech other, with their centers on the x xis. The chrge on sphere
More informationOrthogonal Polynomials and Least-Squares Approximations to Functions
Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationDETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE
Determintion RevAdvMterSci of mechnicl 0(009) -7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING
More informationPhys. 506 Electricity and Magnetism Winter 2004 Prof. G. Raithel Problem Set 2 Total 40 Points. 1. Problem Points
Phys. 56 Electricity nd Mgnetism Winter 4 Prof. G. ithel Problem Set Totl 4 Points 1. Problem 8.6 1 Points : TM mnp : ω mnp = 1 µɛ x mn + p π y with y = L where m, p =, 1,.. nd n = 1,,.. nd x mn is the
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function
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 information4. Calculus of Variations
4. Clculus of Vritions Introduction - Typicl Problems The clculus of vritions generlises the theory of mxim nd minim. Exmple (): Shortest distnce between two points. On given surfce (e.g. plne), nd the
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 informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
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 information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationPHYSICS 116C Homework 4 Solutions
PHYSICS 116C Homework 4 Solutions 1. ( Simple hrmonic oscilltor. Clerly the eqution is of the Sturm-Liouville (SL form with λ = n 2, A(x = 1, B(x =, w(x = 1. Legendre s eqution. Clerly the eqution is of
More information13: 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
More informationMath 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
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 informationAPPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through
More informationEmission of K -, L - and M - Auger Electrons from Cu Atoms. Abstract
Emission of K -, L - nd M - uger Electrons from Cu toms Mohmed ssd bdel-rouf Physics Deprtment, Science College, UEU, l in 17551, United rb Emirtes ssd@ueu.c.e bstrct The emission of uger electrons from
More informationSOLUTION OF QUADRATIC NONLINEAR PROBLEMS WITH MULTIPLE SCALES LINDSTEDT-POINCARE METHOD. Mehmet Pakdemirli and Gözde Sarı
Mthemticl nd Computtionl Applictions, Vol., No., pp. 37-5, 5 http://dx.doi.org/.99/mc-5- SOLUTION OF QUADRATIC NONLINEAR PROBLEMS WITH MULTIPLE SCALES LINDSTEDT-POINCARE METHOD Mehmet Pkdemirli nd Gözde
More informationThe Basic Functional 2 1
2 The Bsic Functionl 2 1 Chpter 2: THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 2.1 Introduction..................... 2 3 2.2 The First Vrition.................. 2 3 2.3 The Euler Eqution..................
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationSUPPLEMENTARY 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
More informationLUMS School of Science and Engineering
LUMS School of Science nd Engineering PH- Solution of ssignment Mrch, 0, 0 Brvis Lttice Answer: We hve given tht c.5(î + ĵ + ˆk) 5 (î + ĵ + ˆk) 0 (î + ĵ + ˆk) c (î + ĵ + ˆk) î + ĵ + ˆk + b + c î, b ĵ nd
More informationMath 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
More informationHomework 4 , (1) 1+( NA +N D , (2)
Homework 4. Problem. Find the resistivity ρ (in ohm-cm) for piece of Si doped with both cceptors (N A = 9 cm 3 ) nd donors (N D = 6 cm 3 ). Since the electron nd hole mobilities depend on the concentrtion
More informationAnalytical Methods Exam: Preparatory Exercises
Anlyticl Methods Exm: Preprtory Exercises Question. Wht does it men tht (X, F, µ) is mesure spce? Show tht µ is monotone, tht is: if E F re mesurble sets then µ(e) µ(f). Question. Discuss if ech of the
More informationPart IB Numerical Analysis
Prt IB Numericl Anlysis Theorems with proof Bsed on lectures by G. Moore Notes tken by Dexter Chu Lent 2016 These notes re not endorsed by the lecturers, nd I hve modified them (often significntly) fter
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 informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationACM 105: Applied Real and Functional Analysis. Solutions to Homework # 2.
ACM 05: Applied Rel nd Functionl Anlysis. Solutions to Homework # 2. Andy Greenberg, Alexei Novikov Problem. Riemnn-Lebesgue Theorem. Theorem (G.F.B. Riemnn, H.L. Lebesgue). If f is n integrble function
More informationAMATH 731: Applied Functional Analysis Fall Additional notes on Fréchet derivatives
AMATH 731: Applied Functionl Anlysis Fll 214 Additionl notes on Fréchet derivtives (To ccompny Section 3.1 of the AMATH 731 Course Notes) Let X,Y be normed liner spces. The Fréchet derivtive of n opertor
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More information