Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry


 Jason Jacobs
 5 years ago
 Views:
Transcription
1 Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry Last updated Thursday, November 7, 25 7:9:55: Copyrght 25 Dan Dll Department of Chemstry, Boston Unversty, Boston MA 225 We have learned how to fnd wave functons and energes by adjustng the total energy so that the wave functon decays to zero n forbdden regons of nfnte wdth or nfnte potental energy. In numercal applcatons of quantum mechancs n chemstry, an alternatve, flexble method of solvng the Schrödnger equaton s to approxmate the wave functon for a partcular system n terms of those for a smlar, socalled model system. A very nce feature of ths approach s that the accuracy of the approxmaton can be systematcally mproved, by ncreasng the number of model wave functons used n the approxmaton. In ths workshop we'll explore ths method, for the example of a partcle confned to an nfnte well wth a slopng bottom. The model system wll be the correspondng nfnte well wth a flat bottom. Part à A curved nfnte well The goal of ths workshop s to determne the wave functon, Y, and energy, E, for the ground state of a partcle n an nfnte well that has a bottom that curves up toward the rght wall. Here s a plot of the curved nfnte square well, and the correspondng flat nfnte square well. energy Fgure. Dstorted nfnte square well potental energy (thck lne) and correspondng undstorted nfnte well potental energy (thn lne). The thck horzontal lne s the ground state energy, 2.8, of the dstorted nfnte well and the thn horzontal lne s the ground state energy, 9.869, of the undstorted nfnte well. Energy and length are n dmensonless unts. The dstorton s H2 xl 5 n dmensonless unts.
2 2 Workshop: Approxmatng energes and wave functons. We know that the ground state energy of a partcle of mass m confned n a nfnte square well of wdth L s h 2 ê H8 ml 2 L. If we assume the partcle s an electron, determne the unts of energy f the wdth s 2. Þ. Answer.5 μ 8 J. 2. Show that ths means that the dstorted well potental energy rses from J at the left wall to 48 μ 8 J at the rght wall. à Exact ground state energy wave functon So that we have a pont of comparson, we can determne the exact energy and wave functon of the dstorted and undstorted well by solvng the curvature form of the Schrödnger equaton, by adjustng the total energy so the wave functon has a sngle loop and decays to zero at each edge of the nfnte well. The result s that the ground state energy of the dstorted square well, E = 2., s about 2% hgher than that of the undstorted well, E = Explan why the ground state energy of the dstorted well must be hgher than that of the undstorted well. Here s a plot of the ground state wave functon, Y HxL (thck curve), of the dstorted and the wave functon, y HxL = è!!! 2 snhp xl (thn curve), of the undstorted well. Y and y Fgure 2. Ground state wave functon, Y HxL (thck curve), of the dstorted well and the wave functon, y HxL = è!!!! 2 snhp xl (thn curve), of the undstorted well. Here s a plot of the curvature, 2 Y ê x 2, of the exact ground state wave functon. Y ê x x Fgure 3. Curvature, 2 Y ê x 2 of the exact ground state wave functon. 4. From the plot of the potental energy and the total energy, does the dstorted well have a forbdden regon between the nfnte potental energy wall at x = and x =? If so,
3 Workshop: Approxmatng energes and wave functons 3 ndcate the correspondng classcal turnng pont and the nflecton pont n the wave functon. Here s a plot of the dfference, Y HxL y HxL, between the exact and the model system ground state wave functons..5 Y y Fgure 4. Wave functon dfference, Y HxL y HxL. 5. Account for the dfference n these two wave functons, n terms of the dfference n the correspondng potental energes. à Frst order approxmaton to energy An alternatve to usng the curvature form of the Schrödnger equaton to determne the energy and wave functon of the dstorted well, s to solve for these quanttes by expandng n the bass of wave functons of the undstorted well. The general form of ths expanson for wave functon, Y a, of the dstorted well wth a loops and energy E a s Y a = y j c j,a. j The smplest approxmaton to the ground state wave functon, Y, of the dstorted well s to nclude just a sngle term, Y = y j c j, ºy c,. j Snce the wave functons of the undstorted well are normalzed, we can ensure that Y s normalzed