Path Integral for a Harmonic Oscillator with Time-Dependent Mass and Frequency

Transcription

1 doi: 0.306/scieceasia ScieceAsia 3 (006: Pah Iegral for a Harmoic Oscillaor wih Time-Depede Mass ad Frequecy Surari Pepore a*, Pogip Wioai a, Taakor Osocha b ad Udom Robkob b a Deparme of Chemisry, Faculy of Sciece, Mahidol Uiversiy, Rama VI Road, Bagkok 0400, Thailad. b Deparme of Physics, Faculy of Sciece, Mahidol Uiversiy, Rama VI Road, Bagkok 0400, Thailad. * Correspodig auhor, g @sude.mahidol.ac.h Received 8 Oc 004 Acceped 6 Feb 006 ABSTRACT: The exac soluios o he ime-depede Schrodiger equaio for a harmoic oscillaor wih ime-depede mass ad frequecy were derived i a geeral form. The quaum mechaical propagaor was calculaed by he Feyma pah iegral mehod, while he wave fucio was derived from he specral represeaio of he obaied propagaor. I was show ha he propagaor ad he wave fucio depeded o he s soluio of a classical oscillaor, i which he ampliude ad phase saisfied he auxiliary equaios. To demosrae he derivaio of he soluio from our auxiliary equaios, expoeial ad periodic fucios of mass wih cosa frequecy were imposed o evaluae he propagaor ad wave fucio for he Caldirola-Kaai ad pulsaig mass oscillaors, respecively. KEYWORDS: Pah iegral, propagaor, wave fucio, a harmoic oscillaor wih ime-depede mass ad frequecy. INTRODUCTION I rece years, he sudy of Hamiloia wih explicily ime-depede coefficies becomes very popular. - The mahemaical challege ad impora applicaios i various areas of physics, such as quaum opics, cosmology, 3 ad aoechology, 4 are he mai reasos for iesive sudies. The mos commo problem i his area is he harmoic oscillaor wih ime-depede frequecy ad/or mass. The harmoic oscillaor wih ime-depede frequecy is he firs exacly solved problem. 5 The sadard mehod for solvig he ime-depede problems is he Lewis- Riesefeld (LR ivaria operaor mehod.,3,5,,5 This mehod is based o cosrucig a ivaria operaor ad wriig Schrodiger s wave fucio i erms of ivaria operaor eigesaes coecig wih imedepede phase facor. However, i he case of a harmoic oscillaor wih ime-depede frequecy ad mass he LRivaria operaor mehod has some difficuly. 6-7 I 99, Daas ad e al. 6 cosruced a ivaria operaor from he caoical rasformaio variables ad rasformed he ivaria operaor o a simple harmoic oscillaor operaor by uiary rasformaio. However, heir wave fucios saisfied he Schrodiger s equaio oly i he case of cosas mass, bu were o applicable i geeral case of imedepede mass. Laer i 997, Pedrosa 7 revised his problem by modifyig he ivaria operaor ad used aoher uiary operaor o iclude he ime-depede mass parameer. His resul preseed he firs wave fucio for he harmoic oscillaor wih imedepede mass ad frequecy. Fially, Cifja 8 proposed a aleraive mehod by assumig he Schrodiger s wave fucio i erms of he Gaussia fucio wih ime-depede coefficies ad usig he space-ime rasformaio o reduced he problem o a simple harmoic oscillaor. He suggesed ha here should be some aemp o develop a easier mehod ha he LR-ivaria operaor mehod o ackle he ime-depede problem. The aim of his paper is o derive he propagaor ad wave fucio for a harmoic oscillaor wih imedepede mass ad frequecy i ay fucio form as described by he Hamiloia p H ( = + m ( ω ( x, ( m ( where m ( ad ω ( are he ime-depede mass ad frequecy, respecively. Our developed mehod is o based o he Hamiloia ad solvig he differeial equaios as described i previously repored aricles, 6-8 bu base o he Lagragia ad solvig he iegral equaio by he Feyma pah iegral approach. 9 I his formulaio he ime-depede Schrodiger s equaio is replaced by iegral equaio

