1 Rabi oscillations. Physical Chemistry V Solution II 8 March 2013
|
|
- Kelly Edwards
- 5 years ago
- Views:
Transcription
1 Physcal Chemstry V Soluton II 8 March Rab oscllatons a The key to ths part of the exercse s correctly substtutng c = b e ωt. You wll need the followng equatons: b = c e ωt 1 db dc = dt dt ωc e ωt. Keepng n mnd that the cosne can be expressed n terms of complex exponentals cosωt = 1 e ωt + e ωt. 3 Settng b 1 = c 1 and ω 1 = 0 as requested, Equaton 1.96 becomes dc 1 dt = V 1 eωt + e ωt c e ωt = V e ωt c V 1 c 4 where we have dropped the e ωt term due to the rotatng wave approxmaton mentoned n the problem. Carryng out the same procedure on Equaton 1.97 from the lecture notes gves dc dt ωc dc dt ωc e ωt = V 1 dc dt eωt + e ωt c 1 + ω c e ωt, = V 1 eωt + 1c 1 + ω c and nally = V 1 c 1 + ω ωc. 5 b In the resonant case ω = ω the problem reduces to a par of symmetrc equatons dc 1 dt dc dt = V 1 c 6 = V 1 c 1. 7 Derentatng Equaton 7 gves d c dt = V 1 dc 1 dt. 8 1 / 8
2 Physcal Chemstry V Soluton II 8 March 013 Insertng Equaton 6 and takng nto account the equalty V 1 = V 1 d c dt = V1 c 9 we see that the tme behavour of c s descrbed by a harmonc oscllaton, whose general soluton reads V1 c t = A sn t + ϕ. 10 Snce the derental equaton par s symmetrc, c 1 must harmoncally oscllate as well. Consderng normalzaton c 1 + c = 1 we end up wth where ϕ s determned by the ntal condtons. V1 c 1 t = cos t + ϕ 11 V1 c t = sn t + ϕ, 1 c c t = sn V1 t + ϕ = 1 cosv 1t + ϕ. 13 The characterstc oscllaton frequency s the Rab frequency V 1. Addtonal Informaton: Physcal meanng and applcatons. In ther work I. Gerhardt et al. were able to montor Rab oscllatons between the ground and the excted state of dbenzantanthrene DBATT [1]. Rab cycles were recorded durng the passage of a long several ns pulse on resonance and the subsequent decay of the excted state populaton by spontaneous emsson to vbratonal levels of the electronc ground state see Fg.1 left. Ths emsson was detected as the Stokes uorescence sgnal whch, to a very good approxmaton, s proportonal to the populaton of the excted state. Ths means that durng the Rab oscllaton the excted state populaton can be montored by measurng the ntensty of the Stokes uorescence sgnal. The result of these studes s presented n Fgure 1. The dashed lne s the exctaton pulse energy. The red lne s the measured, oscllatng Stokes uorescence whch corresponds to an oscllatng excted state populaton. The oscllaton cycles are observed only durng the passage of the laser pulse,.e. whle the lght-matter nteracton s domnated by absorpton and stmulated emsson. The spontaneous emsson channel s used only to record the excted state populaton. After the passage of the laser pulse, the excted state only relaxes through the spontaneous emsson channel. Ths s typcally an exponental decay of an excted state, whch can also be seen n the measured curve. / 8
3 Physcal Chemstry V Soluton II 8 March 013 Fgure 1: Left: Level scheme of a sngle DBATT. Rght: Raw Stokes-shfted uorescence of a sngle molecule red curve and a theoretcal t black curve as a functon of delay wth respect to the optcal pulse. The blue dashed lne dsplays the measured exctaton pulse ntensty [1]. Two oscllaton cycles are observed at ths partcular pulse length ndcated n Fgure 1. One Rab cycle s completed after 3 ns. [1] I. Gerhardt, G. Wrgge, G. Zumofen, J. Hwang, A. Renn, and V. Sandoghdar, Phys. Rev. A 79, Ladder operator formalsm a ˆX can be regarded as the real part of â, ˆP as ts magnary part. Thus ˆX = mω â + â 14 mω ˆP = â â 15 b [â, â ] = ââ â â = mω = mω mω = 1 ˆX + ˆP m ω mω [ ˆX, ˆP ] ˆX ˆP m ω mω [ ˆX, ˆP ] 16 3 / 8