by settng the coeffcent c, equal to one. Ths means that at ths level of approxmaton, we assume the dstorton n the potental energy does not change the wave functon from that of the undstorted potental. Once we have an approxmaton to the wave functon, we can use the Schrödnger equaton to determne the correspondng approxmaton to the energy. H Y º H y º E y Here s how to work wth ths approxmate Schrödnger equaton. Because y s not really an egenfuncton of H, the result of operatng wth H on y wll not be proportonal to y and so we have no proportonalty that we can use to dentfy the approxmate egenvalue. Instead, what we can do s multply both sdes of the second approxmate equalty by y * and then ntegrate over the well. The result s
4 4 Workshop: Approxmatng energes and wave functons y HxL * H HxL y HxL x º E That s, the frst approxmaton to the energy s E º E HL = y HxL * H HxL y HxL x To evaluate the ntegral t s very helpful to express the Hamltonan n terms of the Hamltonan, H HL, of the undstorted potental, that s, to wrte H = H HL + HH  H HL L. The dfference H  H HL s called the perturbaton to the undstorted hamltonan. 6. Show that snce the perturbaton, H  H HL, between the dstorted well and the undstorted well s the potental energy, H2 xl 5, the frst approxmaton to the energy can be expressed as E HL =e + y HxL * H2 xl 5 y HxL x where we use the symbol e j = j 2 h 2 ê H8 ml 2 L for the energes of the undstorted well. 7. The ntegral Ÿ y HxL * H2 xl 5 y HxL x evaluates to Evaluate, n dmensonless unts, the frst approxmaton to the energy of the dstorted well. Answer: Evaluate the percentage error n the frst approxmaton to the energy. Answer: 2.48% à Accuracy of the frst order approxmaton to energy The net result of the smplest, socalled frst order approxmaton scheme s that we can estmate the ground state energy, E, of a system n terms of the energy, e, and ground state wave functon, y, of a model system as E ºe + y HxL * HH  H HL L y HxL x, where the perturbaton H  H HL s the operator correspondng to the dfference between the actual system and the model system. In the example above, H  H HL = H2 xl 5. We have seen that ths very smple approxmaton does a very good job n predctng the energy of the system, n that the error s only 2.48%. Let's see how ths scheme works for a modfed verson of the dstorted well. 9. Show that the frst order estmate of the ground state energy of an nfnte well wth dstorted potental energy H  H HL = H4 xl 5 s E º E HL = The exact ground state energy s E = Ths means that now the percentage error s 52%! Indeed, the zero order energy the energy of the undstorted potental, 9.86 unts s closer to the exact energy than s the frst order approxmaton. Let's see f we can understand why now the error n the frst order approxmaton to the energy s so much greater
5 Workshop: Approxmatng energes and wave functons 5 Here s a plot of the new dstorted potental energy and that of the undstorted well, together wth the correspondng ground state energes, shown as horzontal lnes. energy Fgure 5. Modfed (thck lne) and orgnal (thck, dashed lne) dstorted nfnte square well potental energy and correspondng undstorted nfnte well potental energy (thn lne). The horzontal lnes are the ground state energy, 29.63, of the modfed dstorted nfnte well (thck lne) and the ground state energy, 2.8, of the orgnal dstorted nfnte well (thck, dashed lne). The thn horzontal lne s the ground state energy, 9.869, of the undstorted nfnte well. Energy and length are n dmensonless unts. Here s a plot of the ground state wave functon, Y HxL (thck curve), of the modfed dstorted well, and the wave functon, y HxL = è!!! 2 snhp xl (thn curve), of the undstorted well. Y and y Fgure 6. Ground state wave functon, Y HxL (thck curve), of the modfed dstorted well, wth potental energy H4 xl 5, and the wave functon, y HxL = è!!!! 2 snhp xl (thn curve), of the undstorted well.. From the plot of the potental energy and the total energy, does the modfed dstorted well have a forbdden regon between the nfnte potental energy wall at x = and x =? If so, ndcate the correspondng classcal turnng pont and the nflecton pont n the wave functon.. Compare the exact ground state wave functon for the orgnal dstorted potental energy, H2 xl 5, and the new dstorted potental energy, H4 xl 5. Based on your comparson can you antcpate when the frst order approxmaton to the energy wll work well and when t wll not? à Second order approxmaton to the energy; frst order approxmaton to the wave functon The next smplest approxmaton to the ground state wave functon, Y, of the dstorted well s to nclude two terms n the expanson of the dstorted well wave functon, Y = y j c j, ºy c, +y 2 c 2,. j
