Feynman Path Integral: Formulation Of Quantum Dynamics .

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Feynman Path Integral: Formulation of Quantum DynamicsMasatsugu Sei SuzukiDepartment of Physics, SUNY at Binghamton(Date: January 31, 2021)((Overview))We consider quantum superpositions of possible alternative classical trajectories, ahistory being a path in configuration space, taken between fixed points (a and b). Thebasic idea of the Feynman path integral is a perspective on the fundamental quantummechanical principle of complex linear superposition of such entire spacetime histories.In the quantum world, instead of there being just one classical ‘reality’, represented byone such trajectory (one history), there is a great complex superposition of all these‘alternative realities’ (superposed alternative histories). Accordingly, each history is to beassigned a complex weighting factor, which we refer to as an amplitude, if the total isnormalized to modulus unity, so the squared modulus of an amplitude gives us aprobability. The magic role of the Lagrangian is that it tells us what amplitude is to beassigned to each such history. If we know the Lagrangian L, then we can obtain the actionS, for that history (the action being just the integral of L for that classical history alongthe path). The complex amplitude to be assigned to that particular history is then given bythe deceptively simple formula amplitudebiiexp( S ) exp( Ldt ) .ℏℏaThe total amplitude to get from a to b is the sum of these.((Path integral))What is the path integral in quantum mechanics?The path integral formulation of quantum mechanics is a description of quantumtheory which generalizes the action principle of classical mechanics. It replaces theclassical notion of a single, unique trajectory for a system with a sum, or functionalintegral, over an infinity of possible trajectories to compute a quantum amplitude. Thebasic idea of the path integral formulation can be traced back to Norbert Wiener, whointroduced the Wiener integral for solving problems in diffusion and Brownian motion.This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M.Dirac in his 1933 paper. The complete method was developed in 1948 by RichardFeynman. This formulation has proven crucial to the subsequent development oftheoretical physics, because it is manifestly symmetric between time and space. Unlikeprevious methods, the path-integral allows a physicist to easily change coordinatesbetween very different canonical descriptions of the same quantum system.((The idea of the path integral by Richard P. Feynman))1

R.P. Feynman, The development of the space-time view of quantumelectrodynamics (Nobel Lecture, December 11, 1965).http://www.nobelprize.org/nobel eynman explained how to get the idea of the path integral in his talk of the NobelLecture. The detail is as follows. The sentence is a little revised because of typo.I went to a beer party in the Nassau Tavern in Princeton. There was a gentleman,newly arrived from Europe (Herbert Jehle) who came and sat next to me. Europeans aremuch more serious than we are in America because they think that a good place todiscuss intellectual matters is a beer party. So, he sat by me and asked, «what are youdoing» and so on, and I said, «I’m drinking beer.» Then I realized that he wanted to knowwhat work I was doing and I told him I was struggling with this problem, and I simplyturned to him and said, ((listen, do you know any way of doing quantum mechanics,starting with action - where the action integral comes into the quantum mechanics?»«No», he said, «but Dirac has a paper in which the Lagrangian, at least, comes intoquantum mechanics. I will show it to you tomorrow»Next day we went to the Princeton Library, they have little rooms on the side todiscuss things, and he showed me this paper. What Dirac said was the following: There isin quantum mechanics a very important quantity which carries the wave function fromone time to another, besides the differential equation but equivalent to it, a kind of akernel, which we might call K ( x ' , x) ,which carries the wave function (x ) known attime t, to the wave function (x' ) at time, t . Dirac points out that this function Kwas analogous to the quantity in classical mechanics that you would calculate if you tookthe exponential of i / ℏ , multiplied by the Lagrangian L( xɺ, x) imagining that these twopositions x, x’ corresponded t and t . In other words, K ( x ' , x) is analogous to x ' xexp[i L(, x)] ,ℏ x' xK ( x ' , x ) exp[i L(, x )] .ℏ (1)Professor Jehle showed me this, I read it, he explained it to me, and I said, «what does hemean, they are analogous; what does that mean, analogous? What is the use of that?» Hesaid, «you Americans ! You always want to find a use for everything!» I said, that Ithought that Dirac must mean that they were equal. «No», he explained, «he doesn’tmean they are equal.» «Well», I said, «let’s see what happens if we make them equal.»So I simply put them equal, taking the simplest example where the Lagrangian is1Mxɺ 2 V ( x) , but soon found I had to put a constant of proportionality A in, suitably2adjusted. When I substituted exp(i L / ℏ) for K to get x' x ( x' , t ) A exp[i L(, x)] ( x, t )dx ,ℏ 2(2)