4 Physcal Chemstry V Soluton II 8 March 013 c â â = mω = 1 ω = 1 ω ˆX + ˆP m ω + mω ˆX H ω mω [ ˆX, ˆP ] + ˆP m + ω [ ˆX, ˆP ] The Schrödnger equaton cann generalbe wrtten lke 17 H ψ = ψ 18 t where ψ s not necessarly an egenstate and can also be tme-dependent. The tmendependent case s concerned wth the egenstates of the Hamltonan: H φ = E φ 19 As can be seen from Equaton 17, the tme-ndependent Schrödnger equaton for our case reads lke: where φ n are egenstates wth energy E E n. ω â â + 1 φ = E φ 0 d We want to know the energy of state â n,. e. calculate Hâ n. Our goal s to commute H and â so that we can reduce the problem to the known egenvalue relaton of n. Hâ n = ω commutator = ω = ω = â ω â ââ + â = âh ω n = âe n ω n = E n ωâ n = E n 1 â n n ââ â [â, â ]â + â ââ â â + â n â â n n = E n 1 n / 8
5 Physcal Chemstry V Soluton II 8 March 013 e Have a look at the arbtrary state φ = â ψ. The norm φ φ of any state s postve by denton φ φ 0. From equaton 6 and 7 gven n the exercse sheet one can see that â = â and thus â ψ = âψ = ψ â. 3 We conclude âψ âψ = ψ â â ψ = ψ ˆn ψ 0 4 f From problem.d we see that â n must be proportonal to n 1,. e. â n = c n 1, c C 5 Takng the norm on both sdes leads to n â â n = c n 1 n 1 n ˆn n = c n 1 n 1 n n n = c n 1 n 1 n = c c = n 6 Applyng ths result to the ground state: â 0 = 0 as c = Probng and controllng vbratonal wave packets a Assume that the fs pulse creates a coherent superposton of states φ n q of the 1D harmonc oscllator wth coecents c n t, where q s the spatal vbratonal coordnate and n s the assocated quantum number. From the spatal representaton of ψ0, ψq, 0 = q ψ0 = n c n0φ n q, and the expresson of φ n q n Equaton we obtan ψq, 0 = e q / n c n 0 n n! H nq, 8 5 / 8
6 Physcal Chemstry V Soluton II 8 March 013 where H n q are Hermte polynomals. Replacng c n 0 wth 1/ n n! n the prevous equaton yelds ψq, 0 = e q / n 1 n n! H nq. 9 We easly recognze that the expresson contans the exponental generatng functon of Hermte polynomals H n q, expqw w = n H nqw n /n!, wth w = 1/. Hence ψq, 0 = 4 e π e 1 q b After the pulse rradaton the wave packet evolves accordng to the Hamltonan of the 1D harmonc oscllator to yeld ψq, t = n c n tφ n q = n e Ent/ c n 0φ n q, 31 where E n = ωn+1/ are the energy levels. A few smple steps lead us to the expresson ψq, t = e q / e ωt/ n H n q n! e ωt n. 3 Once more we take advantage of the exponental generatng functon to derve the more compact and elegant form: ψq, t = e ωt/ e ωt exp 4 e 1 q exp ωt. 33 The wave packet has a center of mass q 0 that moves back and forth as q 0 = e ωt,.e. the moton s perodc. c At tme t we just need to supermpose the result of Equaton 30 wth that of Equaton 33 for t = t. Recallng that Ψ t, φ = e φ ψ0 + ψ t we obtan Ψq; t, φ = q Ψ t, φ = e φ 4 e π e 1 q 1 + e ω t/ d The uorescence sgnal s proportonal to Ψ t, φ. e ω t exp 4 e 1 q exp ω t. 34 We smplfy Equaton 34 by assumng ω t 1. Under ths condton we can approxmate usng the power seres expanson e ω t 1 ω t. Wth ω t 1 ths becomes e ω t 1 and we can wrte Ψq; t, φ e φ 4 e π e 1 q e π e ω t e 1 q 1+ω t / 8
