Statistical Physics. Solutions Sheet 5.
|
|
- Sharyl Chandler
- 5 years ago
- Views:
Transcription
1 Sttistic Physics. Soutions Sheet 5. Exercise. HS 04 Prof. Mnfred Sigrist Ide fermionic quntum gs in hrmonic trp In this exercise we study the fermionic spiness ide gs confined in three-dimension hrmonic potenti nd compre it with the cssic cse see Exercise Sheet 3. The eigenenergies of the gs re given by ε = ~ωx + y + z, where = x, y, z, with i {0,,,...}, bes the sttes nd the zero point energy ε0 = 3 ~ω/ ws omitted. The occuption number corresponding to stte is given by n. Consider the high-temperture, ow-density imit z. Derive the grnd cnonic prtition function Zf of this system nd compute the grnd potenti Ωf. Show tht Ωf f4 z, where the function fs z is defined s fs z = = z. s 3 Soution. We begin with the gener definition of the grnd cnonic prtition function within the occuption number formism chpter 3 of the ecture notes nd find Y βε n Y = ze + ze βε. S. Zf = n In order to compute the grnd potenti Ω = /β og Z, we use the series expnsion og + x = x for < x. S. = This expnsion is ppicbe to the ogrithm of the prtition function in S. if ze βε it is wys positive in the high-temperture, ow density imit z. We find z β~ω z og Zf = og + ze = e βε = e =0 = = P 3 if β 0 = z β~ω 3 z = = P e β~ω = = z if β β~ω3 f4 z if β 0 =, f z if β!3 βε S.3 nd obtined both the high nd the ow temperture imits in either cse z must be given. Aterntivey, for the high temperture imit, with the hep of the Euer-Mcurin formu see, e.g., Exercise Sheet 4, we cn pproximte the sum over the oscitor modes by n integr nd find in eding order!3 z β~ω e og Zf = S.4 =0 = Z 3 Z 3 z z d e β~ω = d e β~ω 0 0 = = z 4 = f4 z S.5 = β~ω3 β~ω3 =
2 Either wy, the high temperture expnsion resuts in the grnd potenti Ω f = β β ω 3 f4z. S.6 b Ccute the prtice number N nd the intern energy U s function of N. In order to get U in terms of N insted of deing with the chemic potenti, introduce the prmeter 3 ω N /3 ρ 4 k B T nd rete it to z using the high-temperture, ow-density expnsion of N up to Oz. Interpret the condition ρ. Finy, expnd U up to second order in ρ, reting it to N. Soution. First, we compute the intern energy of the system, U f = β Ω f β, z S.7 where the derivtive hs to be tken t constnt fugcity z = e βµ. Strting from S.6 we find U f = 3 β β ω 3 f4z, which shows tht the intern energy is proportion to the grnd potenti, U f = 3 Ω f. The verge prtice number cn be computed in simir wy, We hve N f = z z og Z f. N f = z z where we used z f4z = f3z. z β ω 3 f4z = β ω 3 f3z, S.8 S.9 S.0 S. In order to rete the intern energy to the prtice number, we strt with the high-temperture, owdensity expnsion of the tot prtice number, N f = β ω f3z z z. S. 3 β ω 3 8 Rewriting this eqution using the prmeter ρ eds to ρ = z z 8. S.3 The condition z therefore impies ρ. Soving this eqution for z, we obtin z = 4 ± 4 ρ. Choosing the reevnt soution nd expnding + x + x x we find 8 z = ρ + ρ 8. Expnding in ρ ows us to de with the prtice number insted of the chemic potenti. S.4 To interpret the condition ρ we first note tht for this system, the Fermi energy foows ɛ F = 3 ω mx, whie the number of occupied sttes is proportion to 3 mx. The chrcteristic energy sce is thus given by 3 ωn /3. Therefore, this condition requires tht the chrcteristic energy sce is much smer thn