6 6 Workshop: Approxmatng energes and wave functons As before, once we have an approxmaton to the wave functon, we can use the Schrödnger equaton to determne the correspondng approxmaton to the energy, H Y º H 8 y c, +y 2 c 2, < º E 8y c, +y 2 c 2, <. At ths pont, though, we have to proceed dfferently, snce we do not know the relatve contrbutons of the two undstorted wave functons, that s, snce we do not know the expanson coeffcents c, and c 2,. The frst step s to transform the Schrödnger equaton nto two algebrac equatons, that we can then try to use to determne the unknown expanson coeffcents. One of the "secret handshakes" of quantum mechancs s the way to carry out the transformaton to algebrac equatons. Ths s done by frst multplyng the approxmate Schrödnger equaton by y * and then ntegratng over the potental energy well. 2. Show that the result of these steps s H, c, + H,2 c 2, º E c,, n terms of the matrx elements H j,k = Ÿ y j HxL * H y k HxL x. In a smlar way we can get another algebrac equaton usng y * 2, H 2, c, + H 2,2 c 2, º E c 2,. The next step s to get numercal values for the matrx elements H j,k. To do ths we need the ntegrals y HxL * H2 xl 5 y 2 HxL x =2.92 and y 2 HxL * H2 xl 5 y 2 HxL x = 4.397, where the values are n dmensonless energy unts. 3. Determne the values of H,, H,2, H 2, and H 2,2, n dmensonless energy unts. Answer: 2.38, 2.92, and Use these values of H j,k, to reduce the Schrödnger equaton to the two algebrac equatons 2.38 c, c 2, = E c,, and c, c 2, = E c 2,.
7 Workshop: Approxmatng energes and wave functons 7 à Ensurng that we can solve the approxmaton: The secular equaton and the second order approxmaton to the energy Usng matrx notaton, the algebrac equatons of the second order approxmaton can be wrtten compactly as Ä HH  E a IL c = j H, H,2 y z  E ÇÅ k H 2, H a j y É z j c,a y z = 2,2 { k { ÖÑ k c 2,a { j E a y z j c,a y z =. k E a { k c 2,a { 5. Carry out the matrx multplcaton n the last equalty to show that ths equaton s equvalent to the two algebrac equatons, wth E replaced by E a and the correspondng second ndex on the expanson coeffcents by a. We'll see n a moment why we have replaced E wth the more general symbol E a, and correspondngly ntroduced the subscrpt a n the expanson coeffcents. If we could fnd the nverse matrx, HH  E a IL , then snce HH  E a IL  HH  E a IL = I, we would have the result HH  E a IL  HH  E a IL c = Ic= c =. That s, each of the expanson coeffcents, and so the approxmaton to the wave functon Y a, would vansh! To prevent ths, we need to not be able to fnd the nverse matrx, HH  E a IL . It turns out that the nverse of a matrx s proportonal to the nverse of ts determnant. So, f we can arrange for the determnant of the matrx H  E a I to be equal to zero, then the nverse matrx wll be nfnte, and so effectvely t wll be undefned for the purposes of numercal computaton. The determnant,» H  E a I», of the matrx H  E a I s ƒ E a E a ƒ = H E al μ H E a L  H2.92L μ H2.92L Ths s a quadratc equaton n the unknown energy, E a. Settng ths quadratc equal to zero, we can solve for the two values of E a that satsfy the requrement that the matrx H  E a I not have an nverse. 6. Show that the two values of the energy are E a= = 2. and E a=2 = What we have done s to fnd the two values of E a for whch we wll not be able to nvert the matrx H  E a I, and so for whch the values of the coeffcents c,a and c 2,a wll not vansh. The energes are found by solvng» H  E a I» =. Ths s known as the secular equaton. Fndng the values of energy that solve the secular equaton s how energy quantzaton arses n fndng wave functons by expandng n a bass. The exact value of the energy of the ground state of the dstorted well s E = 2.8, and that the frst order approxmaton to the ground state energy s The frst, lowest energy that solves the secular equaton, that s, the second order approxmaton to the energy of the ground state, s E a= = 2., and so the second approxmaton to the energy s an mprovement over the frst order approxmaton to the energy, and now qute close to the exact energy, dfferng by only.3 demsnsonless unts (about.25 %).