7 Physcal Chemstry V Soluton II 8 March 013 A few more algebrac operatons and droppng terms of the order of ω t lead to Ψq; t, φ = 4 e π e 1 q 1 e qω t + e φ. 36 From ths we get Ψ t, φ : Ψ t, φ = Ψ t, φψ t, φ e = 4 π e 1 q 1 e qω t + e φ e 4 π e 1 q 1 e qω t + e φ 37 e = π e q 1 e qω t + e φ e qω t + e φ 38 e = π e q e qω t+φ + e qω t+φ e = π e q 1 + cosqω t + φ 40 Ths means, that the uorescence s a snusodal functon of the delay tme t between the pulses and also a functon of the phase φ. Addtonal Informaton: Physcal meanng and applcatons. In contrast to the long some ns resonant exctaton n the case of Rab oscllaton, Brnks et al. appled two broad-band, femtosecond pulses separated by a tme t n order to generate vbratonal wave packets and to observe ther nterference []. They also examned how the phase aects the uorescence sgnal. Accordng to the result from exercse d the uorescence sgnal from the nterference of two vbratonal wave packets has to be a cosne functon. Moreover, the phase φ sgncantly determnes ths nterference. Ths s exactly what Brnks et al. have observed. Fgure shows the uorescence sgnal measured at derent delay tmes t for n-phase φ = 0 and n-antphase φ = π exctaton pulses. One can clearly see the oscllatng behavor as predcted by the cosne functon and also the nuence of the phase factor. In ths case, the n-phase and n-antphase exctaton pulses lead to an nverted molecular response. Vbratonal relaxaton, not consdered n the exercse, s supermposed on the oscllatons. [] Daan Brnks et al. Nature Letters 465, / 8
8 Physcal Chemstry V Soluton II 8 March 013 Fgure : Sngle-molecule uorescence ntensty as a functon of the tme delay between two n-phase φ = 0 and n-antphase φ = π exctaton pulses. Some molecules present uctuatons even for tme delays t as long as 10 fs and wth more than one frequency component []. 8 / 8
SUPPLEMENTARY INFORMATION
do: 0.08/nature09 I. Resonant absorpton of XUV pulses n Kr + usng the reduced densty matrx approach The quantum beats nvestgated n ths paper are the result of nterference between two exctaton paths of
More informationψ = i c i u i c i a i b i u i = i b 0 0 b 0 0
Quantum Mechancs, Advanced Course FMFN/FYSN7 Solutons Sheet Soluton. Lets denote the two operators by  and ˆB, the set of egenstates by { u }, and the egenvalues as  u = a u and ˆB u = b u. Snce the
More information14 The Postulates of Quantum mechanics
14 The Postulates of Quantum mechancs Postulate 1: The state of a system s descrbed completely n terms of a state vector Ψ(r, t), whch s quadratcally ntegrable. Postulate 2: To every physcally observable
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationSnce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t
8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationDesigning Information Devices and Systems II Spring 2018 J. Roychowdhury and M. Maharbiz Discussion 3A
EECS 16B Desgnng Informaton Devces and Systems II Sprng 018 J. Roychowdhury and M. Maharbz Dscusson 3A 1 Phasors We consder snusodal voltages and currents of a specfc form: where, Voltage vt) = V 0 cosωt
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 19 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Devce Applcatons Class 9 Group Theory For Crystals Dee Dagram Radatve Transton Probablty Wgner-Ecart Theorem Selecton Rule Dee Dagram Expermentally determned energy
More informationRate of Absorption and Stimulated Emission
MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More information5.04, Principles of Inorganic Chemistry II MIT Department of Chemistry Lecture 32: Vibrational Spectroscopy and the IR
5.0, Prncples of Inorganc Chemstry II MIT Department of Chemstry Lecture 3: Vbratonal Spectroscopy and the IR Vbratonal spectroscopy s confned to the 00-5000 cm - spectral regon. The absorpton of a photon
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationLevel Crossing Spectroscopy
Level Crossng Spectroscopy October 8, 2008 Contents 1 Theory 1 2 Test set-up 4 3 Laboratory Exercses 4 3.1 Hanle-effect for fne structure.................... 4 3.2 Hanle-effect for hyperfne structure.................
