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 = 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 = - 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. This means that one neglects only first in the Schrodinger one. Relations (11) and (28) are the corresponding the relation (11). ones.in the two approaches. Bohm's approach does not contain the In the interpretation of Bohm, finally, the trajectories exist, relation (4) whereas in the Schrodinger one it is the relation which couples but the set of trajectories is not a subset of the classical set because elements of mechanical and wave theory. The Hamilton-Jacobi equation for the action is not a classical action i.e. neither the classical the function W is at the beginning of Schrodinger's procedure whereas Hamilton-Jacobi equation is satisfied nor the energy relation (1), In "quantum Hamilton-Jacobi equation" for the function S is near the end fact, we shall show that these new "trajectories" are entirely different of Bohm's procedure. from the classical trajectories. The fact that S does not satisfy the classical H-J equation but the Bohm takes the phase of the wave function as the analogue of the quantum one indicates that the Schrodinger equation is not consistent with classical action (we will call it "quantum action") and keeps the re- all the relations used in its "derivation". That might he the hidden reason lation (10). But now there is the new relation (31) between "momentum for the existence of different interpretations of quantum mechanics of the particle" and "quantum action". In Bohm's theory particles are real, whose common feature is that they all contain the Schrbdinger. equation itself. but since the relation (1) is not valid the expression (4) for The statistical interpretation /18/ neglects the whole classical phase velocity is not valid either. background physical picture which led Schrbdinger to his discovery. This interpretation denies the existence of waves as well as the exis- III. Remarks on Bohm's Interpretation tence Of trajectories. Only Eq, (24) remains in which the absolute Already in 1952 Halpern /20/ criticized the contention of Bohm value square of as an overall phase of ty has no physical interpretation. equation: "To obtain the solution of a mechanical problem it is neces- The old quantum theory /19/ is closer to Schrddinger's wave sary to have an action S which depends on the coordinates of the system, picture than the statistical interpretation. It assumes that classical the physical parameters entering into the expression for kinetic trajectories exist. The Sommerfeld conditions of quantizations reduce and potential energy and f nonadditive integration constants. Bohm has the set of all trajectories to a subset in which these conditions are failed to show that such a function can be found... Until a way to satisfied. Since the Schrodinger wave equation also reduces the set of find these integration constants has been devised, the similarity with all allowed energies of the system to a subset, one may say that in the Hamilton-Jacobi theory is purely extraneous, and one cannot talk this theory the classical Hamiltonian (1), the relation between momen- about a mechanical interpretation of the wave equation". tum and classical action (10), the expression for phase velocity (4) Bohm replied /21/ that Hamilton-Jacobi theory was being used only and Schrodinger equation are the basic equations. Waves do not destroy for the purpose of indicating in a simple way how one might arrive classical trajectories, they only reduce the whole set of trajectories at a causal interpretation of the quantum theory, while the theory it- to a subset. In such an approach one does not require that the phase self was to be based directly on the