8 8 Workshop: Approxmatng energes and wave functons Part 2 à Solvng the frst order approxmaton to the wave functon At ths pont we know that j E a y z j c,a y z = k E a { k c 2,a { has a nontrval soluton (that s, other than the expanson coeffcents all beng equal to zero) for two specal values of the energy, E a. 7. Show that the equatons for a = are j H,  E H,2 y z j c, y z = j k H 2, H 2,2  E { k c 2, { k and that the equatons for a =2 are y z j c, y z = { k c 2, { j H,  E 2 H,2 y z j c,2 y z = j k H 2, H 2,2  E 2 { k c 2,2 { k y z j c,2 y z = { k c 2,2 { The next step s to convert these two homogeneous algebrac equatons nto nhomogeneous equatons. The way to do ths s n two steps. The frst step s to set one of the coeffcents, say the frst one, to the value one. The result s j H,  E a= H,2 y k H 2, H 2,2  E a= { z j k j H,  E a=2 H,2 y k H 2, H 2,2  E a=2 { z j k y Hc'L 2, { y Hc'L 2,2 { z = y j z j k { k z = y j z j k { k y Hc'L 2, { z = y Hc'L 2,2 { Ths amounts to solvng for the rato of the two coeffcents, and we ndcate ths by puttng a prme on the other coeffcent of each par. We can later fx the absolute value of the two coeffcents from each set of equatons by requrng that the resultng wave functons be normalzed. z = The second step s to gnore the frst equaton of each set. The remanng equatons HH 2,2  E a= L Hc'L 2, =H 2, HH 2,2  E a=2 L Hc'L 2,2 =H 2, allow us to solve for the remanng coeffcents, Hc'L 2, =HH 2,2  E a= L  H 2, Hc'L 2,2 =HH 2,2  E a=2 L  H 2,.
9 Workshop: Approxmatng energes and wave functons 9 8. Evaluate the coeffcents Hc'L 2, and Hc'L 2,2. Answer:.99 and Show that the normalzaton constant of the frst order approxmaton to the wave functon s í "################ + Hc'L ###### 2 2,a. 2. Show that the normalzed approxmate wave functons are Y ºY,approx HxL =.4 sn Hp xl +.29 sn H2 p xl, Y 2 ºY 2,approx HxL =.29 sn Hp xl .4 sn H2 p xl. Here s a comparson of the approxmate wave functon to the exact wave functon of the dstorted well and to the wave functon of the undstorted well. Y and y Fgure 7. Exact ground state wave functon, Y HxL (thck curve), of the dstorted well, wth potental energy H2 xl 5, approxmaton to the exact wave functon, Y,approx HxL (thck, dashed curve), and the wave functon, y HxL = è!!!! 2 snhp xl (thn curve), of the undstorted well. 2. What do you conclude from the plot? à The other approxmate energy and wave functon. We determned two values of the energy such that the secular determnant vanshes. The second energy s an approxmaton to the frst excted state of the dstorted well. 22. Show that the zero order approxmaton to the energy of the frst excted state s Show that the frst order approxmaton to the energy of the frst excted state, E 2 ºe 2 + y 2 HxL * HH  H HL L y 2 HxL x, gves the value E 2 º The exact value of the energy of the frst excted state of the dstorted well s E 2 = 43.77, and so once agan, the frst order approxmaton to the energy, 43.87, s qute good. The second energy that solves the secular equaton, that s, the second order approxmaton to the energy of the frst excted state, s E a=2 = Snce E a=2 s not as close to the exact energy as the frst order approxmaton, n ths case the frst order approxmaton to the energy s dong a slghtly better job than the second order approxmaton to the energy.