and just calculated things out by Taylor series expansion, out came the Schrödingerequation. So, I turned to Professor Jehle, not really understanding, and said, «well, yousee Professor Dirac meant that they were proportional.» Professor Jehle’s eyes werebugging out-he had taken out a little notebook and was rapidly copying it down from theblackboard, and said, «no, no, this is an important discovery. You Americans are alwaystrying to find out how something can be used. That’s a good way to discover things!» So,I thought I was finding out what Dirac meant, but, as a matter of fact, had made thediscovery that what Dirac thought was analogous, was, in fact, equal. I had then, at least,the connection between the Lagrangian and quantum mechanics, but still with wavefunctions and infinitesimal times.It must have been a day or so later when I was lying in bed thinking about thesethings, that I imagined what would happen if I wanted to calculate the wave function at afinite interval later. I would put one of these factors exp(i L / ℏ) in here, and that wouldgive me the wave functions the next moment, t and then I could substitute that backinto (2) to get another factor of exp(i L / ℏ) and give me the wave function the nextmoment, t 2 , and so on and so on. In that way I found myself thinking of a largenumber of integrals, one after the other in sequence. In the integrand was the product ofthe exponentials, which, of course, was the exponential of the sum of terms like L / ℏ .Now, L is the Lagrangian and is like the time interval dt, so that if you took a sum ofsuch terms, that’s exactly like an integral. That’s like Riemann’s formula for the integral Ldt , you just take the value at each point and add them together. We are to take thelimit as 0 , of course. Therefore, the connection between the wave function of oneinstant and the wave function of another instant a finite time later could be obtained by aninfinite number of integrals, (because goes to zero, of course) of exponential (iS / ℏ)where S is the action expression (3),S L( xɺ , x)dt .(3)At last, I had succeeded in representing quantum mechanics directly in terms of theaction S. This led later on to the idea of the amplitude for a path; that for each possibleway that the particle can go from one point to another in space-time, there’s anamplitude. That amplitude is an exponential of i / ℏ times the action for the path.Amplitudes from various paths superpose by addition. This then is another, a third way,of describing quantum mechanics, which looks quite different than that of Schrödinger orHeisenberg, but which is equivalent to them.1.IntroductionThe time evolution of the quantum state in the Schrödinger picture is given by (t ) Uˆ (t , t ' ) (t ' ) ,or3

x (t ) x Uˆ (t , t ' ) (t ' ) dx' x Uˆ (t , t ' ) x' x ' (t ' ) ,in the x representation, where K(x, t; x’, t’) is referred to the propagator (kernel) andgiven byiK ( x, t ; x' , t ' ) x Uˆ (t , t ' ) x' x exp[ Hˆ (t t ' )] x ' .ℏNote that here we assume that the Hamiltonian Ĥ is independent of time t. Then we getthe formx (t ) dx ' K ( x, t ; x' , t ' ) x' (t ' ) .For the free particle, the propagator is described byK ( x, t ; x' , t ' ) mim( x x ' ) 2exp[].2 iℏ(t t ' )2 ℏ( t t ' )(which will be derived later)((Note))Propagator as a transition amplitudeiK ( x, t; x ' , t ' ) x exp[ Hˆ (t t ' )] x 'ℏii x exp( Hˆ t ) exp( Hˆ t ' ) x 'ℏℏ x, t x ' , t 'Here we defineix, t exp( Hˆ t ) x ,ℏix, t x exp( Hˆ t ) .ℏWe note thatiix, t a x exp( Hˆ t ) a exp( Ea t ) x a ,ℏℏwhereHˆ a Ea a .4