equations of motion -7- -8- md2r7dt2= - V{U (r) + V[r)} (33) The first relation results R(x) = C/(ds/dx)1/2 (39) where v^i) is the classical potential, and Un(r) is the "quantum po- 2 3 3 2 2 2 UQ(x) = -(fi /4m) [(d s/dx ) / (ds/dx) - 3(d s/dx )/2(ds/ax) ] (40) tential". Guided by the Hamilton-Jacob! theory, one guesses tentatively The substitution of the latter expression into the "quantum Hamilton- writes Bohtn, that 1f the momentum were equal to p = v'S(r), this func- Jacobi equation11 (jives 2 2 2 tion would satisfy the equations of motion. Now, Bohm takes p to be E = s' /2m + V(x) +• (fl /4m) [s""/s ' - (3/2) But since S satisfies (27), Bohm finds Furthemore, by taking covariant derivative (36) instead of dp/dt dp/dt = - V(v + U (35) one has replaced the ordinary differential equation for p(t), of Newton, Q by a partial differential equation for p(x,t). Thus the assertion and concludes that Eq. (33) follows. Bohm further concludes in Bohm's reply /21/ that Newton's form of mechanics is the methodo- that this verifies that a particle travelling with a velocity^/ = 7S/B logical basis of his interpretation is not valid. will satisfy the equations of motion (26) and (27). Therefore accordli, The main consequence of the above observations is that, while to Bohm the condition p = VS is a consistent subsidiary condition. classical trajectories depend on initial velocity v and initial posi- Eqs. (34) and (35) should however be written as a covariant deriv- tion r , the trajectory calculated in the causal interpretation from ative mr" = p* = grad S(r,t) depends only on the initial position r" /22/. §f = - v(v+uQ), 5_ = (36) The best way to see the main features of this "causal interpreta- At first sight Bohm's answer seems acceptable. But in his original tion" is to determine explicitly the "associated trajectories" for work /I/ as well as in his answer /21/ Bohm does not always keep solvable quantum systems. This is the subject of the next section. We track of the other equation (26) and in this way gives the impression shall also show that the two ways of introducing "causal" orbits by that U..(r) may be treated on the same footing as the external potential- Eqs.(32) and (33) respectively are in fact not equivalent at least for This impression is misleading because S and R are coupled also stationary states. through Eq. (26) and therefore UQ(~r) in Eq.(27) is not a function given in advance and independent of S or p. IV. Causal trajectories in the stationary states of the The most direct way to see the importance of Eq. (26) is ) is central potential (hydrogen atom, spherical oscillator,...) to consider the one dimensional stationary case. Eqs. (29) become 2 2 It is well-known that stationary solutions of the Schrbdinger Rd s(x)/dx + 2(dR/dx) • (ds/dx) = 0 (37) equation in the case of a central potential V(r) (ds/dx)2/2m + V(x) - (fi2/2m) (d2R/dx2)/R = E (38) -9- -10- and finally 2.2, ) 2 2 [- (n /2m)V + = EiJ; Therefore, according to the causal interpretation in the stationary have the form states the electron moves along the circles lying in the planes parallel iMr,e,f>) = R(r) (43) to the x-y plane for any central potential* Tor the same magnetic where functions quantum nurber M these trajectories do not distinguish between R(r) = A F^(r) different potentials. ©(9) = P"(COB 6) For kinetic energy we find (45) T = p /2m = pJT/2m = m r w~ • sin"6/2m (53) fJM(lp) = exp{iM (45c) i.e. quantum energy is a sum of the kinetic energy, the Coulomb By Bohm's definition the quantum action is proportional to the phase potential energy and the quantum potential energy. of the wave function. Since the functions R(r) and ©( e) are real we Bohm's trajectories for the hydrogen atom in particular are quit, conclude that different from Bohr's (semiclassical) trajectories. Bohr's trajec- s(r) = ftM We see in agreement with Halpern remark /20/ that s(r) does not depend set satisfies the conditions of quantization). The nucleus lies in on three independent non-additive constants. the plane of the orbit. This is not the case with Bohm's orbits whose Using (43) and (45) one easily determines the quantum potential planes in general do not contain the nucleus.This is due to the difference be- 2 UQ(?) = - = E - V -fi M /2nvr"sin*"0 tween quantum action s(r) given in (46) and the classical action given (47) by Taking into account that 2 1 /2 w(r) = w1 (r) + W2(6) +• w3(ip) = / [2m (E-V) -a^/r ] dr grad s = grad ftMUJ = («M/rsin9) «L (48) /(a2 - a2/sin2[))1/2d0 + (55) we see that S satisfies the "quantum Hamilton-Jacobi equation" Bohr trajectories satisfy the relation p* = grad w and the conditions 2 2 2 2 E = (grad s) /2m - (e /r) - (fi /2m) V (R(r)@ (6) /R(r)fi) (8)) (49) of quantization, whereas Bohm trajectories(Fig- 1) satisfy In spherical coordinates mr* reads p = qrad s - grad fiNUp = grad w^with a^ = fiM_ mr = mr u> sin.9 'e^ (50) In order to see further the difference between the quantum and Hence equating (48) to (50) on the basis of the definition (31) of classical actions we write the Hamilton-Jacobi equation for the func- "momentum" we obtain * The circles for the hydrogen atom have also been mentioned by • 2 2 Belifante /26/. y) = fiM/mr sin 0, r = r , 0 = 6 (51) -11- -12- tion w(r) taking into account its structure given in (55) We now come to the second definition of the associated causal momenta i (Vw ) /2m + (Vw2) V = E (56) $ by dp /dt=-V(V + u ). in the central potential for the states given in From Eq, (55) we find (43) we obtain (Vv^)2 = Ow/3r)2= - - 2m(V-E) (57a) , 2,, (57b) 2 /r' = apr* - c^ 2 2 2 2 2 where M and En are the quantum ranters of the particular stationary = Ow3/3(p) /r sin 9 = c<3/r si.n (57c) 2 2 2 2 state under consideration. This is an interesting dynamical system in + (Vw2) = 2m(E-V) - a /r sin 6 (58) its own right. The potential is singular along the z-axis with an inverse By comparing (55) with (46) and (58) with (47) we conclude that in the cube force law (a two-dimensional version of the Dirac's charge-monopole central potential the "quantum action" is just a part of classical ac- system). The equations of motion are tion, whereas quantum potential is associated with the rest of the mr = YV(1/P2) , p2 = x2 + yZ , y = H2M2/2m classical action as follows We have two integrals of motion for this dynamical system, the total s (r) = w_, ex, = fiM 1 59) energy and the z-component of the angular momentum. In cylindrical co- 2 UQ(r) = [(Vw^ + (V (60) ordinates The fact that the "associated" trajectories are all the same for (62) L » mp any central potential is evidently the consequence of the property z Furthermore, z = 0 , or z = vz(0), z = vz(0)t + z(0). Consequently, we that these orbits depend only on the initial coordinates and not on have one first, order equation for P the initial velocities. 2 2 2En • + (11 _ Ii) J_ = K K = 2 - - v in In the ground state of the central potential, in particular, we in m p have £ = 0, M = 0 and consequently s = 0, p = 0 which means that the This equation can be integrated and gives "particle" is immobile. Similarly,in the case of standing waves P= LK{t+C)2 + (L2 - Tl2HZ)/i (63a) u*(x) = sT ' cosimx/2a n = 1, 3... Whence 1/2 t + C (63b) -If -) arctg + C u (x) = a sin7inx/2a n = 2, 4... n Krtf (- En = 'n2"2n2/8ma2 between two impenetrable potential walls at x = -a and x = a, and for The projection of the orbits in the z=const. plane are 2 the bound states in the rectangular potential hole p=l [tg (-j^ (f + (p )) + l] -U |xj T and schematically shown in Fig. 1. (dotted lines). For L =HM {angular = dx/dt = tik. (1-r.) / (1+r. +2r. cos2k.x ) x<0 momentum of the associated particle = quantum angular momentum) the II = dx/dt = Rk^ x>0 (68) equations simpl ify considerably: P= j2—(——^TJ )- There are also special II z TO T It is easy to see that p[ and Pj are always positive. This means that circular orbits p = const., ^p = const. for k,>0 the particle moves only in the positive x-direction. V. Passage through a step-like barrier in the causal The integration of Eq. (68) gives the relation between x interpretation and t In the case of the potential shown in Fig.2. the wave functions 1 (H-r^)x + (r1/k1)sin2klx +C = (fik1/m) (1-r^)t x<0 II in regions I and II have the form x = (fik^/mjt + C x^O (69) ikjX , -ik,x r is,/11 -Fj(x) = We determine the integration constants from the conditions t - 0, x = x ; t = t , x = 0. II ^5 Q Q (64) e where kj - (2m/fiZ)V (65) t-t A = o rlAol- i Vol kj + kj), tj = 2k1/(k1 + k|) (66) t>to (70) When kj is real (IT k /2m>V) the quantum action and its amplitude in regions I and II are 2, 2, If kl is imaginary (1TkJ/2m Rl = tlA01 the wave functions in regions I and II are x < 0 (67) -ik^ -2iarctg(p1'/k1) sj = tlarctg ({1 - r^) ik1x A . + A ,e e x > 0 -P,ol x o1 - ,., 1/0 -i-arctg (p'/k ) II 1 lf. (x) = A e (2k /(k + f> ") ) ' e (72) whereas the quantum action and the associated momentum take the form The expressions for "momentum" follow directly -Tiarctg ( p^ (73) P-l " u ^1" " - (74) Therefore, a curious result is obtained. If the initial energy is less than the potential energy in region II the particle remains at rest at the initial position in region I. -15- -16- For the second definition of causal trajectories, Eq.(35), we find that.since quantum potential in regions I and II 2 2 UQI(x) = 1i k /2m, U (75) is independent of x,the equation d[>/dt - - V{V+UQ) gives arbitrary constant values for momentum in the two regions 1 "E = c p II, 2 r n (76) We note that in this physical situation the causal interpretation (79) does not agree with the statistical interpretation since according The "momenta" now oscillate in time which may perhaps be interpreted to the latter one there is a finite probability for the particle re- as due to reflected waves. But the explicit interpretation of trajec- flection at the boundary of the two regions. On the other hand.Dewdney tories remains. found that among the numerically computed trajectories inthetime depen- dent states there exist both reflected and transmitted trajectories VI. The causal motion of a spinning neutral particle til. Dewdney's result may be understood by analysing the trajectory inside the magnetic field associated with the simplest wave packet which is a superposition of The spinning neutral particle in the magnetic field is described two energy eigenstates: i> by the two-component spinor ^ = (^ ) which satisfies the Pauli -E^) /-n i(s?-E,t)/ft Z equation ik.'x -iE.t/t/Hn ik'x -iE t/tl ± TT ik (" 2m + >ja-BH = lf> ^r (80) +C2R2 e e (77) In the causal interpretation the spinor ^ is written /6,23/ in the form .iU Here Rj,s[ and sj are given in {67) whereas RJJ, s., and s2 are ob- iS/n[ 2 = R e (81) tained from (67) replacing k, by k,: S = S - Et \4 One directly obtains s,, Sj,, p, and p,. One assumes a = 1,2; O1 = -6 sT(x,t) = ft-artg 1 R^ /ft 2 2 = 1 (82) II II C,R.sin(p.'x-E,t)/fitC R sin (p'x-E_t) /R Dewdney et al. showed /6/ that the Pauli equation is equivalent to s (x,t) = ft-arctg 11 i L four partial differential equations for variables 5, R and s -E1t)/K+C2R 2 cos(p2X-E2t) /H 2 (78) (mv /2)+UQ+Us+(2/n)uB-s = 0 (83a) 3p/3t + V(pv) = 0 (83b) -17- Ds/Dt = T + (2/fl)uBxs (83c) -18- Here P = R is the probability density + Barut et.ai. showed /24/ that by writing the state (88) in the form v*= (1/m) {VS - in $ V U is a spin-dependent addition (86) one makes explicit the representations of the translation and rotation to the quantum potential groups to which this state for given k' and k" belongs. The form 2 2 UQ = - {•n /2m)V R/R (90) is consistent with the following transformation relation whereas T is a quantum torque - rj) R (f) = (it1 (91) (87) t = By comparing the form (91) and the requirement (82} of the causal interpretation, we see that symmetry requirements and the requirements Because of the flow derivative Ds/Dt (as we have noted already in (36) of the causal interpretation lead to the same result in the identifi- in connection with Dp/Dt) a quantum torque arises in the spin equation cation of the two parts in the wave function, one of which {Re1 ) (.83c) so that this equation has nothing to do with either the clas- describes the external degrees of freedom and the other ($) describes sical spin precession, nor with the Heisenberg quantum spin equation. the internal spin degrees of freedom. Finally we find that the equation corresponding to Eq. (36) We see that action s(f) is given by in the spin case is s(r) = (11/2) (P + t" )-r (9Z) + $ - -v a We shall now determine S, s and v for stationary states in a By substituting the latter expression into (84) we obtain for the nntentum Z magnetic field along z- axis. p = mv = ' | + (93) The general eigenstate of (80) which propagates in the positive which is in the direction of the current density /25/ x-direction is (94) = Re where 2/2m - = H2k" 2/2m (89) -20- -19- The trajectories associated with the "Newton equ&tion" dp' ( v -V) ^ 2 dt (VS) - (96b) are straigt lines characterized by (with v = vS/m) -+• -f Thus p = VS indicates the flow lines of the wave mechanical cur- P = Po rent density, whereas the real part of the expectation value of the because in the stationary state (90) the second paranthesis vanishes time displacement operator ifi 3^ and flow displacement operator whereas LL, U and z-component of the spin vector 1 are independent {3t + "v-V );(i .e. covariant derivative)give the "Hami ltonian" and of v "Lagrangian" of a "particle" whose momentum is "p = VS, moving in the u = o q total potential V + IL. (k- - V)2 4m In the case with spin we obtain - = (VS -in? V£) + -VR/R (97) i I + Vll.Currents and Flows We think that the best way to interprete the so-called causal we have "associated motions" is to say that they express certain flow properties R2 R1 R +R of the current density (see also Ref.26). We have the following general f 2 Rl+R2 " (98) relations in the spinless case These equations show that one associates the "causal momentum" to the = Re{5 I(J+VIJJ )/ IJJ 41 = V S (95a) translational motion of the spin 1/2 particle and not to the individual spin components. -TWR/R (95b) Furthermore VIII. Coherent States (V + U ) (96a) Q These are time-dependent states which have some properties very close to the classical trajectories. Hence it would be interesting to find the two associated orbits for such states. -21- -22- We consider Schrbdinger's original coherent states 121/ for the harmonic oscillator characterized by a complex number A ("amplitude") P = n $- where |n> are the energy eigenstates of the Hamiltonian and with the classical solutions q , (t) = (p t + q cos» t O OO H = —;F(- ^-5 + x ) (100) P (t) = c Sl t m ) q Sin(1) t (107) cl Po ° "O " " o c o Here x is the dimensionless variable related to the coordinate q of the oscillator by in which p and q. are the initial position and momentum. (Note that 00 1 ,^ x = q// mto /fi 22 -1 1 2 2 Classical action is w = (u) /2) [q-(a -q ) +asin (q/ct)J vJiere a =2E/u ). Hence we find •iut/2 t/2 Thus in the coherent state all four concepts of momenta and po- = the normalized coherent states Conclusion m^ 1/4 -^ The "causal interpretation" of quantum