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Alternative Method for Determining the Feynman Propagator of A

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Alternative Method for Determining the Feynman Propagator of A Sym metry comma I n tegra b i l i ty a nd Geometry : .... Methods a nd App l i ca t i ons .... S I GMA 3 open parenthesis 2007 closing nnoindent Sym metry , I n tegra b i l i ty a nd Geometry : n h f i l l Methods a nd App l i ca t i ons n h f i l l S IGMA3 ( 2007 ) , 1 1 0 , 1 2 pa ges parenthesisSym metry comma , I 1n 1tegra 0 comma b i l i ty 1 a 2 nd pa Geometry ges : Methods a nd App l i ca t i ons S I GMA 3 ( 2007 ) , 1 1 0 , 1 2 pa Alternativeges .. Method .. for .. Determining nnoindenttheAlternative .. FeynmanAlternative .. Propagatornquad .. Method ofMethod a .. Non hyphennquad forRelativisticf o r nquad DeterminingDetermining Quantum .. Mechanical .. Problem to the power of big star nnoindentMarcosthe MOSHINSKYthe Feynmannquad daggerFeynman sub commanquad Propagator EmersonPropagator SADURNInquad dagger ofo f a ..a andnquad Non AdolfoNon DEL -− Relativistic CAMPORelativistic ddagger dagger Instituto de F acute-dotlessi sica Universidad NacionalProblem Aut? acute-o noma de M e-acute xico comma nnoindentQuantumQuantum nquad MechanicalMechanical nquad $ Problem ^f n star g$ ApartadoMarcos Postal MOSHINSKY 20 hyphen 364 commay; Emerson .. 1 0 SADURNIM acute-e xico Dy periodand Adolfo F period DEL comma CAMPO M e-acutez xico E hypheny Instituto mail : de .. m F .. os´{ sica hi at Universidad fi s i c a period Nacional una .. m Autperiodo´ mnoma x comma de sM a d ..e´ xico u r ni , at .. fis i c a period u n .. a m .. period .. mx nnoindent Marcos MOSHINSKY $ ndagger f , g$ Emerson SADURNI $ ndagger $ nquad and Adolfo DEL CAMPO ddagger DepartamentoApartado de Qu acute-dotlessi Postal 20 - 364mica , hyphen 1 0F dotlessi-acuteM e´ xico D sica . 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Vasco comma $ nddagger $ E - mailApdo : periodm .. 644 os hi comma@fi Bilbaos i c comma a . una Spain m . m x , s a d u r ni @ fis i c a . u n a m . mx E hyphenz Departamento mail : .. q f bdee de Qu c a at´{ lgmica period - F e h u´{ sica .. period , e s Universidad del Pa ´{ s Vasco , Apdo . nnoindentReceived644 , August Bilbao$ ndagger 2 , 1 Spain comma$ 2007 Instituto comma in de final F form $ n Novemberacutefnimath 1 3 commag $ 2007 sica semicolon Universidad Published Nacional online November Aut $ 22n commaacutef 2007og $ noma de M $ nacutefeg $ x i c o , Original article is available atE htt - mail p : slash : slashq f bdeew w w c period a @ elg m . i e s hperiod u d . e eslash s j o urna l s slash S IG M A slash 2 0 0 7 slash 1 1 0 slash Received August 2 1 , 2007 , in final form November 1 3 , 2007 ; Published online November 22 , 2007 n centerline fApartado Postal 20 − 364 , nquad 1 0 M $ nacutefeg $ xicoD.F. ,M $ nacutefeg $ x i c o g AbstractOriginal period article .. A is direct available procedure at htt for p : determining / / w w w the . propagator e m i s . associated d e / j with o urna a quantum l s / S IG M A / 2 mechanical0 0 7 / problem 1 1 0 / was given by the Path Integration Procedure of Feynman period .. The Green n centerline fE − mail : nquad m nquad oshi $@ fi$ sica.una nquad m . m x , s a d nquad u r ni function commaAbstract which . is theA direct Fourier procedure Transform for with determining respect to the the propagator time variable associated of the propagator with a quantum comma $ @ $ nquad f i s i c a . u n nquad a m nquad . nquad mx g can be derivedmechanical later period problem In our was approach given by comma the Path with Integration the help ofProcedure a Laplace of transform Feynman comma. The a Green direct way to get the energyfunction dependent , which Green is the function Fourier Transformis presented with comma respect and to the the propagator time variable can ofbe the obtained propagator , nnoindent $ nddagger $ Departamento de Qu $ nacutefnimathg $ mica − F $ nacutefnimathg $ s i c a , nquad Universidad del Pa later with ancan inverse be derived Laplace later transform . In our approach period .. The , with method the help is i of llustrated a Laplace through transform simple , a direct one way to $ nacutefnimathg $ s nquad Vasco , Apdo . nquad 644 , Bilbao , Spain dimensionalget examples the energy and for dependent time independent Green function potentials is presented comma , though and the it propagator can be generalized can be obtained to the derivationlater of more with complicated an inverse Laplace propagators transform period . The method is i llustrated through simple one n centerline fE − mail : nquad qfbdeeca $@$ lg .ehu nquad . e s g Key words :dimensional propagator semicolonexamples and Green for functions time independent semicolon potentials harmonic , oscillator though it can be generalized to 2000 