2 74 ScieceAsia 3 (006 ( x, = K( x, ; x, ( x, dx, ( where kerel Κ or propagaor represes he rasiio probabiliy ampliude for he iiial wave fucio (x, propagaig o he fial wave fucio (x, i he space-ime cofiguraio. As firs saed by Feyma, 9 he propagaor ca be defied by he pah iegral of i Sx [ ( ] Kx (, ; x, = e Dx [ ( ], (3 where he measure D[x(] deoes he sum over all pah bewee (x, ad (x,. The fucio S[x(] is a acio defied by Sx [ ( ] = L( xxd,,, (4 where Lxx (,, is he Lagragia of he sysem. The propagaor ca be relaed o he imedepede Schrodiger s wave fucio by (, ; *, = (, (,. = 0 Kx x x x (5 To solve his formulaio, he propagaor from he Feyma defiiio i Eq. (3 is calculaed ad he he specral represeaio of propagaor i Eq. (5 is applied o derive he ime-depede Schrodiger s wave fucio. I is well kow ha he quaum soluios for he ime-depede problems 6-8 deped o he udeermied auxiliary equaio. Hece, o demosrae he derivaio of he explici form of he propagaor ad wave fucio, he expoeial ad periodic fucio of mass wih cosa frequecy is employed as a example i he cases of a Caldirola- Kaai oscillaor ad a harmoic oscillaor wih srogly pulsaig mass respecively. I secio, our derivaio of he pah iegral for a harmoic oscillaor wih ime-depede mass ad frequecy is described. I secio 3, he wave fucios for a harmoic oscillaor wih ime-depede mass ad frequecy are derived i a geeral form. To demosrae paricular cases, our geeral soluios are applied o derive he propagaor ad wave fucios for he well-kow Caldirola-Kaai oscillaor ad he harmoic oscillaor wih srogly pulsaig mass i secio 4 ad 5, respecively, ad he coclusio is give i secio 6. The Pah Iegral For a Harmoic Oscillaor wih Time-Depede Mass ad Frequecy The Lagragia associaed o he Hamiloia i Eq. ( is L = m( x m( ω ( x. (6 The quadraic Lagragia propagaor ca be separaed io a pure fucio of ime F(, ad he expoeial fucio of classical acio S cl (x, ; x, as suggesed i Ref. 9 Kx x = F e (, ;, / (, ;, (, is cl x x. (7 Calculaio of he fucio F(, preseed by Pauli, Moree, or Joes ad Papadoupoulos 0- ca be performed by he semiclassical approximaio of pah iegral formula ad he applied o Eq. (3 i F (, = Scl ( x, ; x,. π x x (8 Therefore, he crucial issue i propagaor calculaio i his sysem is o obai he classical acio S cl (x, ; x,. By usig he Euler-Lagrage equaio for he Lagragia i Eq. (6, he equaio of moio ca be wrie as η x+ x + ω ( x = 0, (9 η where we defie η ( = m(. For physical reasos, le cosider he soluio o he equaio of he moio i he form x( = ρ( [A cosγ( + B siγ(], (0 where ρ( = α( /η(, γ( refers o he ampliude ad phase of he classical oscillaos ad A ad B are cosas. The fucios α(, γ( ad η( ca be deermied by subsiuig Eq. (0 io Eq. (9 ad α αη αγ αω ( + { Acosγ + Bsiγ} η η η η αγ αγ + { Asiγ Bcosγ} = 0. η η ( Sice cosas A ad B ca o vaish simulaeously, he fucios α(, γ( ad η( have o saisfy he followig auxiliary equaios η( α γ α + ( 0 ( ω α η( = αγ ad γ + = 0. (3 α The cosas A ad B i Eq. (0 ca be deermied by imposig he boudary codiios of x( = x ad x( = x. The classical pah ha coecs he poi of (x, ad (x, ca be wrie as α( η x η x xcl( = si( γ γ si ( γ γ, η(si( γ γ α α (4

