The Feynman path integral


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1 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 that moves n tme. Ths Hlbert space can be descrbed n any bass we choose: coordnate, x, momentum p, or whatever suts our need. It s also possble to regard the state as fxed and the bass as changng n tme. Ths s the Hesenberg pcture.. Hesenberg operators Consder an operator actng on a state, then projected onto any other state at tme t, χ, t χ, Â ψ, t Û t, Â Û t, t0 ψ, χ, Û t, ÂÛ t, ψ, so f we defne a tmedependent Hesenberg operator, Â H Â t Û t, ÂSÛ t, then we get the same predcton by lookng at Â t actng on the fxed ntal state: χ, t ÂS ψ, t χ, ÂH t ψ, We may replace the Schrodnger equaton wth evoluton equatons for operators. dervatve where Û t, e Ĥt t0, Takng the tme dâh dt tû t, ÂSÛ t, + Û t, ÂS tû t, t, ĤÛ ÂSÛ t, Û t, ÂSÛ t, Ĥ ĤÂH ÂHĤ and we have the Hesenberg equaton of moton, dâh dt Accordng to Sakura, ths was frst wrtten by Drac. [Ĥ, Â H ]
2 .2 Hesenberg bass kets The Hesenberg pcture also requres a change n the bass kets. Snce bass kets are egenkets of partcular operators, and the operators are now tmedependng, the egenkets also change. We have Â S a S a a S where a state n the Schrödnger bass s gven by ψ a, t a ψ, t In the Hesenberg pcture, these become Â H t a, t H a a, t H Û Â S Û a, t H a a, t H Â S Û a, t H aû a, t H so we must have Û a, t H a S Invertng, a, t H Û a S we see that the Hesenberg bass evolves oppostely to the Schrodnger state to gve the same result..3 Transton ampltudes Gven the tmedependence of the bass kets, we may ask for the probablty ampltude for a bass ket a, H at tme to be found n another drecton b, H at tme t, b, t a, Ths s called the transton ampltude. For example, the transton ampltude for a system to go from x at tme to x at tme t s x, t x, 2 Propagators We have seen that the tme evoluton of state s gven by ψ, t e Ĥt t0 ψ, when the Hamltonan s ndependent of tme. Insertng an dentty n terms of an energy bass, ψ, t e Ĥt t0 a E a E a ψ, a e Eat t0 E a E a ψ, Now vew the state n a coordnate bass, x ψ, t a e Eat t0 x E a E a ψ, ψ x, t a e Eat t0 x E a E a ψ, 2
3 Insertng one more dentty n the coordnate bass, we have ψ x, t d 3 x e Eat t0 x E a E a x x ψ, a d 3 x e Eat t0 x E a E a x ψ x, a Now defne the propagator K x, t; x, a x E a E a x e Eat t0 so that we have ψ x, t d 3 x K x, t; x, ψ x, Identfyng the propagator for a gven problem separates the ntal wave functon from the potental, allowng a formal soluton for the wave functon at a later tme and arbtrary poston. Holdng x, fxed, u a x s the statonary state wave functon, and e Eat s ts tme dependence, so K x, t; x, satsfes the tmedependent Schrödnger equaton. Also, lm K x, t; x, δ 3 x x t Moreover, the propagator s essentally a Green s functon that ncludes the tme evoluton, gvng the probablty ampltude for a partcle ntally at x at to be found at x at the later tme t. In ths way, the propagator s the transton ampltude for the system. We can make ths explct: K x, t; x, a x E a E a x e Eat t0 x e Ht a E a E a e Ht0 x x Û t, 0 Û, 0 x so removng the dentty a E a E a and dentfyng the Hesenberg bass states, Û, 0 x x, H and Û t, 0 x x, t H we have the transton ampltude: K x, t; x, x, t x, Transton ampltudes, or propagators, have a composton property. If we nsert the dentty operator n the form d 3 x x, t x, t where < t < t, nto the transton ampltude, t becomes an ntegral over a product of transton ampltudes: x, t x, d 3 x x, t x, t x, t x, Ths shows that the probablty ampltude for gong from x, to x, t s the product of the probablty ampltudes for gong from x, to an ntermedate state at tme t and the probablty of gong from that state to x, t, summed over all possble ntermedate postons. Ths s just lke the composton of condtonal probabltes: P A gven B P A gven C P C gven B C but t s sgnfcant that t apples to probablty ampltudes nstead of probabltes. Ths fact underles Bell s theorem. 3