((Heisenberg picture))The physical meaning of the ket x, t :The operator in the Heisenberg's picture is given byiixˆ H exp( Hˆ t ) xˆ exp( Hˆ t ) ,ℏℏiiixˆ H x, t exp( Hˆ t ) xˆ exp( Hˆ t ) exp( Hˆ t ) xℏℏℏi ˆ exp( Ht ) xˆ xℏi exp( Hˆ t ) x xℏi x exp( Hˆ t ) x x x, tℏThis means that x, t is the eigenket of the Heisenberg operator x̂H with the eigenvaluex.We note thatiℏ S (t ) exp( Hˆ t ) H .Then we getix S (t ) x exp( Hˆ t ) H x, t H .ℏThis implies thatx, t x, tH,x x S.where S means Schrödinger picture and H means Heisenberg picture.2.PropagatorWe are now ready to evaluate the transition amplitude for a finite time intervalK ( x, t ; x ' , t ' ) x, t x ' , t 'iii x exp[ Hˆ t )] exp[ Hˆ t )]. exp[ Hˆ t )] x 'ℏℏℏ5

where t t t'(in the limit of N )NtDtt0twhere t0 t' in this figure.We next insert complete sets of position states (closure relation)K ( x, t ; x' , t ' ) x, t x ' , t 'i dx1 dx2 . dx N 2 dx N 1 x exp( Hˆ t ) x N 1ℏii x N 1 exp( Hˆ t ) x N 2 . x3 exp( Hˆ t ) x2ℏℏii x2 exp( Hˆ t ) x1 x1 exp( Hˆ t ) x 'ℏℏThis expression says that the amplitude is the integral of the amplitude of all N-leggedpaths.((Note))( x ' , t ' ), ( x1 , t1 ), ( x2 , t 2 ), ( x3 , t 3 ), ( x4 , t 4 ),.( x N 4 , t N 4 ), ( x N 3 , t N 3 ), ( x N 2 , t N 2 ), ( x N 1 , t N 1 ), ( x, t )witht' t 0 t1 t2 t3 t 4 . t N 3 t N 2 t N 1 t tN6

We definex ' x0 ,x xN ,t ' t0 ,t tN .We need to calculate the propagator for one sub-intervalxi exp[ i ˆH t ) xi 1 ,ℏwhere i 1, 2, , N, andpˆ 2Hˆ V ( xˆ ) ,2mThen we haveiixi exp( Hˆ t ) xi 1 dpi xi pi pi exp( tHˆ ) xi 1ℏℏi dpi xi pi pi 1̂ tHˆ xi 1 O(( t ) 2 )ℏipˆ 2 dpi xi pi pi 1̂ t ( V ( xˆ )) xi 1 O (( t )2 )ℏ2mipˆ 2i dpi xi pi [ pi 1̂ t ( ) xi 1 pi tV ( xˆ ) xi 1 ] O (( t )2 )ℏ2mℏ2ipi dpi xi pi [ pi 1̂ t ( i ) xi 1 pi tV ( xi 1 ) xi 1 ] O (( t )2 )ℏ2mℏ2ip dpi xi pi pi 1̂ t[ i V ( xi 1 )] xi 1 O(( t ) 2 )ℏ2m7

