Harmonic oscillator approximation

Transcription

1 armonc ocllator approxmaton armonc ocllator approxmaton Euaton to be olved We are fndng a mnmum of the functon under the retrcton where W P, P,..., P, Q, Q,..., Q P, P,..., P, Q, Q,..., Q lnwgner functon E W P m P m m m Q Q Q Q. Scalng To decreae number of parameter, let u perform calng p P / m m m Q Q Q Q / and rewrte a

2 armonc ocllator approxmaton W p p. 3 Pont of extremum euaton Extremum pont are found from the euaton W that euvalent to a et of p p,,...,. 4 The econd euaton of 4 euvalent to. 4a ote that dw de W dp W d dp d d. p de de p de de de Two nd of extrema If p then /, /. 5,,..., Coordnate 5 correpond to real momentum p f E.

3 armonc ocllator approxmaton 3 If the coordnate are enumerated o that... then no other mlar extremum for p where,3,..., can be a mnmum of W becaue there ext a pont wth the ame coordnate and nterchanged momemta wth le W. Extremum pont wth all zero momenta can be found by olvng an euaton for, E 6 and then fndng coordnate ung e. 4a. Choong mnmum of W Frtly, we chooe an extremum wth mnmum W among the econd nd of extrema zeromomenta. If an extremum of the frt nd 5 ext then t W hould be compared wth the econd-nd mnmum to mae a proper choce of the global mnmum. Conder here the frt tep n detal. Let u defne functon F U. 7 The problem of fndng of the econd-nd mnmum reduce to fndng a root of the euaton F E wth mnmum of U that euvalent to mnmum of the functon R U / F. In ome regon, th functon can be etmated a, R ~,,,...,. 8,

4 armonc ocllator approxmaton 4 where / and /. For llutraton, let u plot graph of the functon tang for example } {,,, } and { 3 4 { 4 } {,,3, } : F R For example at E 5 the euaton F E ha four real root wth mnmum of R and hence W U for the mallet root. 95. Sngularte of the functon nvere to F,.e. a a functon of E are determned from the d euaton d F. In a partcular cae of, there one real ngularty / 3 3 / / 3 / 3 / 3 / 3 9 / 3 / 3 and a par of complex-conugate ngularte

5 armonc ocllator approxmaton 5, / 3 3 / / 3 / 3 / 3 / 3 / 3 / 3 ± π / 3 e. 9a ± π / 3 e For arbtrary, there no cloed-form formula. We can etmate mnma and generally, complex tatonary pont of the functon F by aumng that they appear a a reult of nterference of ome couple of neghbor pole, correpondng to a couple of term n the um 9. For example, for the above example wth 4, the econd local mnmum of F at appear a a reult of uxtapoton of two term, the econd and the thrd, and can be etmated ung e. 9 for the econd and the thrd coordnate and dregardng another coordnate. It gve whch very cloe to the exact poton of the mnmum at The followng table lt all ngularte found by formula 9 n comparon wth exact ngularte. Pole n 7 taen nto account,, 3 3, 4 Sngularte, e ± ± ±.434 Exact ngularte ± ± ±.43 Our aumpton gve accurate predcton for almot all ngularte except only one par of ngularte. Large E behavor t large energe, there are par of extremum pont,

6 armonc ocllator approxmaton 6, 3 4 O W O O O, where,...,,,, and E ±/. an llutraton, n our prevou example, there are four par of root of the euaton E F at 4 E, and the mallet root correpond to the mnmum of W : O O/ For, radu of convergence of thee expanon dtance from the orgn to the mot dtant ngularty, max,, * n n F E where n are gven by formula 9. For arbtrary, t can be etmated a, * * max E E where, * E the radu of convergence for wth only two non-zero adacent term n 7 wth ndexe and here, the freuence are arranged n acendng or decendng order.

7 armonc ocllator approxmaton 7 Small E behavor t mall energe, there a par of real extrema at / µ W W µ µ, / where W and µ ± E /W negatve gn correpond to mnmum of W. There are alo complex tatonary pont. For the above example, appearance of two root of the euaton F E for mall E 3 llutrated on the fgure below the lower graph of the functon R how that the mallet root ha mnmum of W : O/ O 5O/ O

8 armonc ocllator approxmaton 8 lmot clacally allowed tranton mall W formula. If E E V,,...,, then the mnmum decrbed by the followng δ, δ, 3 E W δ where δ E E / E and wa defned above. Perturbaton theory for an anharmonc amltonan Formal perturbaton theory for general anharmoncty To fnd tatonary pont, t neceary to olve multaneou euaton In th ecton, we conder an anharmonc amltonan n the form mall parameter, formula 3, and 6,, f nto power ere W. 4 E where a and W are uadratc functon of momenta and coordnate gven by g 4,,, l l ome anharmoncty whch typcally l... ere, we expand unnown momenta, coordnate, and

