The path integral approach in the frame work of causal interpretation

Transcription

1 Annales de la Fondation Louis de Broglie, Volue 28 no 1, The path integral approach in the frae work of causal interpretation M. Abolhasani 1,2 and M. Golshani 1,2 1 Institute for Studies in Theoretical Physics and Matheatics (IPM), P.O.Box , Tehran, Iran 2 Departent of Physics, Sharif University of Technology, P.O.Box , Tehran, Iran. E-ail: ajida@theory.ip.ac.ir ABSTRACT. We try to clarify the relation of Feynan paths -which appear in the Feynan path integral foralis- with de Broglie-Boh paths. To accoplish this, we introduce a Bohian path integral and use it to obtain the propagator of a free particle and a general syste. 1 Introduction In the coon interpretation of quantu echanics, there is no picture of paths. In spite of this, the Feynan approach to quantu echanics[1], uses the concept of path as a atheatical tool to calculate a propagator for the wave function: K(x, t; x o,t o )= D[x] e i S[x] (1) where S is the classical action and the integral is taken over all possible paths between the two points (x, t) and (x o,t o ). One can show that (1) satisfies the Schrödinger equation and can be used as a substitute for it. This eans that, if we have an arbitrary initial wave function, we can obtain it at any other tie by the use of (1). On the other hand, in the causal interpretation of quantu echanics[2,3], it is claied that particles ove on the paths which are given by x = I( ψ ψ )= S (2)

2 2 M. Abolhasani and M. Golshani where S is the phase of the wave function (ψ = Re i S ). In this paper, the authors want to clarify the relation between Feynan paths and the de Broglie-Boh paths[4]. First, we obtain a path integral forulation of quantu echanics in the fraework of the causal interpretation. We call it Bohian path integral (BPI). Then, we introduce Fourier-Boh paths as a tool to calculate BPI. In this way, we are naturally lead to Feynan path integrals (FPI). Our discussion could be considered as an approach for obtaining FPI fro the Schrödinger equation. 2 The Bohian path integral To obtain BPI we consider ψ at two points of a Bohian path. For siplicity, we first consider two points with infinitesall distance d x. Then, we have ψ( x, t) ψ( x + d x, t + dt) =ψ( x, t)+ dt + ψ( x, t t) dx As a result of our assuption that these two points are on a Bohian path and using (2), we have dx S = dt. With the use of this expression for d x and the Schrödinger equation for ψ t we obtain: { [ ψ( x + d x, t + dt) = 1+ i ( S) ] 2 2 (Q + V ) } dt 2 S 2 dt ψ( x, t) 2 R where Q = 2 2 R is the so-called quantu potential (R = ψ ). Now, we can write the expression in {} as an exponential. Then, for the two points (x, t) and (x o,t o ), with a finite distance on a Bohian path, we have ψ( x, t) =exp { i x,t } x,t 2 S L q dt x o,t o x o,t o 2 dt ψ( x o,t o ) (3) where L q = ( S) 2 2 (Q+V ) and the integrals are taken over the Bohian path which initiates at ( x o,t o ) and ends at ( x, t). In fact, we have integrated the Schrödinger equation on a Bohian path. If we substitute Re i S for ψ, in the Schrödinger equation, we obtain two equations fro