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More information1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:
68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse
More informationAdvanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017)
Advanced rcuts Topcs - Part by Dr. olton (Fall 07) Part : Some thngs you should already know from Physcs 0 and 45 These are all thngs that you should have learned n Physcs 0 and/or 45. Ths secton s organzed
More informationRobert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations
Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationDegenerate PT. ψ φ λψ. When two zeroth order states are degenerate (or near degenerate), cannot use simple PT.
Degenerate PT When two zeroth order states are degenerate (or near degenerate), cannot use smple PT. Degenerate PT desgned to deal wth such cases Suppose the energy level of nterest s r fold degenerate
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More informationFrequency dependence of the permittivity
Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationMolecular structure: Diatomic molecules in the rigid rotor and harmonic oscillator approximations Notes on Quantum Mechanics
Molecular structure: Datomc molecules n the rgd rotor and harmonc oscllator approxmatons Notes on Quantum Mechancs http://quantum.bu.edu/notes/quantummechancs/molecularstructuredatomc.pdf Last updated
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationSupplementary Information for Observation of Parity-Time Symmetry in. Optically Induced Atomic Lattices
Supplementary Informaton for Observaton of Party-Tme Symmetry n Optcally Induced Atomc attces Zhaoyang Zhang 1,, Yq Zhang, Jteng Sheng 3, u Yang 1, 4, Mohammad-Al Mr 5, Demetros N. Chrstodouldes 5, Bng
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationTitle: Radiative transitions and spectral broadening
Lecture 6 Ttle: Radatve transtons and spectral broadenng Objectves The spectral lnes emtted by atomc vapors at moderate temperature and pressure show the wavelength spread around the central frequency.
More informationThe non-negativity of probabilities and the collapse of state
The non-negatvty of probabltes and the collapse of state Slobodan Prvanovć Insttute of Physcs, P.O. Box 57, 11080 Belgrade, Serba Abstract The dynamcal equaton, beng the combnaton of Schrödnger and Louvlle
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationDepartment of Chemistry Purdue University Garth J. Simpson
Objectves: 1. Develop a smple conceptual 1D model for NLO effects. Extend to 3D and relate to computatonal chemcal calculatons of adabatc NLO polarzabltes. 2. Introduce Sum-Over-States (SOS) approaches
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationMathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt
Appled Mathematcs Volume 2, Artcle ID 4539, pages do:.55/2/4539 Research Artcle A Treatment of the Absorpton Spectrum for a Multphoton V -Type Three-Level Atom Interactng wth a Squeezed Coherent Feld n
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More information8. Superfluid to Mott-insulator transition
8. Superflud to Mott-nsulator transton Overvew Optcal lattce potentals Soluton of the Schrödnger equaton for perodc potentals Band structure Bloch oscllaton of bosonc and fermonc atoms n optcal lattces
More informationΔ x. u(x,t) Fig. Schematic view of elastic bar undergoing axial motions
ME67 - Handout 4 Vbratons of Contnuous Systems Axal vbratons of elastc bars The fgure shows a unform elastc bar of length and cross secton A. The bar materal propertes are ts densty ρ and elastc modulus
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationBoundaries, Near-field Optics
Boundares, Near-feld Optcs Fve boundary condtons at an nterface Fresnel Equatons : Transmsson and Reflecton Coeffcents Transmttance and Reflectance Brewster s condton a consequence of Impedance matchng
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationProf. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model
EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationSpin. Introduction. Michael Fowler 11/26/06
Spn Mchael Fowler /6/6 Introducton The Stern Gerlach experment for the smplest possble atom, hydrogen n ts ground state, demonstrated unambguously that the component of the magnetc moment of the atom along
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15
NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound
More informationThe optimal delay of the second test is therefore approximately 210 hours earlier than =2.
THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple
More information( ) 1/ 2. ( P SO2 )( P O2 ) 1/ 2.
Chemstry 360 Dr. Jean M. Standard Problem Set 9 Solutons. The followng chemcal reacton converts sulfur doxde to sulfur troxde. SO ( g) + O ( g) SO 3 ( l). (a.) Wrte the expresson for K eq for ths reacton.
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationGrover s Algorithm + Quantum Zeno Effect + Vaidman
Grover s Algorthm + Quantum Zeno Effect + Vadman CS 294-2 Bomb 10/12/04 Fall 2004 Lecture 11 Grover s algorthm Recall that Grover s algorthm for searchng over a space of sze wors as follows: consder the
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationApplied Nuclear Physics (Fall 2004) Lecture 23 (12/3/04) Nuclear Reactions: Energetics and Compound Nucleus
.101 Appled Nuclear Physcs (Fall 004) Lecture 3 (1/3/04) Nuclear Reactons: Energetcs and Compound Nucleus References: W. E. Meyerhof, Elements of Nuclear Physcs (McGraw-Hll, New York, 1967), Chap 5. Among
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationThis model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:
1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationUniversity of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014
Lecture 16 8/4/14 Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 214. Real Vapors and Fugacty Henry s Law accounts or the propertes o extremely dlute soluton. s shown n Fgure
More informationG = G 1 + G 2 + G 3 G 2 +G 3 G1 G2 G3. Network (a) Network (b) Network (c) Network (d)
Massachusetts Insttute of Technology Department of Electrcal Engneerng and Computer Scence 6.002 í Electronc Crcuts Homework 2 Soluton Handout F98023 Exercse 21: Determne the conductance of each network
More informationPhysics 4B. A positive value is obtained, so the current is counterclockwise around the circuit.
Physcs 4B Solutons to Chapter 7 HW Chapter 7: Questons:, 8, 0 Problems:,,, 45, 48,,, 7, 9 Queston 7- (a) no (b) yes (c) all te Queston 7-8 0 μc Queston 7-0, c;, a;, d; 4, b Problem 7- (a) Let be the current
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More information( ) = ( ) + ( 0) ) ( )
EETOMAGNETI OMPATIBIITY HANDBOOK 1 hapter 9: Transent Behavor n the Tme Doman 9.1 Desgn a crcut usng reasonable values for the components that s capable of provdng a tme delay of 100 ms to a dgtal sgnal.
More information9 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 - Chapter 9R -Davd Klenfeld - Fall 2005 9 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys a set
More information4. INTERACTION OF LIGHT WITH MATTER
Andre Tokmakoff, MIT Department of Chemstry, 3/8/7 4-1 4. INTERACTION OF LIGHT WITH MATTER One of the most mportant topcs n tme-dependent quantum mechancs for chemsts s the descrpton of spectroscopy, whch
More informationBe true to your work, your word, and your friend.