3 ScieceAsia 3 ( where he oaios γ, α ad η refer o γ(, α( ad η( respecively. The acio ca be calculaed from he ime iegraio of he Lagragia from o. Sx (, ; x, = Lxxd (,,. (5 For he acio of our geeral sysem, he Lagragia i Eq. (6 is subsiued io Eq. (5, ad he iegraed by pars of he firs erm o he righ had side of Eq. (6 usig he equaio of moio i Eq. (9. The classical acio ca be wrie as m m Scl( x, ; x, = x cl x cl x cl x cl.. (6 Subsiuig he classical pahs of Eq. (4 io Eq. (6, he classical acio becomes mx αη αη mx αη αη Scl( x, ; x, = αη αη + ( co ( csc (. m γ x + m γ x γ γ ηη γγ xx γ γ (7 By subsiuig he above classical acio io Eq. (8, he pre-expoeial facor ca be obaied as ηη γγ F (, =. πi si( γ γ (8 From Eqs. (7, (7 ad (8, he propagaor for he harmoic oscillaor wih a ime-depede mass ad frequecy ca be expressed by ηη γγ Kx (, ; x, = πi si( γ γ αη αη mx i αη exp αη αη mx αη i exp cos( γ γ si( γ γ ηη γγ xx ( m γ x + m γ x (9 This resul ca be verified by assumig η = η = m = cos. This reduces o he propagaor for a harmoic oscillaor wih ime-depede frequecy ad cosa mass as m γγ K( x, ; x, = πi si( γ γ im α x α x exp α α ( γ x + γ x im exp cos( γ γ si( γ γ xx γγ (0 This propagaor is i agreeme wih he resul of Khadekar ad Lawade 3 by usig he Feyma polygoal mehod ad more recely wih he resul of Yeo 4 e al. by expadig he wave fucio obaied from he LR-ivaria mehod. Furhermore, his resul ca be reduced o he case of he simple harmoic oscillaor propagaor, by seig α, ω ad η = η = m o be cosas. The auxiliary equaios become 0 or γ γ = ω(. ( γα+ ωα= Subsiuig hese parameers io Eq. (9, he well-kow propagaor for a simple harmoic oscillaor, as appearig i Feyma ad Hibbs, 9 ca be obaied as mω K( x, ; x, = πi siω( imω ( ( exp cos ( x x ω xx +. siω ( Wave Fucio For a Harmoic Oscillaor wih Time-Depede Mass ad Frequecy I his secio, he ime-depede Schrodiger s wave fucio is calculaed from he specral represeaio of he propagaor i Eq. (5 by defiig as = i e φ, φ = γ γ, (3 si φ =, i (4 + cos φ =, (5 m γ m γ a= x, b = x. (6 The geeral propagaor i Eq. (9 ca be wrie

4 76 ScieceAsia 3 (006 ηη γγ K( x, ; x, = ( π i αη αη αη αη exp mx mx αη αη + exp ab ( a + b. (7 Now usig he ideiy + = +, (8 ( he propagaor ca be rewrie as ηη γγ K( x, ; x, = ( π i αη αη αη αη exp mx mx αη αη ab ( a + b exp ( a + b exp. (9 By exployig he Mehler s formula 5 ( ( ab a + b exp = H( a H( b, (30 = 0! where H ( x is he Hermie polyomial, he propagaor becomes ηη γγ Kx (, ; x, = π αη αη mx i αη exp αη αη mx αη exp ( m γ x + m γ x m γ H x H = 0 (3 iφ + m γ e x! Comparig he propagaor i Eq. (3 wih Eq. (5, he wave fucio for a harmoic oscillaor wih imedepede mass ad frequecy ca be expressed as m ( γ ( ( x, = exp i + γ (! π im( α( η( α( η( exp + i γ ( x α( η( m ( γ ( H x. (3 Sice each ( x, saisfies he ime-depede Schrodiger s equaio, he geeral soluio ca be wrie as = 0 Ψ ( x, = C ( x,, (33 where C are cosas. The geeral wave fucio i Eq. (3 ca be verified by seig αωη,, = η = m o be cosa i he auxiliary equaio as meioed i Eq. (. The wave fucio ca be reduced o mω ( x, = exp i + ω! π mω mω x H x exp, (34 which is he wave fucio for a simple harmoic oscillaor appearig i he ex-book o quaum mechaics. 