4 3 The Feynman path ntegral We consder a partcle wth Hamltonan of the form Ĥ p2 2m + V x. Applyng the composton property N tmes n gong from x 0, to x N, t N, N x N, t N x 0, d 3 x x N, t N x N, t N x, t x 0, Now look at one of the transton ampltudes, x +, t + x, t x +, t e Ĥt+ t x, t x, t e p x+ x e Ĥt+ t x, t Let N be suffcently large that t + t t becomes nfntesmal. To evaluate the translaton operator and the Hamltonan, we nsert a momentum bass, x +, t + x, t d 3 p x +, t p, t p, t e Ĥt+ t x, t d 3 p x +, t p, t p, t Ĥ t x, t d 3 p x +, t p, t p 2 2m t V x t p, t x, t Now, usng p, t x, t 2π 3/2 e p x the nfntesmal transton ampltude becomes x +, t + x, t d 3 p x +, t p, t p, t e Ĥt+ t x, t d 3 p x +, t p, t p, t Ĥ t x, t 2π 3 d 3 p e p x+ p 2 2m t V x t 2π 3 d 3 p e p x+ x e p 2 2m t V x t [ p x + x p2 2π 3 d 3 p exp 2π 3 d 3 p exp 2π 3 d 3 p exp L p, x dt where we fnd the Hamltonan replaced by the Lagrangan, [ p dx ] dt H dt L p, x dt p ẋ H dt Notce that all operators have been replaced by egenvalues. e p x ] 2m t V x t 4
5 Now reassemble the full, fnte transton ampltude: N x N, t N x 0, 2π 3N 2π 3N and replacng the sum of nfntesmals by an ntegral, exp N N d 3 x d 3 p N L p, x dt exp d 3 x d 3 p t N exp L p, x dt exp L p, x dt exp S [x t, p t] N L p, x dt where S [x t, p t] s the acton functonal n terms of both poston and momentum. Fnally, we defne the functonal ntegral to be the sum over all ntervenng paths, here n both confguraton and momentum spaces: D [x t] D [p t] N 2π 3N /2 N 2π 3N /2 Wth ths notaton, the transton ampltude, or propagator, s gven by x N, t N x 0, D [x t] D [p t] exp S [x t, p t] Ths s the Feynman path ntegral. Notce agan that the acton here s wrtten as an ndependent functonal of poston and momentum. The nfnte products of ntermedate ntegrals may be nterpreted as meanng that the phase exp S [x t] s to be summed over every value of poston and momentum. As we shall see from examples, the result nvolves some curous normalzatons, but the formulaton s very powerful because t may be mmedately generalzed to feld theory. Any theory of felds Φ havng an acton functonal may be quantzed by averagng exp S [Φ] over all feld confguratons. Φ x, t f Φ x, t D [Φ x, t] exp S [Φ x, t, Π x, t] d 3 x d 3 p S [Φ x, t] t f L Φ x, t, Π x, t d 4 x t Here, the poston and tme are smply parameters, whle the feld and ts conjugate mometum are the dynamcal varables. The most mportant advantage of the path ntegral formulaton s that t allows for a systematc perturbaton theory. If we wrte the partcle Lagrangan as and expand the exponental x N, t N x 0, D [x t] L L 0 + V D [p t] exp t N t N L 0 dt + V dt + 5
6 t s possble to evaluate the potental terms order by order. The same expanson apples to feld theory, Φ x, t f Φ x, t D [Φ x, t] exp S 0 [Φ x, t] + t N V dt + allowng term by term approxmaton. Ultmately, each term n the expanson nvolves dfferent powers of the potental. We keep track of the large number of requred ntegrals by sets of Feynman dagrams, each dagram correspondng to a partcular set of ntegrals. Typcally, equvalence to other methods holds, but s not demanded. The path ntegral s an ndependent model for quantzaton. 4 The momentum ntegrals For the form of Hamltonan we have chosen, Ĥ p2 2m + V x, t s possble to do all of the momentum ntegrals. Each one s smply a Gaussan: 2π 3 d 3 p exp L p, x dt 2π 3 d 3 p exp p x + x p2 t 2m V x dt 2π 3 d 3 p exp p m dx 2 + m 2m dt 2 x + x 2 V x dt 2π 3 exp m 2 v2 V x dt d 3 p exp 2m p mv 2 Lettng y p m x + x, the ntegral becomes d 3 y exp 2m y2 The magnary unt does not really cause any problem. Addng an nfntesmal part for convergence we have d 3 y exp ε y2 d 3 y exp ε + 2m 2m y2 Each of the three Gaussans gves so lm ε 0 d 3 y exp dy exp αy 2 ε y2 2m The full th ntegral s therefore, 2π 3 d 3 p exp L p, x dt π α 3/2 2mπ lm ε 0 ε + 2πm 3/2 m 3/2 exp 2π 2 mv2 V x dt Combnng these n the full path ntegral, we have x N, t N x 0, N d 3 x 2π 3N d 3 p exp N 3N/2 d 3 x m 2π 3 exp N t N L p, x dt 2 mv2 V x dt 6
7 and replacng the sum of nfntesmals by an ntegral, and defnng the functonal ntegral measure to be the transton ampltude s N D [x t] d 3 x m 2π 3 3N/2 x N, t N x 0, D [x t] exp t N L x, ẋ dt D [x t] exp S [x t] where S [x t] s now the usual confguraton space acton. Ths s the usual form of the Feynman path ntegral. 7
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