3 the therm energy high-temperture imit. This mens tht we consider tempertures t which the verge occuption of the sttes is much smer thn one ow-density imit. We write the intern energy up to second order in ρ s U = 3 β β ω f4z = 3 z z 3 β β ω 3 6 = 3 ρ + ρ β β ω 3 6 = 3 3 N + N β ω 3 ω = 3N + N, S.5 β 6 6 where we recover the equiprtition w in eding order nd the positive first order quntum corrections N ω/ 3 distinguishing the fermions from the cssic ide gs. c Compute the het cpcity C. Which quntity hs to be fixed in order to do this? Soution. Since our system does not hve voume s thermodynmic vribe we hve to compute the specific het C N by fixing the number of prtices. Hence, s strting point we use the expression S.5 for the inner energy, where we cn keep N fixed: U C N = = 3Nk B 3 ω T N 8 N. S.6 d Compute the isotherm compressibiity κ T. Soution. By definition κ T = v N, N µ T S.7 where v = N. Therefore, κ T = v z N µ N = v z T N βz f 3z = vβ fz β ω 3 z f 3z N 3 ω 8 N. S.8 e Interpret your resuts for U, C, nd κ T by compring them with the corresponding resuts for the cssic Botzmnn gs. How do the quntum corrections infuence the fermionic system? Soution. In summry we hve found up to first order in ρ: 3 ω U = 3N + N, S.9 6 C N = 3Nk B 3 ω 8 N, S.0 κ T = N 3 ω 8 N, S. These resuts s function of temperture re potted in Fig. ; ech for the cssic nd the fermionic cse. Note tht our expnsions up to second order in ρ re stricy speking ony vid for ρ, whie the pots extend to rger vues of ρ to emphsize the trends. We see tht 3
4 U kbt 3N CN kbt 3N B ΚT kbt N k B T Figure : Thermodynmics of the fermionic gs dhsed, bue compred to the cssic gs soid, bck. Note tht these quntities re computed within the high-temperture, ow-density pproximtion nd re therefore not exct resuts. Sti, they cn be used to observe trends. We set N ω 3 = 00. B In first order in ρ the resuts for the cssic Botzmnn gs in hrmonic trp re recovered cf. Exercise Sheet 3. Due to quntum corrections, the intern energy U for fermions is higher thn the cssic ide gs. This cn be understood by tking quntum sttistics into ccount. Fermions re not owed to occupy the sme stte mutipy Pui. Lowering the temperture, the system tends to occupy ow energy sttes with growing probbiity. Whie the cssic system does not cre bout mutipe occupncies, in the fermionic system the doube occupncy is forbidden nd occuption of ow-energy sttes is thus reduced, incresing the inner energy U f compred to the cssic gs. 4
5 Exercise. Sommerfed expnsion nd density of sttes Consider thermy equiibrted system of non-intercting fermions with singe prtice sttes beed by the quntum numbers nd corresponding energies ε. Work in the grnd cnonic ensembe nd write the prtice nd energy densities in the form n = fε = dε gεfε, 5 u = ε fε = dε εgεfε, 6 where fε is the Fermi-Dirc distribution function. Wht is gε? Soution. In therm equiibrium, the occuption probbiity of prticur stte with energy ε is given by the Fermi-Dirc distribution fε = e ε µ/ +, nd we find from the definition of n nd u n = fε = dεfε δε ε = dε gεfε, u = ε fε = dε εfε δε ε = dε εgεfε, with gε = δε ε = ωε, nd ωε s in the ecture notes. Tht is, gε is the density of energy eves divided