Chemstry 13 NT Be true to your work, your word, and your frend. Henry Davd Thoreau 1 Chem 13 NT Chemcal Equlbrum Module Usng the Equlbrum Constant Interpretng the Equlbrum Constant Predctng the Drecton
More informationHow Differential Equations Arise. Newton s Second Law of Motion
page 1 CHAPTER 1 Frst-Order Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons
More informationQuantum Mechanics I - Session 4
Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More informationIntroduction to Super-radiance and Laser
Introducton to Super-radance and Laser Jong Hu Department of Physcs and Astronomy, Oho Unversty Abstract Brefly dscuss the absorpton and emsson processes wth the energy levels of an atom. Introduce and
More informationSolution 1 for USTC class Physics of Quantum Information
Soluton 1 for 018 019 USTC class Physcs of Quantum Informaton Shua Zhao, Xn-Yu Xu and Ka Chen Natonal Laboratory for Physcal Scences at Mcroscale and Department of Modern Physcs, Unversty of Scence and
More informationCausal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
More informationDynamics of a Superconducting Qubit Coupled to an LC Resonator
Dynamcs of a Superconductng Qubt Coupled to an LC Resonator Y Yang Abstract: We nvestgate the dynamcs of a current-based Josephson juncton quantum bt or qubt coupled to an LC resonator. The Hamltonan of
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationSolutions for Homework #9
Solutons for Hoewor #9 PROBEM. (P. 3 on page 379 n the note) Consder a sprng ounted rgd bar of total ass and length, to whch an addtonal ass s luped at the rghtost end. he syste has no dapng. Fnd the natural
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationΔ x. u(x,t) Fig. Schematic view of elastic bar undergoing axial motions
ME67 - Handout 4 Vbratons of Contnuous Systems Axal vbratons of elastc bars The fgure shows a unform elastc bar of length and cross secton A. The bar materal propertes are ts densty ρ and elastc modulus
More informationNon-interacting Spin-1/2 Particles in Non-commuting External Magnetic Fields
EJTP 6, No. 0 009) 43 56 Electronc Journal of Theoretcal Physcs Non-nteractng Spn-1/ Partcles n Non-commutng External Magnetc Felds Kunle Adegoke Physcs Department, Obafem Awolowo Unversty, Ile-Ife, Ngera
More informationMEASUREMENT OF MOMENT OF INERTIA
1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us
More informationPerfect Fluid Cosmological Model in the Frame Work Lyra s Manifold
Prespacetme Journal December 06 Volume 7 Issue 6 pp. 095-099 Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold Perfect Flud Cosmologcal Model n the Frame Work Lyra
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationHomework 4. 1 Electromagnetic surface waves (55 pts.) Nano Optics, Fall Semester 2015 Photonics Laboratory, ETH Zürich
Homework 4 Contact: frmmerm@ethz.ch Due date: December 04, 015 Nano Optcs, Fall Semester 015 Photoncs Laboratory, ETH Zürch www.photoncs.ethz.ch The goal of ths problem set s to understand how surface
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons
More information8.592J: Solutions for Assignment 7 Spring 2005
8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1
More informationLagrangian Field Theory
Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,
More informationSupporting Information Part 1. DFTB3: Extension of the self-consistent-charge. density-functional tight-binding method (SCC-DFTB)
Supportng Informaton Part 1 DFTB3: Extenson of the self-consstent-charge densty-functonal tght-ndng method SCC-DFTB Mchael Gaus, Qang Cu, and Marcus Elstner, Insttute of Physcal Chemstry, Karlsruhe Insttute
More informationAPPENDIX 2 FITTING A STRAIGHT LINE TO OBSERVATIONS
Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 1 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS In the physcal measurements we often make a seres of measurements of the dependent
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More informationLecture 2: Numerical Methods for Differentiations and Integrations
Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015
Lecture 2. 1/07/15-1/09/15 Unversty of Washngton Department of Chemstry Chemstry 453 Wnter Quarter 2015 We are not talkng about truth. We are talkng about somethng that seems lke truth. The truth we want
More information