where pi (i 1, 2, 3, , N), or2iipˆxi exp( Hˆ t ) xi 1 dpi xi pi pi xi 1 [1 t ( i V ( xi 1 ))]ℏℏ2m1ii dpi exp[ pi ( xi xi 1 )[1 tE ( pi , xi 1 )] ℏℏ2 ℏii1 dpi exp[ pi ( xi xi 1 )] exp[ tE ( pi , xi 1 )] 2 ℏℏℏ1i( x xi 1 ) dpi exp[ { pi i t E ( pi , xi 1 ) t}] 2 ℏℏ t1i( x xi 1 ) dpi exp[ { pi i E ( pi , xi 1 )} t ] 2 ℏℏ twhereE( pi , x i 1 ) pi 2 V(xi 1 ) .2mThen we haveK ( x, t ; x ' , t ' ) x, t x ' , t 'dp1 dp2dpdp. N 1 N N 2 ℏ 2 ℏ2 ℏ 2 ℏ2Ni( x xi 1 )p exp[ { pi i ( i V ( xi 1 ))} t ]ℏ i 1 t2m lim dx1 dx2 . dx N 1 We note thatdpi( xi xi 1 ) pidpipiii 2 ℏ exp[ ℏ { pi t 2m } t} 2 ℏ exp[ ℏ { pi ( xi xi 1 ) i 2mℏ t}mim t xi xi 1 2 exp[() ]2 ℏi t2ℏ t22((Mathematica))8

Clear@"Global "D; f1 ExpB p x 2π ——Integrate@f1, 8 p, , D êêSimplify@ , 8— 0, m 0, Im @ tD 0 D &1p22m— t F;m x22 t —2π t —mThen we haveK ( x, t ; x ' , t ' ) lim dx1 dx2 . dx N 1 (N m N /2)2 ℏi tNim x xi 1 2 exp[ t { ( i) V ( xi 1 )}]ℏ i 1 2 tNotice that as N and therefore t 0 , the argument of the exponent becomes thestandard definition of a Riemann integralNim x xi 1 2ilim t { ( i) V ( xi 1 )} dtL( x, xɺ ) ,N ℏ tℏ t'i 1 2 t 0twhere L is the Lagrangian (which is described by the difference between the kineticenergy and the potential energy)9

xOL( x, xɺ ) t1m( xɺ ) 2 V ( x) .2It is convenient to express the remaining infinite number of position integrals using theshorthand notation D[ x(t )] lim dx dx . dxN 12N 1(m N/2) .2 ℏi tThus we haveiK ( x, t ; x ' , t ' ) x, t x' , t ' D[ x (t )] exp{ S [ x(t )]} ,ℏwheretS [ x (t )] dtL( x , xɺ ) .t'The unit of S is [erg sec].10

When two points at (ti, xi) and (tf, xf) are fixed as shown the figure below, forconvenience, we usetfS [ x (t )] dtL( x, xɺ ) .tixx2x1tt1t2This expression is known as Feynman’s path integral (configuration space path integral).S [ x (t )] is the value of the action evaluated for a particular path taken by the particle. Ifone wants to know the quantum mechanical amplitude for a point particle at x’, at time t’to reach a position x, at time t, one integrates over all possible paths connecting the pointswith a weight factor given by the classical action for each path. This formulation iscompletely equivalent to the usual formulation of quantum mechanics.The expression for K ( x, t; x ' , t ' ) x, t x ' , t ' may be written, in some loose sense, asx N x, t N t x0 x' , t 0 t ' exp[( all ) path exp(iS ( N ,0)]ℏiS path 1ℏ) exp(11iS path 2ℏ) exp(iS path nℏ)

where the sum is to be taken over an innumerably infinite sets of paths.xxcl Ht Lt(a)Classical caseSuppose that ℏ 0 (classical case), the weight factor exp[iS / ℏ] oscillates veryviolently. So there is a tendency for cancellation among various contribution fromneighboring paths. The classical path (in the limit of ℏ 0 ) is the path of least action,for which the action is an extremum. The constructive interference occurs in a verynarrow strip containing the classical path. This is nothing but the derivation of EulerLagrange equation from the classical action. Thus the classical trajectory dominates thepath integral in the small ħ limit.In the classical approximation ( S ℏ )x N x, t N t x0 x' , t 0 t ' "smooth function" exp(iS cl).ℏ(1)But at an atomic level, S may be compared with ħ, and then all trajectory must be addedin x N x, t N t x0 x ' , t 0 t ' in detail. No particular trajectory is of overwhelmingimportance, and of course Eq.(1) is not necessarily a good approximation.12