10 Workshop: Approxmatng energes and wave functons Here s a comparson of the normalzed approxmate wave functon, Y 2,approx, to the exact wave functon of the dstorted well and to the wave functon of the undstorted well, for the frst excted state. Y and y Fgure 8. Exact frst excted state wave functon, Y 2 HxL (thck curve), of the dstorted well, wth potental energy H2 xl 5, approxmaton to the exact wave functon, Y 2,approx HxL, (thck, dashed curve), and the wave functon, y 2 HxL = è!!!! 2 snh2 p xl (thn curve), of the undstorted well. The exact and approxmate wave functons have been multpled by  to match the phase of the wave functon of the undstorted well. 24. What do you conclude from the plot? à Accuracy of the second order approxmaton to energy and the frst order approxmaton to the wave functon Let's see how the second order approxmaton works for a modfed verson of the dstorted well. 29. Show that the equatons for the expanson coeffcents are now j H,  E a H,2 y z j c,a y z = j E a y z j c,a y z = k H 2, H 2,2  E a { k c 2,a { k E a { k c 2,a { The values of E a that solve the secular equaton correspondng to these equatons are E a= = 3.52 and E a=2 = As before, the lower of these two values s the second order approxmaton to the ground state energy. Proceedng n the same way as before, you can fnd that the expanson coeffcents, relatve to c,a =, are Hc'L 2, =.629 and Hc'L 2,2 =.59, and so that the normalzed approxmaton to the ground state wave functon s Y º.2 snhp xl snh2 p xl. Here s a comparson of the normalzed approxmate wave functon to the exact wave functon of the new dstorted well and to the wave functon of the undstorted well.
11 Workshop: Approxmatng energes and wave functons Fgure 9. Exact ground state wave functon, Y HxL (thck curve), of the modfed dstorted well, wth potental energy H4 xl 5, approxmaton to the exact wave functon (thck, dashed curve), and the wave functon, y HxL = è!!!! 2 snhp xl, (thn curve) of the undstorted well. 25. What do you conclude from the plot? Recall that the exact value of the energy of the ground state of the modfed dstorted well s E = 29.62, and that the frst order approxmaton to the ground state energy s 9.27; that s, the frst order approxmaton to the energy was very poor for ths case. The frst, lowest energy that solves the secular equaton, that s, the second order approxmaton to the energy of the ground state, s E a= = 3.52, and so the second order approxmaton to the energy s qute good, and a major mprovement over the frst order approxmaton to the energy. Here s a table consstng of () the ground state energy of the undstorted well, the frst order and second order approxmatons to the energy of the dstorted well, and the exact energy of the dstorted well, and (2) the ground state energy of the undstorted well, the frst order and second order approxmatons to the energy of the modfed dstorted wells, and the exact energy of the modfed dstorted well. zero order frst order second order exact H2xL 5 dstorton H4xL 5 dstorton Approxmate and exact energes for the alternatve dstorted wells. The dstorton s the devaton H  H HL from the nfnte square well. 26. Why do you suppose n ths case, for the modfed verson of the dstorted well, wth potental energy H4 xl 5, the frst order approxmaton s so poor whereas the frst order approxmaton was so good for the orgnal verson of the dstorted well, wth potental energy H2 xl Based the table, and the analyses of the wave functon of the undstorted well, and the frst order approxmate wave functons and exact wave functons of the dstorted wells, shown n fgures 7 and 9, propose gudelnes on how to best approxmate energes and wave functons n terms of those of a reference system.