6-7 I should be oed ha he geeral wave fucio i Eq. (3 is slighly differe from repored by he resul of Pedrosa 7 ad Cifja 8 because of differe α( oaios. I order o comparig, le = ρ( η( ad γ ( = 0 d m ( ρ ( he he auxiliary equaio i Eq. ( becomes m ( ρ( + ρ( + ω ( ρ( =. 3 (35 m ( m( ρ ( ad he wave fucio ca be wrie as

5 ScieceAsia 3 ( ( x, 4 =! πρ ( d exp i+ 0 m ( ρ ( exp im( ρ( i + x ( ( ρ m ρ ( x H, ρ( which agrees wih heir resuls. 7-8 (36 The Caldirola-Kaai Oscillaor I his secio, he applicaio of he soluio of our auxiliary Eqs. ( ad (3 is demosraed for derivig he explici form of he propagaor ad wave fucio. The sysem seleced as a example is he quaum damped harmoic oscillaor or he Caldirola-Kaai oscillaor. 8-9 By iroducig he mass law, he imedepede mass ca be wrie as m( = me r, (37 where m is he cosa mass ad r is he cosa dampig coefficie. The Caldirola-Kaai Hamiloia ca be obaied by he Hamiloia i Eq. ( wih cosas ω p H = e + mω e x m r r (. (38 I order o obai he propagaor of his sysem, he explici forms of he fucio α( ad γ( i Eq. (9 have o be derived. By subsiuig he ime-depede mass i Eq. (37 io he auxiliary Eqs. ( ad (3, i ca be derived ha α = Ω ad (39 γ( = Ω, (40 where Ω is he reduce frequecy defied by r Ω = ω. (4 4 By subsiuig Eqs. (37, (39, (40 ad η ( = m( io Eq. (9, he propagaor for he Caldirola-Kaai oscillaor ca be obaied as r ( + / mωe K( x, ; x, = πi siω( imω r r x x e exp co Ω( ( e x + e x siω r ( + / ( irm exp ( e r x e r x. 4 (4 The wave fucio for his sysem ca be calculaed by Eq. (3 as mω 4 r ( x, = exp i Ω +! π 4 r m ir r mω exp e x H e x Ω+. 4 (43 The obaied propagaor ad wave fucio i Eq. (4 ad (43 are i he same form as ha repored by Jausis e al. 30 The Harmoic Oscillaor wih Srogly Pulsaig Mass The oher well kow example of a ime-depede mass oscillaor is a harmoic oscillaor wih srogly pulsaig mass. 3 This oscillaor ca be applied i coecio wih he elecromageic field i a Fabry- Pero caviy i coac wih a reservoir of resoa wo-level aoms. The periodic release ad reabsorpio of phoo ca be represeed by a oscillaor of periodically flucuaig eergy. I oher words, i ca be represeed by a periodically varyig mass as m( = m cos ν, (44 where ν is he frequecy of mass. I his case he Hamiloia becomes p H ( = secν+ mcos νω x. (45 m By subsiuig he mass law io he auxiliary Eqs. ( ad (3, we ca ge α = ad (46 Ω γ( = Ω, (47 where he augmeed frequecy Ω is defied by Ω = ω + ν. (48 By subsiuig hese Eqs. (44, (46 ad (47 io he geeral propagaor i Eq. (9, he propagaor for a harmoic oscillaor wih srogly pulsaig mass ca be derived as mωcosν cosν K( x, ; x, = π i siω(

