LECTURE NOTES ON PATH INTEGRALS

Transcription

1 Mts Wllin LECTURE NOTES ON PATH INTEGRALS References John Townsend, A Modern Introduction to Quntum Mechnics (University Science Books 2000), Ch. 8. J. J. Skuri, Modern Quntum Mechnics (revised ed, Addison-Wesley 1994), Sec R. Shnkr, Principles of Quntum Mechnics (2ed, Plenum 1994), Ch. 8 nd 21. H. Kleinert, Pth Integrls in Quntum Mechnics, Sttistics nd Polymer Physics (World Scientific 1990). Advntges with the pth integrl formultion of QM: Gives wy to think bout the mening of QM. Includes the clssicl limit explicitly. Deep connection to sttisticl mechnics. Disdvntge: Is often more complicted thn the Heisenberg-Schrödinger formultion for solving ctul problems. 1 The propgtor Consider the time evolution of n opertor A tht commutes with H, which is ssumed time-independent. Common eigenkets: A = H = E (1) Assume n initil stte Time evolution opertor ψ, t 0 = ψ, t 0 (2) U(t, t 0 ) = e ī h H(t t 0) (3) ψ, t = U(t, t 0 ) ψ, t 0 = e ī h H(t t 0) ψ, t 0 = 1 e ī h E(t t 0) ψ, t 0 (4)

2 Let us go to the position representtion. For simplicity we write ll formuls for the one-dimensionl cse, but x cn in generl denote the position vector in d-dimensions. We get ψ(x, t) = x ψ, t = e ī h E(t t 0) x ψ, t 0 (5) Insert dx 0 x 0 x 0 = 1 to lso get ψ(x 0, t 0 ) = x 0 ψ, t 0 in the position bsis: ψ(x, t) = x ψ, t = dx 0 e ī h E(t t0) x x 0 ψ(x 0, t 0 ) ψ(x, t) = dx 0 K(x, t, x 0, t 0 )ψ(x 0, t 0 ) (6) (note tht x = energy eigenfunction). The integrl kernel is clled the propgtor: K(x, t, x 0, t 0 ) = x x 0 e ī h E(t t 0) (7) K completely determines the time evolution of the initil stte. The time evolution is hence perfectly cusl: the time evolution is unique if the system is left lone. However, s soon s mesurement is mde the system chnges in n uncontrollble wy into one of the eigensttes of the opertor corresponding to the observble being mesured. 2 Properties of K 1. For t > t 0, K(x, t, x 0, t 0 ) stisfies the time-dependent Schrödinger eqution (TDSE) in the vribles x, t, with fixed x 0, t 0. This is obvious since x e ih(t t 0)/ h is the energy eigenfunction t x, t. 2. At the initil time, t = t 0 : K(x, t, x 0, t 0 ) = x x 0 e ī h E(t t 0) x x 0 = x x 0 = δ(x x 0 ) (8) 3. K(x, t, x 0, t 0 ) = x x 0 e ī h E(t t 0) = x e ī h H(t t 0) x 0 K(x, t, x 0, t 0 ) = x e ī h H(t t 0) x 0 (9) Hence K is the wve function t x, t of prticle loclized exctly t x 0 t time t 0. 2

3 4. From Eq. (6) we see tht K behves just like Green function. Indeed, if we define the cusl propgtor s K c (x, t, x 0, t 0 ) = θ(t t 0 )K(x, t, x 0, t 0 ) (10) where θ(t) is the Heviside function (θ(t) = 0 for t < 0, = 1 for t > 0), then ( h2 2 2m x + V (x) i h ) K 2 c = i hδ(t t 0 )δ(x x 0 ) (11) t The RHS comes from the fct tht dθ(t) dt = δ(t), which gives i h θ(t t 0) K(x, t, x 0, t 0 ) = i hδ(t t 0 )K(x, t 0, x 0, t 0 ) = i hδ(t t 0 )δ(x x 0 ) t (12) where the lst equlity uses Property 2 bove. 3 The free propgtor For free prticle the momentum ˆp nd Hmiltonin H = ˆp2 commute. Common 2m eigensttes for ˆp nd H re plne wves: ˆp p = p p (Townsend Ch. 3). The propgtor is strightforwrd to compute: = H p = p2 1 p x p = e ī h px (13) 2m 2π h K(x, t, x 0, t 0 ) = x e ī h 2m (t t0) x 0 dp x e ī ˆp 2 h 2m (t t0) p p x 0 = 1 }{{} 2π h =e h ī p 2 2m (t t 0 ) p ˆp 2 dp e ī h px e ī h p 2 2m (t t0) e ī h px 0 This is Gussin integrl tht is computed s usul by completing the squre in the exponent: i h p(x x 0) ip2 2m h (t t 0) = ī ( p m t t0 h 2m 2 The momentum integrl gives x x 0 t t0 ) 2 + im(x x 0) 2 2 h(t t 0 ) dp e ī p 2 (t t 0 ) h 2m = 2πm h i(t t 0 ) (14) 3