3 The path integral approach in the fraework of causal... 3 the real and iagenery parts of the resulting equation. The real part is a quantu Hailton-Jacobi equation which includes Q. By integrating this equation on the Bohian path we obtain the first exponential in (3). The iaginary part is a continuity equation for R 2. By integrating this equation on the Bohian path we obtain the second exponential in (3). Now, the first ter gives the phase of wave function at ( x, t) as a result of its phase at ( x o,t o ). As one ay expect, here the classical action of FPI is replaced by the quantu action (which includes Q). On the other hand, since there is no interference between the paths, in order to have an evolution for R = ψ, a real exponent is needed. The second ter is a proper ter for this purpose. As one ay expect the divergence of paths ( S ) appears in this ter. Note that, BPI can be used to obtain ψ( x, t) fro ψ( x o,t o ), instead of the Schrödinger equation as is the case for FPI. If we have ψ( x, t o ) we can obtain ψ( x, t o + dt) by the use of (3) and by repeatation of this process for other dt s we obtain ψ( x, t o + t), where t is finite. That is, we use the word BPI for the relation (3), although it is not a real path integral. Because, the ters in the curly braces that are used to propagate the wave function depend explicitly on the values of the wave function in the region where it is being propagated. Nevertheless, we prefer to use the word BPI for (3). Note that we use the terinology BPI for the relation (3), although it is not a path integral in the usual sense. Because, the ters in the curly braces which are supposedly propagateing the wave function depend explicitly on the value of the wave function in the region where it is being propagated. Nevertheless, we prefer to use the terinology BPI for (3). In the following section we show that one can obtain ψ( x, t) analytically for a free wave packet. We also obtain K(x, t; x o,t o ) for this case. 3 the propagator of a free wave packet In this section we use BPI to obtain the tie evolution of a free wave packet. For this special case, one can do it analytically. Suppose ψ o (x) is a wave packet in one diension. It can be written as ψ o (x) = ϕ(k)e ikx dk (4)

4 4 M. Abolhasani and M. Golshani We consider the evolution of this packet as a result of the evolution of its Fourier coponents e ikx. For e ikx states BPI becoes very siple. In fact, for these states we have 2 S =0, and Q =0 This eans that Bohian paths for the plane waves are straight lines with the gradient v = k in space-tie. Furtherore, the quantu of action for BPI changes to the classical action. In this way, our path integral becoes ore siilar to FPI. We nae this paths, the Fourier- Boh paths (FBP). Now, we try to obtain the wave function at (x, t). For each coponent e ikx only one FBP has contribution at (x, t), and this initiates fro a special point at (x o,t o ). If teporarily we consider k to be discrete, one can write ψ(x, t) as a cobination of all such contributions: ψ(x, t) = i e i S i ϕ(ki )e ikixo i x t k i t o x oi Fig. 1 Several Fourier Coponents of the Bohian paths end at x where x oi = x ki (t t o) (Fig.1), and the suation is taken over all k i s. On other hand, S i (action on the ith path) is given by S i = t t o p 2 2 dt = t t o k 2 i 2 2 dt = k2 i 2 2 (t t o)

5 The path integral approach in the fraework of causal... 5 If we replace the suation by integration and replace x o by x k (t t o) we obtain: ψ(x, t) = dke i k2 2 (t to) ϕ(k)e ikx (5) which is the sae thing as one would expect for ψ(x, t) fro the Schrödinger equation. If we want to obtain K(x, t; x o,t o ) for this case, we ust consider ψ o (x) = δ(x x o ). For this state we have ϕ(k) =e ikxo /2π. By substituting ϕ(k) in (5) one obtains { i(x K(x, t; x o,t o )= 2πi (t t o ) exp xo ) 2 } 2 (t t o ) t t o x x o Fig. 2 Continuous lines show a typical Feynan path ; broken lines indicate Bohian paths Note that, here instead of having one Bohian path at each points x o, we have infinite FBP (each path for a definite k). It is the cost that we pay for the eliination of 2 S and Q ters in BPI. In other words, for obtaining inforation about the evolution of R = ψ with the help of phase, we ust ake use of the infinite FBP at each point x, instead of one Bohian path. This job is done by choosing ψ o (x) as (4), and by taking its evolution as the evolution of its coponents e ikx.

