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The Quantum Potential And''causal''trajectories For REFERENCE IC/88/161 \ ^' I 6 AUG INTERNATIONAL CENTRE FOR / Co THEORETICAL PHYSICS THE QUANTUM POTENTIAL AND "CAUSAL" TRAJECTORIES FOR STATIONARY STATES AND FOR COHERENT STATES A.O. Barut and M. Bo2ic INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/88/161 I. Introduction International Atomic Energy Agency During the last decade Bohm's causal trajectories based on the and concept of the so called "quantum potential" /I/ have been evaluated United Nations Educational Scientific and Cultural Organization in a number of cases, the double slit /2,3/, tunnelling through the INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS barrier /4,5/, the Stern-Gerlach magnet /6/, neutron interferometer /8,9/, EPRB experiment /10/. In all these works the trajectories have been evaluated numerically for the time dependent wave packet of the Schrodinger equation. THE QUANTUM POTENTIAL AND "CAUSAL" TRAJECTORIES FOR STATIONARY STATES AND FOR COHERENT STATES * In the present paper we study explicitly the quantum potential and the associated trajectories for stationary states and for coherent states. We show how the quantum potential arises from the classical A.O. Barut ** and M. totii *** action. The classical action is a sum of two parts S ,= S^ + Sz> one International Centre for Theoretical Physics, Trieste, Italy. part Sj becomes the quantum action, the other part S2 is shifted to the quantum potential. We also show that the two definitions of causal ABSTRACT trajectories do not coincide for stationary states. We show for stationary states in a central potential that the quantum The content of our paper is as follows. Section II is devoted to action S is only a part of the classical action W and derive an expression for the "quantum potential" IL in terms of the other part. The association a comparative study of Schrodinger discovery of the wave equation and of momenta of some "particles" in the causal interpretation of quantum Bohm's "causal" interpretation. In Section III we analyze critically mechanics by p = VS and by dp'/it = - y(V+U_) gives for stationary states very different orbits which have no relation to classical orbits but express Bohm's method. In Sections IV and V we determine the associated tra- some flow properties of the quantum mechanical current. For coherent states, jectories in the stationary states for a central potential(hydrogen on the other hand, p and p1 as well as the quantum mechanical average p atom, spherical oscillator) for the rectangular potential barrier, and and classical momenta, all four, lead to essentially the same trajectories except for different integration constants. The spinning particle is also for tunnelling through a barrier. Section VI is devoted to stationary considered. states of the spinning neutral particle in the magnetic field. Currents and flows are treated in Section VII and finally.we discuss the KIRAMARE - TRIESTE coherent states in Section VIII. July 1988 II. Bohm's interpretation versus Schrb'dinqer discovery In introducing the concept of quantum potential /I/ Bohm treated * To be submitted for publication. ** Permanent address: Department of Physics, University of Colorado, CB 390, Schrodinger equation as a postulate. The sequence of steps leading to Boulder, CO 803O9, USA. causal trajectories, is as we are going to show, approximately *** Permanent address: Institute of Physics, P.O. Box 57, Beograd, Yugoslavia. -2- r the inverse of the Schrb'dinger's original sequence of steps or /11,12/, one from classical trajectories to wave equation, the other I 1/2 ]grad w| = [2m(E - V)J ' (9) from wave equation to some new associated trajectories which have noth- The relation ing to do with the original classical trajectories. p - mv = grad W = grad w (IT) SchrSdinger's discovery of wave mechanics represents one specific is satisfied in the above procedure. synthesis of mechanics, wave theory and Planck hypothesis. Basic mechan- In this method,the relation between the mechanical and the wave ical concepts are the classical Hamilton and Lagrange functions for motion is expressed by assuming the action W to be proportional to the a particle phase of the wave H = (p2/2m) + V(r) = T + V (1) H?,t) = A exp(iW/K) = A exp(-iEt/K + iw/K) . (11) L = <p"2/2m) - Vlt) = T - V . (2) Further to the motion of the particle,one associates the motion of And the basic wave concept is