4 (A note on Gussin integrls: The clcultion bove uses the result π dx e x2 = which is vlid if hs positive rel prt, or if is purely imginry: π dx e ix2 = i where is now rel. Here we must select the root i = exp(iπ/4). These results re derived in App. D in Townsends book.) Hence we get K(x, t, x 0, t 0 ) = 1 2πm h 2π h i(t t 0 ) e im(x x0) 2 h(t t 0 ) m K(x, t, x 0, t 0 ) = 2πi h(t t 0 ) e im(x x0) h(t t 0 ) (15) Comprison with diffusion: The free TDSE turns into the diffusion eqution D 2 P (x, t) = t P (x, t) if we identify D h nd t it. Then we recognize 2m Eq. (15) s the Green function for the diffusion eqution. Conclusion: Quntum mechnics for free prticle is like diffusion in imginry time. 4 Connection to sttisticl mechnics Let us tke t 0 = 0, nd select the finl point equl to the strting point, x = x 0, nd then integrte over ll x. We obtin: dx K(x, t, x, 0) = dx x x e ī h Et = e ī h Et (16) (where the lst step ssumes tht the energy eigensttes re normlized: dx x x = dx x 2 = 1.) This expression smells very much like the prtition function in sttisticl mechnics. In fct, if we set β = it/ h, where β > 0 is rel vrible nd time the time vrible t becomes imginry, then we obtin precisely the prtition function Z = e βe (17) Hence the quntum problem in imginry time is equivlent to sttisticl mechnics problem t inverse temperture β = 1/k B T. This perhps sounds fr out t first, but it is ctully very fruitful nd beutiful connection becuse it llows results from QM nd sttmech to be exchnged. 4

5 5 Propgtor s trnsition mplitude K(x, t, x 0, t 0 ) = x x 0 e ī h E(t t0) = x e ī h Ht e ī h Ht 0 x 0 = x, t x 0, t 0 (18) This is the probbility mplitude for prticle initilly t x 0, t 0 to be found t x, t. This is clled the trnsition mplitude for going from the stte x 0, t 0 to x, t. Renme x x 2, t t 2, nd introduce n intermedite time t 0 < t 1 < t 2. Insert the completeness reltion t time t 1 : dx 1 x 1, t 1 x 1, t 1 = 1 (19) into x 2, t 2 x 0, t 0 : x 2, t 2 x 0, t 0 = dx 1 x 2, t 2 x 1, t 1 x 1, t 1 x 0, t 0 (20) This is clled the composition property. (This reltion is clled the Chpmn- Kolmogoroff eqution in probbility theory, nd the Smoluchovsky eqution in diffusion theory). 6 Trnsition mplitude s sum over ll pths Divide the time between the initil time t 0 nd the finl time t N into N equl timesteps: t n t n 1 = t = t N t 0. Use the composition property (20) repetedly N s follows: x N, t N x 0, t 0 = dx N 1 dx 1 x N, t N x N 1, t N 1 x 1, t 1 x 0, t 0 (21) This expression cn be visulized pictorilly s sum over ll pths between the fixed spcetime strting point nd end point; see Fig. 8.4, P. 222 in Townsends book. 7 Feynmn-Dirc pth integrl formul Assume time-independent Hmiltonin of the form H = p2 + V (x). Henceforth set t 0 = 0, nd rerrnge the rguments of K s follows: K(x, x 0, t) = 2m x e ī h Ht x 0. 5

6 Now use e ī h Ht = ( ) N. e ī h H t N Assume smll timesteps, t = t/n 0, nd clculte to first order in t. We obtin the so clled Trotter formul: e ī h H t = 1 ī ( h H t = 1 ī p 2 ) ( h 2m t 1 i ) h V (x) t = e ī p 2 h 2m t e ī h V (x) t (22) where ll equlities re vlid to first order in t, i.e. for N. (However, for singulr potentils, e.g. the Coulomb potentil V (r) 1/r, this computtion is in trouble. See Kleinerts book for discussion.) We thus obtin for N : K(x, x 0, t) = x e ī h 2m t e ī h }{{ V (x) t } x 0 (23) N times Insert the resolution of unity dx x x = 1 between ll N opertors. Consider typicl mtrix element: p 2 x n e ī h 2m t e ī h V (x) t x n 1 = (free propgtor) e }{{} = x n 1 e h ī V (x n 1 ) t m = 2πi h t e im(xn xn 1) 2 2 h t e ī h V (x n 1) t where we used Eq. (15) for the free propgtor. p 2 ī h V (x n 1) = Now collect ll such fctors. Renme the finl point x = x N : ( ) [ m N/2 N 1 ] [ ( i t N m(xn x n 1 ) 2 )] K(x N, x 0, t) = dx n exp V (x 2πi h t n=1 h n=1 2( t) 2 n 1 ) (25) This discrete formul cn, in continuum nottion for N, be written K(x, x 0, t) = D[x]e ī h S(x,x 0,t) (26) where S(x, x 0, t) = t 0 dt L is the clssicl ction, nd L = 1 2 mẋ2 V is the Lgrngin. The integrtion mesure is defined s ( D[x] = lim N m 2πi h t ) N/2 N 1 n=1 (24) dx n (27) The pth integrl formul (26) is mentioned s remrk in Dircs QM book. In trying to mke sense of Dircs remrk, Feynmn worked out his lterntive formultion of QM. Remeber tht K is solution to the TDSE. Hence QM cn be formulted in terms of the clssicl ction, just s clssicl mechnics. 6

