Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry
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1 Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry Last updated Thursday, November 7, 25 7:9:5-5: 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, so-called 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, so-called 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 - H-2.92L μ H-2.92L 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 non-trval 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 {
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