Mathematics Subject Classificationthe derivation : 8 1 V of 35 more semicolon complicated 8 1 Q 5propagators . nnoindent Received August 2 1 , 2007 , in final form November 1 3 , 2007 ; Published online November 22 , 2007 1 .. Introduction Key words : propagator ; Green functions ; harmonic oscillator It i s well know that quantum2000 mechanics Mathematics acquired Subject it s Classification final formulation : 8 in 1 V 1 925 35 ; endash 8 1 Q 5 1 926 through nnoindentfundamental1Original Introduction papers of articleSchr o-dieresis isavailable dinger and Heisenberg at http period : / .. /www Originally . these emi papers s . appeared d e / as j two o urna l s /SIGMA/ 2 0 0 7 / 1 1 0 / independentIt i s well views know of that the structure quantum of quantummechanics mechanics acquired comma it s final but in formulation 1927 Schr dieresis-o in 1 925 dinger { 1 926 established through n centerline f Abstract . nquad A direct procedure for determining the propagator associated with a quantum g theirfundamental equivalence comma papers and of since Schr theno¨ dinger one or the and other Heisenberg of the papers . mentioned Originally have these been used papers t o appeared analyzeas two quantum independent mechanical views systems of commathe structure .. depending of quantum on which mechanics method gave , the but most in 1927 convenient Schr o¨ dinger n centerlinewayestablished of solvingf mechanical the their problem equivalence period problem .. Thus , was and the given since existence then by of the one alternative Path or the Integration procedures other of the t o solveProcedure papers a given mentioned problem of Feynman have . nquad The Green g canbeen b e quite used fruitful t o in deriving solutions of it period n centerlineIn theanalyze 1 940f quoterightquantumfunction smechanical Richard , which Feynman is systems the comma Fourier , depending and lat Transform er many on which others with method comma respect derived gave to the a the propagator most time convenient for variable quantum of the propagator , g mechanicalway of problems solving the through problem a path . integration Thus the procedure existence period of .. alternative In contrast procedureswith the Hamiltonian t o solve a given n centerlineemphasisproblem inf thecan can original be b e derived quite formulation fruitful later of quantum in . deriving In our mechanics solutions approach comma of , itFeynmans with . the approach help couldof a be Laplace re hyphen transform , a direct way to g ferred tIn o as the Lagrangian 1 940 ' and s Richard it emphasized Feynman the propagator , and lat K eropen many parenthesis others x comma, derived t comma a propagator x to the power for of prime comma t to the powern centerline ofquantum primef closingget mechanical the parenthesis energy problems which dependent takes through the Green wave a path function integration is presentedprocedure . , and In contrast the propagator with the can be obtained g functionHamiltonian psi open parenthesis x to the power of prime comma t to the power of prime closing parenthesis at the point x to the power of primen centerline andemphasis t imef later t into the the with power original an of prime inverseformulation t o the Laplace point of quantum x comma transform at mechanics t ime . tn commaquad , FeynmansThe i period method e approach period is i could llustrated be re through simple one g Equation:- open parenthesis 1 closing parenthesis .. psi open parenthesis x comma t closing parenthesis = integral K open parenthesis x n centerline f dimensional examples and for time independent potentials , though it can be generalized to g commaferred t comma t o x as to Lagrangian the power of and prime it comma emphasized t to the the power propagator of prime closingK(x; t; parenthesis x0; t0) which psi takes open the parenthesis wave x to the power of prime commafunction t to the power (x0; of t0) primeat the closing point parenthesisx0 and t dx ime to thet0 t power o the of point primex; periodat t ime t; i . e . n centerlineWhile this propagatorf the derivation could b e derived of more by the complicated standard methods propagators of quantum . mechanicsg comma Feynman invented a procedure by summing all t ime dependent paths connecting points x comma x to the power of prime and n centerline fKey words : propagator ; GreenZ functions ; harmonic oscillator g hline (x; t) = K(x; t; x0; t0) (x0; t0)dx0: (1) big star sub This paper is a contribution to the Proceedings of the Seventh International Conference .. quotedblleft Symmetry in n centerline f2000 Mathematics Subject Classification : 8 1 V 35 ; 8 1 Q 5 g NonlinearWhile .. Mathematical this propagator .. Physics could quotedblright b e derived .. by open the parenthesis standard June methods .. 24 of endash quantum 30 comma mechanics .
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