7 8 Clssicl limit of QM Clssicl mechnics is given by the principle of lest ction: A clssicl prticle follows the clssicl pth which minimizes (or t lest extremizes) the clssicl ction: δs = 0. The Euler-Lgrnge vritionl eqution for this problem gives Lgrnges eqution of motion: d dt of motion, m d2 x dt 2 = V x. L = L ẋ x. This is equivlent to Newtons eqution The clssicl limit follows utomticlly from the pth integrl formul (26). For S/ h ll pths cncel by destructive interference, except ner the clssicl pth tht miminizes S. Further discussion of this is given in Townsends book, Sec Quntum interference experiments 9.1 Electron double slit experiment In this experiment, low-intensity bem of electrons is llowed to pss through double slit nd detected behind the slit t detector screen. Ech electron tht hits the detector screen is detected t single loction, so there is no doubt tht electrons re prticles. Even if the bem intensity is so low tht on verge only single electron t the time is in flight, n interference pttern is built up on the detector screen fter tens of thousnds of counts hve been detected. This is in cler violtion of clssicl physics. According to the pth integrl formul, ech electron tkes both possible pths simultneously, nd the interference between those two pths in the pth integrl gives the observed interference pttern. (Other pths thn the stright lines re cnceled by destructive interference; this is wht mkes it possible to im the electron bem in TV.) Long fter this thought experiment ws first described, the experiment ws performed, nd the interference pttern verified. References: A. Tonomur et l., Americn Journl of Physics 57, (1989). OH: Buildup of electron interference pttern. 7

8 9.2 Interference pttern in CD dt disc A rinbow colored interference pttern cn be seen by eye in CD dt disc. The interference is suggestive of wves of light, but quntum mechniclly light is mde up of qunt, i.e. quntum prticles clled photons. Ech photon tkes ll pths nd is thus reflected t ll possible plces on the CD. The photon ction is the energy hν times the time of flight =distnce/(speed of light). This mkes different colors interfere constructively or destructively t different ngles with respect to the position of the eye, which explins the rinbow pttern. DEMO: Rinbow interference pttern in Win 95 disc. 10 More sttisticl mechnics Set x N = x 0 = x nd integrte over x: Z = Tr e ī h where Tr = dx D[x]. Imginry time: it/ h = τ/ h = β, t 0 dt L (28) L = 1 x 2 md2 dt V (x) = 1 x 2 2 md2 V (x) = H (29) dτ 2 In sttisticl mechnics the Hmiltonin is often n integrl over spce of Hmiltonin density h(x): H = d d x h(x) (30) where we explicitly mrked the dimensionlity d of the system in the integrtion symbol. Hence the prtition function Z becomes Z = Tr e 1 h hβ dτ d d x h(x) 0 (31) High-temperture limit, T, β 0: Z Tr e βh (32) e.g. the usul clssicl prtition function, s expected. Low-T limit, T 0, β : Z = Tr e 1 h dτ d d x h(x) 0 (33) 8

9 In the thermodynmic limit, where the system size is infinite, this is formlly prtition function for system in d + 1-dimensions, since in ddition to the d-dimensionl integrtions over ll spce, there is n dditionl infinite integrl over imginry time. Hence quntum mechnics problem in d-dimensions in the groundstte t T = 0 is equivlent to d + 1-dimensionl clssicl sttisticl mechnics problem t temperture k B T = h. This reltionship hs some very interesting nd remrkble consequences. For exmple, if the clssicl problem hs phse trnsition, this implies tht the quntum system hs corresponding chnge of groundstte t T = 0. Such phse trnsition is driven by quntum fluctutions rther thn by therml fluctutions, nd is clled quntum phse trnsition. QPT is hot reserch topic. Exmples of recently studied QPT: Mgnetic QPT in LiHoF 4 mgnetic lloys Hevy fermion mteril CeCu 6 x Au x. Quntum Hll systems: 2D electron gs in semiconductor heterostructures. Spin fluctutions in L2CuO 4. Locliztion in Fermi nd Bose systems. QPT in high temperture superconductors. Reference: S. Schdev, Quntum Phse Trnsitions, Cmbridge U. Press (1999). 9