by the voume. S. S.3 S.4 S.5 b The bove expressions for n nd u re of the form dε Hεfε. 7 For tempertures T kb which is typicy the cse for mets, Hε is sowy vrying in the region where df dε 0 significnty nd the Sommerfed expnsion µ dε Hεfε = dε Hε+ π 6 H µ+ 7π H kb T 6 µ+o 8 µ becomes hndy. Mke use of this expnsion up to O k B T µ to expnd n nd u in T. Hint: Use in sef-consistent wy tht µ T in eding order in T nd expnd µ dε Hε εf dε Hε + µ H. 9 For reference on the Sommerfed expnsion see, e.g., Ashcroft, N. W. nd Mermin N. D., Soid Stte Physics, Hot, Rinehrt nd Winston,
6 Soution. From the Sommerfed expnsion we obtin n = u = µ µ dε gε + π 6 kbt g ε + O dε εgε + π 6 kbt [µg ε + gµ] + O 4 kbt S.6 µ 4 kbt. S.7 As we know tht im T 0 µ =, the remining integrs cn be expnded round the chemic potenti see hint: εf ] n dε gε + [µ g + π 6 kbt g, S.8 εf [ ] u dε gε + µ g + π 6 kbt g + π 6 kbt g. S.9 For this to be sef-consistent, we need to check tht the chemic potenti µ = + αt + OT 3, which we see next. µ c Find the chemic potenti µ nd the specific het c v t constnt density n. Soution. As the density is constnt nd we hve t T = 0 tht εf n = dε gεfε = dε gε, S.30 it foows, from Eq. S.8, tht µ = π 6 kbt g g. S.3 Simiiry, u = u 0 + π 6 kbt g, S.3 nd therefore u c v = T n = π 3 k BT g. S.33 From this we see the importnt resuts tht i the chemic potenti vries with respect to temperture in dependence of g ε nd ii tht het mesurments re too to ccess the density of sttes t the Fermi energy. d Determine gε for the cse of free Fermi gs nd ccute its chemic potenti nd specific het from the previous resuts. Compre your resut for the specific het with the one for cssic gs. Soution. For free fermions, the be corresponds to the momentum of the prtices k nd the energies ε re given by the dispersion retion ε k = k m. With the hep of the step function Θ, the density of sttes cn be written s cf. ecture notes S.34 gε = 3/ m ε / Θε, S.35 4π or terntivey gε = 3 / n ε Θε. S.36 6
7 Using this resut we obtin s for the ow-temperture, high-density imit in the ecture [ ] µ = ε f π kbt, S.37 c v = π kbt nk B. For cssic ide gs Mxwe-Botzmnn distribution we find S.38 c v = 3 nkb, S.39 which mens tht in the fermionic cse the specific het is surpressed by fctor π. S.40 The origin of this surpression ies in the Fermi-Dirc distribution nd the Pui principe. For ow tempertures note tht this cn esiy be sever hunderd Kevin for eectrons in mets ony frction of the fermions, nmey the ones round the Fermi energy, get thermy excited nd contribute to the het cpcity, wheres for cssic gs the prtices cn contribute. e For the free Fermi gs g > 0. This does not need to be the true in more compex systems such s soids cf., e.g., semiconductors. Wht re the consequences of g 0? Soution. The sign of the density of sttes determines whether the chemic potenti increses, decreses or stys constnt with respect to the temperture. A negtive sign woud ed to n increse in the chemic potenti by incresing temperture. 7
Fluid Flow through a Tube
. Theory through Tube In this experiment we wi determine how we physic retionship (so ced w ), nmey Poiseue s eqution, ppies. In the suppementry reding mteri this eqution ws derived s p Q 8 where Q is
More information1. The vibrating string problem revisited.