12 2 Workshop: Approxmatng energes and wave functons à The approxmate wave functons "dagonalze" the hamltonan (optonal) Ths last part explores some general propertes of the procedure used to fnd the second order energes and the correspondng frst order wave functons, for the example of the well wth dstorton H2 xl 5. You may omt ths part f you are pressed for tme, leavng t for study outsde of workshop. 28. Make use of the orthonormalty of the bass functons, y and y 2, to show that Ÿ Y2,approx HxL Y,approx HxL x =. 29. Show that the matrx of the hamltonan of the dstorted well n the bass of ts approxmate wave functons, Y,approx and Y 2,approx, s a dagonal matrx, and that the elements on the dagonal are the second order approxmatons to the energy. The last queston llustrates the general result that solvng the Schrödnger equaton by expanson n a bass s equvalent to fndng the combnatons of bass functons n terms of whch the matrx of the hamltonan s dagonal. It s for ths reason that soluton to the Schrödnger equaton s often referred to as dagonalzng the hamltonan. Formally, the process of dagonalzaton s carred out by a partcular knd of matrx multplcaton known as a smlarty transformaton. A smlarty transformaton of a matrx H by a matrx U s defned as the matrx product U  HU. In ths case, U s the matrx whose frst row s the expanson coeffcents c j,a= and whose second row s the expanson coeffcents c j,a=2, U = j c, c 2, y z = y j z k c,2 c 2,2 { k { The matrx U has the specal property that ts transpose, U é, s ts nverse. Ths property s called orthogonalty. 3. Demonstrate that U s an orthogonal matrx, by showng that U é U = I. 3. Show that U dagonalzes the hamltonan matrx, H = y j z. k { That s, show that U  HU = j E a= y z. k E a=2 {
Molecular 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 informationLinear system of the Schrödinger equation Notes on Quantum Mechanics
Lnear sstem of the Schrödnger equaton Notes on Quantum Mechancs htt://quantum.bu.edu/notes/quantummechancs/lnearsstems.df Last udated Wednesda, October 9, 003 :0:08 Corght 003 Dan Dll (dan@bu.edu) Deartment
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 informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationSalmon: Lectures on partial differential equations. Consider the general linear, secondorder PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of secondorder equatons There are general methods for classfyng hgherorder partal dfferental equatons. One s very general (applyng even to
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 005000 cm  spectral regon. The absorpton of a photon
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons Â and ˆb avalable. Then the best thng we can do s to solve Âˆx ˆb exactly whch
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 information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3d rotaton around the xaxs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3d rotaton around the yaxs:
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 information9 Derivation of Rate Equations from SingleCell Conductance (HodgkinHuxleylike) Equations
Physcs 171/271  Chapter 9R Davd Klenfeld  Fall 2005 9 Dervaton of Rate Equatons from SngleCell Conductance (HodgknHuxleylke) Equatons We consder a network of many neurons, each of whch obeys a set
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 informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
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 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 informationEffects of Ignoring Correlations When Computing Sample ChiSquare. John W. Fowler February 26, 2012
Effects of Ignorng Correlatons When Computng Sample ChSquare John W. Fowler February 6, 0 It can happen that chsquare must be computed for a sample whose elements are correlated to an unknown extent.