Quantum case.What about the case for the finite value of S / ℏ (corresponding to the quantum case)?The phase exp[iS / ℏ] does not vary very much as we deviate slightly from the classicalpath. As a result, as long as we stay near the classical path, constructive interferencebetween neighboring paths is possible. The path integral is an infinite-slit experiment.Because one cannot specify which path the particle choose, even when one knows whatthe initial and final positions are. The trajectory can deviate from the classical trajectoryif the difference in the action is roughly within ħ.(b)((REFERENCE))R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (extended edition)Dover 2005.((Note))We can use the Baker-Campbell-Haudorff theorem for the derivation of the Feynmanpath integral. We have a HamiltonianHˆ Tˆ Vˆwhere Tˆ is the kinetic energy and Vˆ ( xˆ ) is the potential energy. We consideriiexp( Hˆ t ) exp[ (Tˆ Vˆ ) t ]ℏℏˆ exp( P Qˆ )whereiPˆ Tˆ t ,ℏiQˆ Vˆ tℏWe use the Baker-Campbell-Hausdorff theorem111exp( Pˆ ) exp(Qˆ ) exp{Pˆ Qˆ [ Pˆ , Qˆ ] [ Pˆ ,[ Pˆ , Qˆ ]] [Qˆ ,[ Pˆ , Qˆ ]] .}21212withi[ Pˆ , Qˆ ] ( )2 ( t )2 [Tˆ , Vˆ ]ℏi[ Pˆ ,[ Pˆ , Qˆ ]] ( )3 ( t )3[Tˆ ,[Tˆ , Vˆ ]]ℏ13

i[Qˆ ,[ Pˆ , Qˆ ]] ( )3 ( t )3[Vˆ ,[Tˆ , Vˆ ]]ℏIn the limit of t 0 , we getiexp( Hˆ t ) exp( Pˆ Qˆ )ℏ exp( Pˆ ) exp(Qˆ )ii exp( Tˆ t ) exp( Vˆ t )ℏℏUsing this, we get the matrix elementiiix j exp( Hˆ t ) x j 1 x j exp( Tˆ t ) exp( Vˆ t ) x j 1ℏℏℏi t 2i x j exp( pˆ ) x j 1 exp( V ( x j 1 ) t )2mℏℏor1/22i ˆi m ( x j 1 x j ) m x j exp( H t ) x j 1 V ( x j 1 )] t )] exp{ [ℏℏ 2( t )2 2 iℏ t 3.Free particle propagatorIn this case, there is no potential energy.K ( x, t; x' , t ' ) lim dx1 dx2 . dx N 1 (N Nm N/2im x x) exp[ t { ( i i 1 ) 2}] ,2 ℏi tℏ i 1 2 torK(x,t;x' ,t' ) lim dx1 dx2 . dx N 1N (m m) N / 2 exp[{(x1 x' )2 (x 2 x1 )2 (x 3 x 2 )2 . (x x N 1 )2}]2 ℏi t2 ℏi tWe need to calculate the integrals,f1 ( m 2/ 2 m) dx1 exp[{( x1 x' ) 2 ( x 2 x1 ) 2 ]2 ℏi t2ℏi t m 1/ 2 m) exp[( x 2 x' ) 2 ]ℏi t4ℏi t4 1(14

g1 f 1 (m 1/ 2 m) exp[( x3 x2 ) 2 ]2 ℏi t2ℏi t f2 g dx12 g2 f2 (1m 1/ 2 m() exp[(x x' ) 2 ]6 ℏi t6ℏi t 3m 1 /2 m) exp[(x4 x3 )2 ]2 ℏi t2ℏi t f3 g2 dx 3 1m 1/ 2 m() exp[(x 4 x' )2 ]8 ℏi t8ℏi t.m m(x x' )21/ 2K(x,t;x' ,t' ) lim () exp[],N 2 ℏNi t2ℏiN torK(x,t;x' ,t' ) (m m(x x' )21/ 2) exp[],2 ℏi(t t' )2ℏi(t t' )where we uset t' N t in the last part.((Mathematica))15