Weeks 7 8: S eprtion of Vribes In the pst few weeks we hve expored the possibiity of soving first nd second order PDEs by trnsforming them into simper forms ( method of chrcteristics. Unfortuntey, this
More informationMAGIC058 & MATH64062: Partial Differential Equations 1
MAGIC58 & MATH646: Prti Differenti Equtions 1 Section 4 Fourier series 4.1 Preiminry definitions Definition: Periodic function A function f( is sid to be periodic, with period p if, for, f( + p = f( where
More informationMath 124B January 24, 2012
Mth 24B Jnury 24, 22 Viktor Grigoryn 5 Convergence of Fourier series Strting from the method of seprtion of vribes for the homogeneous Dirichet nd Neumnn boundry vue probems, we studied the eigenvue probem
More informationIn this appendix, we evaluate the derivative of Eq. 9 in the main text, i.e., we need to calculate
Supporting Tet Evoution of the Averge Synptic Updte Rue In this ppendi e evute the derivtive of Eq. 9 in the min tet i.e. e need to ccute Py ( ) Py ( Y ) og γ og. [] P( y Y ) P% ( y Y ) Before e strt et
More informationMutipy by r sin RT P to get sin R r r R + T sin (sin T )+ P P = (7) ffi So we hve P P ffi = m (8) choose m re so tht P is sinusoi. If we put this in b
Topic 4: Lpce Eqution in Spheric Co-orintes n Mutipoe Expnsion Reing Assignment: Jckson Chpter 3.-3.5. Lpce Eqution in Spheric Coorintes Review of spheric por coorintes: x = r sin cos ffi y = r sin sin
More informationarxiv: v1 [math.co] 5 Jun 2015
First non-trivi upper bound on the circur chromtic number of the pne. Konstnty Junosz-Szniwski, Fcuty of Mthemtics nd Informtion Science, Wrsw University of Technoogy, Pond Abstrct rxiv:1506.01886v1 [mth.co]
More informationEnergy Balance of Solar Collector
Renewbe Energy Grou Gret Ides Grow Better Beow Zero! Wecome! Energy Bnce of Sor Coector Mohmd Khrseh E-mi:m.Khrseh@gmi.com Renewbe Energy Grou Gret Ides Grow Better Beow Zero! Liuid Ft Pte Coectors. Het
More informationNuclear Astrophysics Lecture 1. L. Buchmann TRIUMF
Nucer Astrophysics Lecture L. Buchmnn TRIUMF Jnury 06 Kot Eduction is wht remins fter you hve forgotten everything you hve erned. iε Jnury 06 Kot Nucer Astrophysics The uminous structure of the Universe
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More information4 The dynamical FRW universe
4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which
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 informationComplete Description of the Thelen2003Muscle Model
Compete Description o the he23usce ode Chnd John One o the stndrd musce modes used in OpenSim is the he23usce ctutor Unortuntey, to my knowedge, no other pper or document, incuding the he, 23 pper describing
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 informationCHM Physical Chemistry I Chapter 1 - Supplementary Material
CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion
More informationData Structures and Algorithms CMPSC 465
Dt Structures nd Algorithms CMPSC 465 LECTURE 10 Solving recurrences Mster theorem Adm Smith S. Rskhodnikov nd A. Smith; bsed on slides by E. Demine nd C. Leiserson Review questions Guess the solution
More informationSuggested Solution to Assignment 5
MATH 4 (5-6) prti diferenti equtions Suggested Soution to Assignment 5 Exercise 5.. () (b) A m = A m = = ( )m+ mπ x sin mπx dx = x mπ cos mπx + + 4( )m 4 m π. 4x cos mπx dx mπ x cos mπxdx = x mπ sin mπx
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 informationLecture 6 Notes, Electromagnetic Theory I Dr. Christopher S. Baird University of Massachusetts Lowell
Lecture 6 Notes, Eectrognetic Theory I Dr. Christopher S. Bird University of Msschusetts Lowe. Associted Legendre Poynois - We now return to soving the Lpce eqution in spheric coordintes when there is
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationPhysicsAndMathsTutor.com
M Dynmics - Dmped nd forced hrmonic motion. A P α B A ight estic spring hs ntur ength nd moduus of esticity mg. One end of the spring is ttched to point A on pne tht is incined to the horizont t n nge
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
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 informationWe 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
More informationHomework Assignment #5 Solutions
Physics 506 Winter 008 Textbook probems: Ch. 9: 9., 9., 9.4 Ch. 10: 10.1 Homework Assignment #5 Soutions 9. A spheric hoe of rdius in conducting medium cn serve s n eectromgnetic resonnt cvity. Assuming
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 information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationCoordinate Geometry. Coordinate Geometry. Curriculum Ready ACMNA: 178, 214, 294.