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 informationNoninteracting Spin1/2 Particles in Noncommuting External Magnetic Fields
EJTP 6, No. 0 009) 43 56 Electronc Journal of Theoretcal Physcs Nonnteractng Spn1/ Partcles n Noncommutng External Magnetc Felds Kunle Adegoke Physcs Department, Obafem Awolowo Unversty, IleIfe, Ngera
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the righthand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationLECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
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 informationFrom BiotSavart Law to Divergence of B (1)
From BotSavart Law to Dvergence of B (1) Let s prove that BotSavart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of BotSavart. The dervatve s wth respect to
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat manyelectron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
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 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 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 informationSection 3.6 Complex Zeros
04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there
More informationRobert Eisberg Second edition CH 09 Multielectron atoms ground states and xray excitations
Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and xray exctatons 901 By gong through the procedure ndcated n the text, develop the tmendependent Schroednger equaton
More informationψ ij has the eigenvalue
Moller Plesset Perturbaton Theory In MollerPlesset (MP) perturbaton theory one taes the unperturbed Hamltonan for an atom or molecule as the sum of the one partcle Foc operators H F() where the egenfunctons
More information1 GSW Iterative Techniques for y = Ax
1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn
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 informationwhere the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the snglepartcle potental and V nt
Physcs 543 Quantum Mechancs II Fall 998 HartreeFock and the Selfconsstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.
More information8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before
.1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
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 Warmup Show that the egenvalues of a Hermtan operator Â are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
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 Â and ˆB, the set of egenstates by { u }, and the egenvalues as Â u = a u and ˆB u = b u. Snce the
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationLinear Regression Analysis: Terminology and Notation
ECON 35*  Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (twovarable) lnear regresson model. It s represented
More informationLinear Feature Engineering 11
Lnear Feature Engneerng 11 2 LeastSquares 2.1 Smple leastsquares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19
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 informationP A = (P P + P )A = P (I P T (P P ))A = P (A P T (P P )A) Hence if we let E = P T (P P A), We have that
Backward Error Analyss for House holder Reectors We want to show that multplcaton by householder reectors s backward stable. In partcular we wsh to show fl(p A) = P (A) = P (A + E where P = I 2vv T s the
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
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 informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture  30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
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 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 information1 Derivation of Rate Equations from SingleCell Conductance (HodgkinHuxleylike) Equations
Physcs 171/271 Davd Klenfeld  Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from SngleCell Conductance (HodgknHuxleylke) Equatons We consder a network of many neurons, each of whch obeys
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More information8 Derivation of Network Rate Equations from Single Cell Conductance Equations
Physcs 178/278  Davd Klenfeld  Wnter 2015 8 Dervaton of Network Rate Equatons from Sngle Cell Conductance Equatons We consder a network of many neurons, each of whch obeys a set of conductancebased,
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
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 informationNONCENTRAL 7POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NONCENTRAL 7POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationThe exponential map of GL(N)
The exponental map of GLN arxv:hepth/9604049v 9 Apr 996 Alexander Laufer Department of physcs Unversty of Konstanz P.O. 5560 M 678 78434 KONSTANZ Aprl 9, 996 Abstract A fnte expanson of the exponental
More informationMEM 255 Introduction to Control Systems Review: Basics of Linear Algebra
MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A
More informationFTCS Solution to the Heat Equation
FTCS Soluton to the Heat Equaton ME 448/548 Notes Gerald Recktenwald Portland State Unversty Department of Mechancal Engneerng gerry@pdx.edu ME 448/548: FTCS Soluton to the Heat Equaton Overvew 1. Use
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06FnteDfference Method (Onedmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationOnesided finitedifference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note Onesded fntedfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
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 informationECEN 667 Power System Stability Lecture 21: Modal Analysis