6 6 M. Abolhasani and M. Golshani Are Feynan paths different fro our FBP? We don t think so. Note that, in both Feynan and Fourier-Boh path integrals, fro each point of the space at tie t o infinite paths originate and these have all possible gradients. Furtherore, to each point of the space at tie t end infinite paths with all possible gradients. On other hand, at each point of the space, at a tie t 1 (t o t 1 t), we have infinite paths with all possible gradients. In fact, since the infinity of Feynan paths and the infinity of FBP are of the sae order, we could ake each Feynan paths fro the cobination of several FBP (Fig.2). In other words, the infinite FBP s which appear in the Fourier-Boh path integral are equivalent with the infinite Feynan paths which appear in the FPI. The only difference is that, in the BPI each path which initiates at x o, with the gradient v i = ki, is ultiplied by one coponent of ψ o(x o ), i.e. ϕ(k i )e ikix. But, in FPI the sae path is ultiplied by all coponents of ψ o (x o ), i.e. i ϕ(k i)e ikix. Now, we know that the two ethods give the sae result for the evolution of ψ. Thus, one ust be able to show that they are equivalent. The solution is given by the infinity of the nuber of paths. For each path which initiates with the gradient v i = ki and an action S, there are other paths with the sae action but with different initial gradients. This eans that in the FBP integral we have ters like...e i S ϕ(k1 )e ik1xo + e i S ϕ(k2 )e ik2xo e i S ϕ(kn )e iknxo +... n =...+ e i S ϕ(k i )e ikixo... which is what appears in the FPI. 4 the propagator of a general syste Here, we use BPI to obtain propagator for wave function of a syste with potential V (x). In this case, we can not consider the evolution of the wave packet (ψ) as a result of the evolution of its Fourier coponents (e ikx ) for a finite period of tie. But for a infinitesall period of tie, it is correct yet. Then, fro (3) and (4) for a infinitesall period of tie (t t o ), we have: i ψ(x, t) = { { i 2 k 2 } } exp 2 V (x)] (t t o ) ϕ(k)e ikx dk (6)

7 The path integral approach in the fraework of causal... 7 where for Fourier coponents we take 2 S = 0, and Q =0.Ifwewant to obtain K(x, t; x o,t o ) for this case, we ust consider ψ o (x) =δ(x x o ) that is ϕ(k) =e ikxo /2π. By substituting in (6) one obtains: K(x, t; x o,t o )= 2πi (t t o ) exp { i [ (x xo ) 2 2(t t o ) V ( x + x o )(t t o )] } (7) 2 This is propagator of the syste for the infinitesall period t t o. If we want to obtain propagator for a finite period of tie we ust use Feynan path integral ethod. Here too, we begin with Bohian paths, but ultiately we coe to Feynan paths as a tools. If we discuss several-particle systes instead of one-particle systes, the wave function exists in configuration space. In this case, in the fraework of de Broglie-Boh theory of quantu echanics, the question arises as to whether such wave could be a real wave. But we know that the de Broglie-Boh paths in both versions of this theory are considered as real paths. Note, that the wave function of a several-particle syste in the fraework of the standard interpretation of quantu echanics exists in the configuration space. But, Feynan paths (which are not consider as real paths) can be defind in the real space. Then, cosidering a several-particle syste to see if it has any new result concerning the relation of de Broglie-Boh paths and Feynan paths has any new point. 5 Conclusion We introduced Bohian path integral and used it to obtain the propagator of a free particle. Then, we showed to what extent the FBP are like the Feynan paths and we gave a heuristic arguent to show the equvalnce of the. In fact, we introduced FBP as a atheatical tool as is the case for Feynan paths. Our discussion is an attept to throw light on the proble of why Feynan s technique works, using Boh s causal interpretation. Besides, the clarification of the relation between Bohian trajectories and Feynan s paths ay be helpful in the calculation of the propagator for the Schrödinger equation. It ay be useful to add that the propagator for soe siple systes are calculated recently[5,6,7].

8 8 M. Abolhasani and M. Golshani References [1] R. P. Feynan, Rew. Mod. Phys. 20, 367 (1948). [2] L. de Broglie, J.Phys. (Paris) 6, 225 (1927). [3] D. Boh, Phys. Rev. 85, , (1952). [4] P. Holland, The Quantu Theory of otion. Cabridge, Cabridge University press (1993). [5] T. O. de Carvalho, Phys. Rev. A47, 2562 (1993). [6] M. A. M. de Aguiar, Phys. Rev. A48, 2567 (1993). [7] R. D. Nevels etal, Phys. Rev. A48, 3445 (1993). (Manuscrit reçu le 21 octobre 2000)