Coordinte Geometr Coordinte Geometr Curricuum Red ACMNA: 78, 4, 94 www.mthetics.com Coordinte COORDINATE Geometr GEOMETRY Shpes ou ve seen in geometr re put onto es nd nsed using gebr. Epect bit of both
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2009
University of Wshington Deprtment of Chemistry Chemistry Winter Qurter 9 Homework Assignment ; Due t pm on //9 6., 6., 6., 8., 8. 6. The wve function in question is: ψ u cu ( ψs ψsb * cu ( ψs ψsb cu (
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationModule 6: LINEAR TRANSFORMATIONS
Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for
More information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts
More informationTHERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION
XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es
More informationLinear Systems COS 323
Liner Systems COS 33 Soving Liner Systems of Equtions Define iner system Singurities in iner systems Gussin Eimintion: A gener purpose method Nïve Guss Guss with pivoting Asymptotic nysis Tringur systems
More information+ x 2 dω 2 = c 2 dt 2 +a(t) [ 2 dr 2 + S 1 κx 2 /R0
Notes for Cosmology course, fll 2005 Cosmic Dynmics Prelude [ ds 2 = c 2 dt 2 +(t) 2 dx 2 ] + x 2 dω 2 = c 2 dt 2 +(t) [ 2 dr 2 + S 1 κx 2 /R0 2 κ (r) 2 dω 2] nd x = S κ (r) = r, R 0 sin(r/r 0 ), R 0 sinh(r/r
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 informationPhysics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15
Physics H - Introductory Quntum Physics I Homework #8 - Solutions Fll 4 Due 5:1 PM, Mondy 4/11/15 [55 points totl] Journl questions. Briefly shre your thoughts on the following questions: Of the mteril
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose nti-derivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationStability condition of finite difference solution for viscoelastic wave equations
Erthq Sci (009): 79 85 79 Doi: 0007/s589-009-079- Stbiity condition of finite difference soution for viscoestic wve equtions Chengyu Sun, Yunfei Xio Xingyo Yin nd Hongcho Peng Chin University of Petroeum,
More informationTutorial 4. b a. h(f) = a b a ln 1. b a dx = ln(b a) nats = log(b a) bits. = ln λ + 1 nats. = log e λ bits. = ln 1 2 ln λ + 1. nats. = ln 2e. bits.
Tutoril 4 Exercises on Differentil Entropy. Evlute the differentil entropy h(x) f ln f for the following: () The uniform distribution, f(x) b. (b) The exponentil density, f(x) λe λx, x 0. (c) The Lplce
More informationIntroduction to statically indeterminate structures
Sttics of Buiding Structures I., EASUS Introduction to stticy indeterminte structures Deprtment of Structur echnics Fcuty of Civi Engineering, VŠB-Technic University of Ostrv Outine of Lecture Stticy indeterminte
More informationRel Gses 1. Gses (N, CO ) which don t obey gs lws or gs eqution P=RT t ll pressure nd tempertures re clled rel gses.. Rel gses obey gs lws t extremely low pressure nd high temperture. Rel gses devited
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 informationStudy Guide Final Exam. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines
Msschusetts Institute of Technology Deprtment of Physics 8.0T Fll 004 Study Guide Finl Exm The finl exm will consist of two sections. Section : multiple choice concept questions. There my be few concept
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More information1 Linear Least Squares
Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More 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 informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
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 informationLecture 13 - Linking E, ϕ, and ρ
Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More information2. My instructor s name is T. Snee (1 pt)
Chemistry 342 Exm #1, Feb. 15, 2019 Version 1 MY NAME IS: Extr Credit#1 1. At prissy Hrvrd, E. J. Corey is Nobel Prize (1990 winning chemist whom ll students cll (two letters 2. My instructor s nme is
More informationCandidates must show on each answer book the type of calculator used.