9 armonc ocllator approxmaton 9 p p p The frt-order correcton to harmonc approxmaton wll be found here formally for a general anharmoncty. For hortne, momenta and coordnate wll be unfed to a ngle -component et...,,..., p, p,..., p,,,...,, o e. 4 are rewrtten a W E,,,...,. 6 ow, both de of e. 6 wll be expanded n power of. In the econd euaton of 6, For hortne, let u wrte everywhere for ome functon,,,,. The econd euaton of 6 now read... E. 8 In zero order, t euvalent to unperturbed euaton E, and n the frt order,. 9 In a mlar manner, the frt euaton of 6 expanded a

10 armonc ocllator approxmaton W W,...,. In zero order, t euvalent to unperturbed euaton W, and n the frt order, W,,. Soluton of G, where { } arrve to G a matrx recprocal to { W, }, G,. By ubttutng e. nto e. 9, we. 3 From e. 3, determned a G, G. 4, Summary of reult for an anharmonc perturbaton For general anharmonc correcton, the frt anharmonc correcton to coordnate of pont of launch determned a follow. t the frt tep, unperturbed coordnate wth together are calculated accordng to the precedng chapter. Then, the matrx of the econd dervatve { W, } at the pont, calculated, and t nvere matrx { } G

11 armonc ocllator approxmaton found. fter that, the correcton calculated by formula. calculated by formula 4. Fnally, correcton are Perturbaton of harmonc potental by a cubc polynomal Matrxe of the econd dervatve of and W ee formula 3 are dagonal, δ, and W δ where,,..., we dregard here dervatve over momenta becaue we hall conder perturbaton ndependent of momenta, o all matrxe have ze x ntead of x. For the cae when the perturbaton term contan only coordnate and ha a form of a general cubc polynomal, 6,, f, 5 the frt dervatve of are,, f. 6 E. 4 and are conderably mplfed n our cae, 6 3 g,, g g f, f. 7 where g are dagonal matrx element of the matrx G from the prevou ecton, and tand for the coordnate correcton, not the dplacement.

12 armonc ocllator approxmaton pplcaton to benzene molecule Phycal parameter We chooe a ample molecular ytem to tet above approxmaton and to etmate ther applcablty range. Let u conder two-mode ytem wth the followng phycal parameter Q Q Q Q o.397 o.84, o.43 o.86 m m 6 m 836 m e e, 8 cm 39 cm 93 cm 33 cm 8 thee data are for S S tranton n benzene molecule, wth coordnate and correpondng to CC and C bond. Scaled parameter are.39, , n atomc unt. ote that dregardng the dplacement that utfed for large energe, we have W E / where a maxmum of four freuence from 9,.e.. It mean that momenta and the frt coordnate are zero, and only non-zero. For mall dplacement or large energy, oluton gven by euaton wth and / E. The expanon converge when energy larger than.5. Sum of three term of the expanon E. for W hown on the followng fgure, together wth the exact W.

13 armonc ocllator approxmaton 3 'SQGQF RI ORJDULWKP RI :LJQU IQFWLRQ RQ QUJ\ [DFW /DUJ DSSU : There, t hown alo the econd tatonary pont above the mnmum. For the phycal energy E.77, the large-energy approxmaton very accurate. mode an example of the ytem wth four varable, we condered nonymmetrcal vbratonal e g wth freuence 68, 356, 599, 78 and 5, 377, 454, 48 cm. Snce all dplacement for that mode are zero, the oluton mply gven by the frt term of the largeenergy approxmaton. We condered a hypothetcal ytem wth the ame freuence and four eual non-zero dplacement.39, the ame a from the frt example. We found that largeenergy expanon converge for th ytem for energe larger than.3, o that phycal energy ~. le wthn the range of applcablty of th approxmaton. Thee two example how that the radu of applcablty of order of. for typcal molecule. ppendx. Proof of a mathematcal neualty Let u prove that f F F and < then U < U, where functon F and U are defned by E. 7 a a coneuence of th neualty, the root of the euaton F E wth mnmum of U the mnmal root.

14 armonc ocllator approxmaton 4 Introducng ymmetrcal varable and t, we rewrte ncrement of the functon a t U U F F, where. Multplyng the euaton F F by /, we arrve to an euaton /. Snce 4 > t, then t / >, and t / <. Subttutng the latter neualty nto E. and tang nto account that >, we fnd > t. 3 Comparng 3 wth the econd euaton n we conclude that > U U end of the proof.