mechanics by quantum po- tential has a priori an appealing feature. Distinct from the statisti- Hence the quantum action (phase) is cal interpretation, it emphasizes the importance of the phase of the 2 S = -h[(cu t/2) + ( |A| /4) sin2($-w t) • j A | x • sin (*-ujQt) ] (1 02) wave function and in this way aims to solve the difficulties in the The associated motion is then characterization of quantum mechanical reality. But as we showed, the methodological basis of this interpretation does not lie in classical p = VS =/ Hi" |A sin((J>-uj t) q =/ Fimiij |A|cos { resulting orbits of the "causal" interpretation have nothing to do The quantum potential is with semiclassical picture,rather they model certain flow patterns of 2 U = (tWQ/2) [1 - (x -|A|cos(4>-0Jot)) ] [1 D4) the quantum current. In coherent states which are time-dependent, how- Hence the associated motion of the second kind is ever, the orbits of the classical motion, the average quantum motion (1 15a) dp'/dt = -d(V+UQ)/dq = -ioc and those coming from p = VS and dp'/dt - - v(V+Ug) all coincide ex- which gives cept for different integration constants. But in general the "causal" q' =|A ,1/2 (105b) trajectories cannot give a full picture of the behaviour of the quantum We compare these with the quantum mechanical averages in the coherent system. They must be supplemented by complicated probability densities state of such trajectories. -23- -24- ACKNOWLEDGMENTS References One of the authors (M.B.) is grateful to P.R. Holland and 1. D. Bohm. Phys.Rev. 85,166; 85,180 (1952) A. Kyprianidis for discussion, help and information on references. The authors would like to thank Professor Abdus Salam, the International 2. 0.0. Bohm and B.O. Hiley, Found.Phys. 5, 93 (1975) Atomic Energy and UNESCO for hospitality at the International Centre for 3. C. Philippidis, C. Dewdney and B. Hiley, Nuov.Cim B5£, 15 (1979) Theoretical Physics, Trieste, 4. C. Dewdney and 8.J. Hiley, Found.Phys. 12, 27 (1982) 5. C. Dewdney, Phys.Lett. 1Q9A, 377 (1985) 6. C. Dewdney, P.R. Holland and A. Kyprianidis, Phys.Lett. AU9, 259 (1986) 7. C. Dewdney, in Quantum Uncertainties, Recent and Future Experiments and Interpretations, edited by W.M. Honig, D.W. Kraft and t. Pana- rella, Plenum Publishing Corporation, 1987 8. C. Dewdney. P.R. Holland and A. Kyprianidis, Phys.Lett. A121, 105 (1987) 9. G. Badurek, H. Rauch and D. Tuppinger, Ann.New York Acad.Sci. 480 (1986) 133 10. C. Dewdney, P.R. Holland and A. Kyprianidis, J.Phys.A: 20 (1987) 4717 11. E. Schrb'dinger, Ann.Physik. 79, 361; 79, 489; 80, 437; 81, 109 (1926) 12. E. Schrodinger, Phys.Rev. 28, 1049 (1926) 13. E. Schrbdinger, Four Lectures on Wave Mechanics (Blackie & Son, London, 1929) 14. L. de Broglie, Compt.Rend. 183,447 (1926); 184,273 (1927); 185,380 (1927) 15. L. de Broglie, C.R. Acad.Sci. Paris 183,447 (1926), 184,273 (1927) 16. E. Madelung, Z.f.Phys. 40,322(1926) 17. I. Fenyes, Z.Phys. 132,81 (1952) 18. M. Born, Z. fur Physik. 37 863 (1926) 19. S.I. Tomonaga, Quantum mechanics, (North-Holland, Amsterdam, 1962) Volume I. 20. 0. Halpern, Phys.Rev. 87,389 (1952) 21. D. Bohm, Phys.Rev. 87,389 (1952) -25- -26- 12. A. Kyprianidis, to be published. 23. P.R. Holland, to appear in Physics Reports 24. A.O. Barut, M. Bozic, Z. Marie and H. Rauch, Z.F. Physik Figure captions A238.1 (1987) 25. A.O. Barut and M. Bozic, Physica B (1988). Figure 1. 26. A. Bel infante, A Survey of Hidden Variables Theories, (Pergamon The draft of semidassical (---) and of two types of "causal" trajec- Press, Oxford, 1973) tories; p = vs (- -), dp'/dt = -V(v + Un) ( ) for the central 27. E. Schrb'dinger, Naturwiss. 14,664 (1926) potential, Figure 2. Step like magnetic barrier. -27- -28- associatedd ^causal trajectoriestj " p=VS Semiclassical y trajectories lV[x) IJ I'ig.f. -29-