the wave function 4>(r,t) required to surfaces of constant W and therefore the propagation of the wave. If satisfy the wave (d'Alembert) equation W is the value of W on a particular surface for a given t, and AI|J - Ji'/u = 0 r (3) dW * Edt, then the infinitesimal normal 1/2 where u is the wave (phase) velocity. The synthesis of mechanical and dn = dWQ/[2m(E-v)] (12) wave notions are realized now by deriving a relationship between the determines the velocity of motion of any surface at any one of its phase velocity and the kinetic and total energies of the particle points by 1/2 1/2 1/2 u = dn/dt = E/[2m(E-V)] (13) u = E/{2m{E-VM = E/(2mT) (4) which is the desired eq. (4). in two different ways. The second method uses /13/ the analogy between the principle of The first method uses the Hamilton-Jacobi equation for the least action action W B _ 1/1 i :-> 6/[2m(E-V) ] ' 'ds = 0 (14) W = / (T - V)dt (5) A to and Fermat's principle in optics which taken as a function of the upper limit t and of the final value B 5/ds/u = O (15) of the coordinates x,y,z satisfies the Hamilton-Jacobi partial diffe- A This analogy leads to rential equation 1/2 3W/3t + (V2W/2m) + V(r,t) = 0 • (6) u = C/ [2m(E-V)] (16) Then one identifies the group velocity of the wave of frequency v If the potential does not depend on time, Eq.(6) can be solved by given by 1/v = d(v/u)/d\J (17) setting W(r,t) = - Et + w(r> (7) with the particle velocity v = p/m. The result is the value of the constant C, This substitution leads to , 2 C = E d w/2i| + V(r) = E (8) -k- -3- i.e. the relation (4) again. tion can be written as , Schrodinger then looked for those solutions of the wave equation i|j(r,t) = R(r,t) exp(is(r,t)/-fi) (25) (3) whose time dependence is of the form Eq. (24) is equivalent to two partial differential equations ijj (r ,t) = <|>(r) exp(-iEt/K) , (18) for the real functions R(r,t) and S(r,t) where the constant K must have the physical dimension of action (energy 3R(r,t)/3t = - (1/2m)[RV2S(f,t) + 2VRVS(r,t)] (26) time). Now since the frequency of the wave is obviously 3S(r,t)/at = - [(VS(r,t))2/2m + V(r) - (ft2/2m) V^t^t) /R(r,t)] (2 7) \> = E/2^K (19) These equations were used earlier by de Broglie /15/. Madelung gave Schrodinger "could not resist the temptation" /12/ to use Planck's re- them a hydrodynamical interpretation /16/. They were also used by Fenyes lation in a Stochastic interpretation /17/. The corresponding set of equations E = h-V (20) for the functions R(r) and s(^) associated with stationary i.e. to put K equal to h/2 -n - Ti solution By combining the latter relation with (4) one finds for the wave- .|i(r,t) = E(r) exp(-iEt/fi) exp (is (r) /M) (28) length read X = h/[2m(E-V)]1/2 = h/p (21) R(r)V2s(r) + 2VR(r)7s(r) = 0 (29a) 2 2 2 which is the de Broglie relation originally derived relativisticaiiy E = (Vs(?)) /2m + V(r) - (« /2m) V R(?)/R(f) (29b) /14/, Schrodinger's derivation is nonrelativistic. Eq. (29b) and the corresponding equation (27) look like the Hamilton- By introducing Eqs, (4), (18) and (20) into d'Alembert -Jacobi equation for the functions s(r*) and S(r*,t), respectively, with Eq. (3) Schrodinger found the equation for amplitude t>(r) 2 Z + .2 an additional term -(fl /2m) V R( r)/R(r). Eohm gave to this term the V 0(r) + (Ti /2m) (E-V) • $ (r) = 0 • (22) name "quantum potential" In order to get the equation for a function iji (r,t) whose time 2 2 UQ = - <fi /2Tn)V R(?)/R(r) (30) dependence is not restricted to the dependence of the form exp(-1Et/tl). With the action S(r,t) which satisfies (26) and (27) Bohm associates Schrodinger later in the fourth paper of Ref. 11 used the property of now a "momentum" by the stationary solution p = p(r,t) = grad S(r,t) (31) EiMr,t> = ifi 3iMr,t)/3t (23) to eliminate the term E <t>("r) from (22) and arrived at and in this way obtains the causal interpretation of quantum mechanics in which particles move along one of the trajectories determined by -<Ti2/2m) (24) integrating the equation m dT = p^-t' <32> To each initial coordinate corresponds one trajectory. As stated at the beginning Bohm took the latter equation as a pos- We see that Eq. (24) is the last in Schrodinger's and the tulate and applied the following reasoning. Since every complex func- first one in Bohm's approach. The relation (31) is the last one in -5- -6- Bohm's approach whereas the corresponding relation (10) is one of the of ijj (r,t) is equal to the action.
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