UNIVERSITY OF EAST ANGLIA School of Mthemtics My/June UG Exmintion 2007 2008 ELECTRICITY AND MAGNETISM Time llowed: 3 hours Attempt FIVE questions. Cndidtes must show on ech nswer book the type of clcultor
More informationA5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s
4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
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 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 informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
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 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 informationFig. 1. Open-Loop and Closed-Loop Systems with Plant Variations
ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses
More informationChapter 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
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 information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationClassical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011
Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
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 informationRFID Technologies HF Part I
RFID Technoogies HF Prt I Diprtimento di Ingegneri de Informzione e Scienze Mtemtiche Ampere s w h(r,t) ĉ d = j(r,t) ˆnds t C brt (, ) = µ hrt (, ) S S d(r,t) ˆnds j(r,t) d(r,t) ds ˆn Ø Biot-Svrt (sttic
More informationChapter 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
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 informationarxiv: v1 [cond-mat.supr-con] 27 Dec 2018
Ccution of currents round tringur indenttion by the hodogrph method J. I. Avi 1, B. Venderheyden 2, A. V. Sihnek 3, S. Meinte 1 1 ICTEAM, Université cthoique de Louvin, B-1348, Louvin-L-Neuve, Begium 2
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
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 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 informationReview 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
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationV. More General Eigenfunction Series. A. The Sturm-Liouville Family of Problems
V. More Gener Eigenfunction Series c 214, Phiip D. Loewen A. The Sturm-Liouvie Fmiy of Probems Given n re interv, b], functions p(x), q(x), r(x), nd constnts c, c 1, d, d 1, consider (ODE) (p(x)y (x))
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationMTH3101 Spring 2017 HW Assignment 6: Chap. 5: Sec. 65, #6-8; Sec. 68, #5, 7; Sec. 72, #8; Sec. 73, #5, 6. The due date for this assignment is 4/06/17.
MTH30 Spring 07 HW Assignment 6: Chp. 5: Sec. 65, #6-8; Sec. 68, #5, 7; Sec. 7, #8; Sec. 73, #5, 6. The due dte for this ssignment is 4/06/7. Sec. 65: #6. Wht is the lrgest circle within which the Mclurin
More informationSufficient condition on noise correlations for scalable quantum computing
Sufficient condition on noise correltions for sclble quntum computing John Presill, 2 Februry 202 Is quntum computing sclble? The ccurcy threshold theorem for quntum computtion estblishes tht sclbility
More informationPhysics 2D Lecture Slides Lecture 30: March 9th 2005
Physics D Lecture Sides Lecture 3: Mrch 9th 5 Vivek Shrm UCSD Physics Mgnetic Quntum Number m L = r p (Right Hnd Rue) Cssicy, direction & Mgnitude of L wys we defined QM: Cn/Does L hve definite direction?
More informationLecture 2: January 27
CS 684: Algorithmic Gme Theory Spring 217 Lecturer: Év Trdos Lecture 2: Jnury 27 Scrie: Alert Julius Liu 2.1 Logistics Scrie notes must e sumitted within 24 hours of the corresponding lecture for full
More informationCHAPTER 20: Second Law of Thermodynamics
CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het
More informationA. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions
More informationdt. However, we might also be curious about dy
Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For
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 informationIdentification of Axial Forces on Statically Indeterminate Pin-Jointed Trusses by a Nondestructive Mechanical Test
Send Orders of Reprints t reprints@benthmscience.net 50 The Open Civi Engineering Journ, 0, 7, 50-57 Open Access Identifiction of Axi Forces on Stticy Indeterminte Pin-Jointed Trusses by Nondestructive
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 informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationMatrices, Moments and Quadrature, cont d
Jim Lmbers MAT 285 Summer Session 2015-16 Lecture 2 Notes Mtrices, Moments nd Qudrture, cont d We hve described how Jcobi mtrices cn be used to compute nodes nd weights for Gussin qudrture rules for generl
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More information