6 78 ScieceAsia 3 (006 imν exp cos a cos a cos νx + cos νx imω exp cos Ω(. siω( cosν cosν x x (49 This propagaor ca be simplified by seig ν = 0, ad Ω = ω. The resul is reduced o he simple harmoic oscillaor propagaor. By subsiuig Eqs. (46 ad (47 io Eq.(3, he wave fucio for a harmoic oscillaor wih srogly pulsaig mass ca be obaied as m ( Ω ( x, = exp i + Ω! π im( m( Ω ( + Ω exp ν a ν i x H x. 4 (50 This resul agrees wih he wave fucio of Colegrave ad Abdalla. 3 CONCLUSION ( ν ν x ν νx ( I his paper we successfully calculaed he exac propagaor ad wave fucio for a harmoic oscillaor wih ime-depede mass ad frequecy by he Feyma pah iegral formulaio. The resulig propagaor i Eq. (9 ca be reduced o he propagaor for a harmoic oscillaor wih ime-depede frequecy ad cosa mass which agrees wih he resul of Khadekar ad Lawade 3 ad Yeo e al. 4 as show i Eq. (0. Moreover, our propagaor i Eq. (9 ca be reduced o he simple harmoic oscillaor propagaor 9 by seig he mass ad frequecy o be cosas as show i Eq. (. The resulig wave fucios i Eq. (3 ca be also reduced o he simple harmoic oscillaor wave fucios 6-7 by seig he ime-depede mass ad frequecy fucio o be cosas. The crucial resul i our calculaio is o express he geeral soluio of a ime-depede mass ad frequecy oscillaor as meioed i Eq. (0. This soluio i Eq. (0 has he ime-depede phase ad ampliude which saisfy he auxiliary Eqs. ( ad (3. By modifyig he oaios, he auxiliary Eq. ( become he well kow Piey equaio 3 as show i Eq. (35. I secios 4 ad 5, we have show he usefuless of auxiliary equaios for derivig he explici form of he propagaor ad wave fucio i he case of he Caldirola-Kaai ad srogly pulsaig mass oscillaor. I order o compare our approach wih oher works, he ampliude ρ ( i Eq. (0 correspods o he space rasformaio of x = ρ( y, ad he phase d facor γ ( = 0 m ( ρ ( is he ime rasformaio, boh of he Cifja approach. 8 I he Pedrosa 7 approach he ampliude ρ ( is he crucial facor o cosruc a ivaria ad uiary operaor. Surprisigly, by he differe formulaios of mahemaics, he ρ( fucio i he hree approaches ca saisfy he same oliear differeial equaio or he Piey equaio. Fially, i ca be cocluded here ha he pah iegral is he effecive ad sraigh forward mehod for solvig he ime-depede problems because i requires he same procedure i solvig ime-idepede problems wihou employig ay rasformaio compared wih he oher mehods. ACKNOWLEDGEMENTS The auhors would like o ac kowledge he PERCH- ADB for fiacial suppor. REFERENCES. Sog DY (000 Uiary relaio bewee harmoic oscillaor of ime-depede frequecy ad a simple harmoic oscillaor wih ad wihou a iverse-square poeial. Phys Rev A 6, Kim SP ad Page DN (00 Classical ad quaum aciophase variables for ime-depede oscillaors. Phys Rev A 64, Choi JR ad Zhag S (00 Thermodyamics of he sadard quaum harmoic oscillaor of ime-depede frequecy wih ad wihou iverse quadraic poeial. J Phys A 35, Sog DY (003 Uiary relaio for he ime-depede SU(, sysems. Phys Rev A 68, Cessa HM ad Gausi MF (003 Cohere saes for he ime-depede harmoic oscillaor: he sep fucio. Phys Le A 3, Yeo KH, Um CI, ad George TF (003 Time-depede geeral quaum quadraic Hamiloia sysem. Phys Rev A 68, Sog DY ad Klauder JR (003 Geeraliaio of he Darboux rasformaio ad geeralied harmoic oscillaors. J Phys A 36, Abdalla MS ad Leach PGL (003 Liear quadraic ivarias for he rasformed Tavis-Cummigs model. J Phys A 36, Li QG (004 Time evoluio, cyclic soluios ad

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