Free particle propagator;e DtClear@"Global "D;exp : exp ê. 8Complex@re , im DComplex@re , im D ;h1 IHx1 x 'L Hx2 x1L M êê Expand;222— ε 22πf0@x1 D : mExpB m2—εh1F;f1 Integrate@f0@x1D, 8x1, , D êêmSimplifyB , Im BF 0F &ε—m Hx2 x′ L24ε—2g1 f1 mε—π1— ε 22πmExpB m2—εHx3 x2L2 F êê Simplify;f2 ‡ g1 x2 êê SimplifyB , Im B m ε—F 0F &m Hx3 x′ L26ε—ε—6πmg2 f21— ε 22πmExpB m2—εHx4 x3L2 F êê Simplify;f3 ‡ g2 x3 êê SimplifyB , Im B m ε—m Hx4 x′ L28ε—2g3 f3 mε—2π1— ε 22πmExpB m2—εHx5 x4L2 F êê Simplify;f4 ‡ g3 x4 êê SimplifyB , Im B m ε—m Hx5 x′ L210 ε —10 πF 0F &ε—m16F 0F &

4.Gaussian path integralThe simplest path integral corresponds to the vase where the dynamical variablesappear at the most up to quadratic order in the Lagrangian (the free particle, simpleharmonics are examples of such systems). Then the probability amplitude associated withthe transition from the points ( xi , t i ) to ( x f , t f ) is the sum over all paths with the actionas a phase angle, namely,iK ( x f , t f ; xi , t i ) exp[ S cl ]F (t f , ti ) ,ℏwhere S cl is the classical action associated with each path,tfS cl dtL( xcl , xɺcl , t ) ,tiwith the Lagrangian L( x, xɺ, t ) described by the Gaussian form,L( x, xɺ , t ) a(t ) xɺ 2 b(t ) xɺx c(t ) x 2 d (t ) xɺ e(t ) x f (t )If the Lagrangian has no explicit time dependence, then we getF (t f , ti ) F (t f t i ) .For simplicity, we use this theorem without proof.iK ( x f , t f ; xi , ti ) exp[ Scl ]F (t f ti ) .ℏ((Proof)) R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals.Let xcl (t ) be the classical path between the specified end points. This is the path which isan extremum for the action S. We can represent x(t ) in terms of xcl (t ) and a newfunction y (t ) ;x(t ) xcl (t ) y (t ) ,where y (ti ) y (t f ) 0 . At each t, the variables x(t ) and y (t ) differ by theconstant xcl (t ) (Of course, this is a different constant for each value of t). Thus, clearly,dxi dyi ,17

for each specific point ti in the subdivision of time. In general, we may say thatDx (t ) Dy (t ) .The integral for the action can be written astfS [t f , ti ] L[ xɺ (t ), x (t )), t ]dt ,tiwithL( xɺ, x, t ) a (t ) xɺ 2 b(t ) xɺx c(t ) x 2 d (t ) xɺ e(t ) x f (t ) .We expand L( xɺ, x, t ) in a Taylor expansion around xcl , xɺcl . This series terminates afterthe second term because of the Gaussian form of Lagrangian. ThenL( xɺ, x, t ) L( xɺcl , xcl , t ) L L1 2 L 2 2L 2L 2ɺɺ x y xɺ

(Date: January 31, 2021) ((Overview)) . What is the path integral in quantum mechanics? The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mecha

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