ECEN 667 Power System Stablty Lecture 21: Modal Analyss Prof. Tom Overbye Dept. of Electrcal and Computer Engneerng Texas A&M Unversty, overbye@tamu.edu 1 Announcements Read Chapter 8 Homework 7 s posted;
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 informationOpen string operator quantization
Open strng operator quantzaton Requred readng: Zwebach 4 Suggested readng: Polchnsk 3 Green, Schwarz, & Wtten 3 upto eq 33 The lghtcone strng as a feld theory: Today we wll dscuss the quantzaton of an
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More information8 Derivation of Network Rate Equations from Single Cell Conductance Equations
Physcs 178/278  Davd Klenfeld  Wnter 2019 8 Dervaton of Network Rate Equatons from Sngle Cell Conductance Equatons Our goal to derve the form of the abstract quanttes n rate equatons, such as synaptc
More informationNorms, Condition Numbers, Eigenvalues and Eigenvectors
Norms, Condton Numbers, Egenvalues and Egenvectors 1 Norms A norm s a measure of the sze of a matrx or a vector For vectors the common norms are: N a 2 = ( x 2 1/2 the Eucldean Norm (1a b 1 = =1 N x (1b
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 nonnegatve Hence the equatons x, x, x + 7 0 etc are not
More information12. The HamiltonJacobi Equation Michael Fowler
1. The HamltonJacob 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 informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationCS4495/6495 Introduction to Computer Vision. 3CL3 Calibrating cameras
CS4495/6495 Introducton to Computer Vson 3CL3 Calbratng cameras Fnally (last tme): Camera parameters Projecton equaton the cumulatve effect of all parameters: M (3x4) f s x ' 1 0 0 0 c R 0 I T 3 3 3 x1
More informationDynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence)
/24/27 Prevew Fbonacc Sequence Longest Common Subsequence Dynamc programmng s a method for solvng complex problems by breakng them down nto smpler subproblems. It s applcable to problems exhbtng the propertes
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationFormal solvers of the RT equation
Formal solvers of the RT equaton Formal RT solvers Runge Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan
More informationGrover s Algorithm + Quantum Zeno Effect + Vaidman
Grover s Algorthm + Quantum Zeno Effect + Vadman CS 2942 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 informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the faketest data; fxed
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 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: Manybody 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 informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 OO 4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 OO 5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More informationTimeVarying Systems and Computations Lecture 6
TmeVaryng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy
More information7. Products and matrix elements
7. Products and matrx elements 1 7. Products and matrx elements Based on the propertes of group representatons, a number of useful results can be derved. Consder a vector space V wth an nner product ψ
More informationAsymptotics of the Solution of a Boundary Value. Problem for OneCharacteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5  HIKARI Ltd, www.mhar.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for OneCharacterstc Dfferental Equaton
More informationChapter 8. Potential Energy and Conservation of Energy
Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and nonconservatve forces Mechancal Energy Conservaton of Mechancal
More informationThe Dirac Equation for a Oneelectron atom. In this section we will derive the Dirac equation for a oneelectron atom.
The Drac Equaton for a Oneelectron atom In ths secton we wll derve the Drac equaton for a oneelectron atom. Accordng to Ensten the energy of a artcle wth rest mass m movng wth a velocty V s gven by E
More informationarxiv:condmat/ v2 [condmat.meshall] 3 Jan 2006
arxv:condmat/0210519v2 [condmat.meshall] 3 Jan 2006 Non Equlbrum Green s Functons for Dummes: Introducton to the One Partcle NEGF equatons Magnus Paulsson Dept. of mcro and nanotechnology, NanoDTU,
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationUnit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationThis model contains two bonds per unit cell (one along the xdirection and the other along y). So we can rewrite the Hamiltonian as:
1 Problem set #1 1.1. A oneband model on a square lattce Fg. 1 Consder a square lattce wth only nearestneghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationLOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin
Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, MH Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for ConsumptionSaving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for ConsumptonSavng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationTemperature. Chapter Heat Engine
Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationMATH Sensitivity of Eigenvalue Problems
MATH 537 Senstvty of Egenvalue Problems Prelmnares Let A be an n n matrx, and let λ be an egenvalue of A, correspondngly there are vectors x and y such that Ax = λx and y H A = λy H Then x s called A
More informationG4023 MidTerm Exam #1 Solutions
Exam1Solutons.nb 1 G03 MdTerm Exam #1 Solutons 1Oct0, 1:10 p.m to :5 p.m n 1 Pupn Ths exam s openbook, opennotes. You may also use prntouts of the homework solutons and a calculator. 1 (30 ponts,
More information