aa r X i v : . [ qu a n t - ph ] N ov On quantum potential dynamics
Sheldon Goldstein ∗ and Ward Struyve † November 6, 2014
Abstract
Non-relativistic de Broglie-Bohm theory describes particles moving under the guid-ance of the wave function. In de Broglie’s original formulation, the particle dy-namics is given by a first-order differential equation. In Bohm’s reformulation, itis given by Newton’s law of motion with an extra potential that depends on thewave function—the quantum potential—together with a constraint on the possiblevelocities. It was recently argued, mainly by numerical simulations, that relaxingthis velocity constraint leads to a physically untenable theory. We provide furtherevidence for this by showing that for various wave functions the particles tend toescape the wave packet. In particular, we show that for a central classical potentialand bound energy eigenstates the particle motion is often unbounded. This workseems particularly relevant for ways of simulating wave function evolution basedon Bohm’s formulation of the de Broglie-Bohm theory. Namely, the simulationsmay become unstable due to deviations from the velocity constraint.
Non-relativistic de Broglie-Bohm theory (also called Bohmian mechanics) [1–3] de-scribes point-particles moving under the guidance of the wave function. In the case ofspinless particles, with positions X k , k = 1 , . . . , n , and configuration X = ( X , . . . , X N ),the equations of motion are given by d X k ( t ) dt = 1 m k ∇ k S ( X ( t ) , t ) , (1)where the wave function ψ = | ψ ( x, t ) | e i S ( x,t ) / ~ , with x = ( x , . . . , x n ), satisfies the non-relativistic Schr¨odinger equationi ~ ∂ t ψ = − n X k =1 ~ m k ∇ k + V ! ψ . (2) ∗ Departments of Mathematics, Physics and Philosophy, Rutgers University, Hill Center, 110 Frel-inghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: [email protected] † Departments of Mathematics and Philosophy, Rutgers University, Hill Center, 110 FrelinghuysenRoad, Piscataway, NJ 08854-8019, USA. E-mail: [email protected] ψ the particle distribution is given by | ψ | .This distribution—the quantum equilibrium distribution —is preserved by the particledynamics, i.e., it does not change its form as a function of ψ .This theory was originally discovered by de Broglie in the late 20’s [4] and redis-covered by Bohm in the early 50’s [5, 6]. Unlike de Broglie, Bohm did not regard theequation of motion (1) as fundamental. Instead, he proposed the second-order differen-tial equation m k d X k ( t ) dt = − ∇ k ( V + Q )( X ( t ) , t ) , (3)which is Newton’s law of motion with an extra ψ -dependent potential Q —the quantumpotential —given by Q = − N X k =1 ~ m k ∇ k | ψ || ψ | . (4)The equation (3) is referred to as Bohm’s dynamics in [7]. We prefer to call it the quantum potential dynamics (QPD for short). In addition, Bohm assumed the constraint d X k dt (0) = 1 m k ∇ k S ( X (0) ,
0) (5)on the initial velocities. The QPD implies that this constraint is preserved in time, i.e.,if (5) holds then d X k ( t ) /dt = ∇ k S ( X ( t ) , t ) /m k for all times t .Bohm thought modifications of his theory might be required in order to understandphenomena over distances smaller than 10 − cm. In particular, he entertained the idea ofrelaxing the constraint on the velocities [5]. However, he did not suggest this relaxationwithout further modifications of the theory. Rather, he considered either modifyingthe Schr¨odinger equation or the Newtonian equation in order to ensure that over timearbitrary initial velocities would tend to those given by the de Broglie-Bohm theory.Thus his modified QPD could reasonably be expected to yield predictions in at leastapproximate agreement with those of standard quantum mechanics. (And while we arehere ignoring spin, it should be noted that the formulation of the QPD for particles withspin is problematical [1].)In [7] the possibility was considered of relaxing the constraint on the initial velocities without additional modifications to the equations of motion. Numerical simulationswere performed for the one-dimensional harmonic oscillator and for the non-relativistichydrogen atom, for particular superpositions of energy eigenstates. In the case of theharmonic oscillator it was found that for initial positions outside the bulk of the wavepacket (i.e., where | ψ | is appreciably different from zero) and for sufficiently largeinitial velocities the particles seem to escape to infinity. In addition, an example wasprovided of a phase-space distribution corresponding to initial momenta larger thanthose of the de Broglie-Bohm theory for which the particles seem to be escaping. It wasfurther argued analytically that for initial positions x → ∞ and for large enough initialvelocities the particles escape to infinity. (However, unless the particles can move so2ar outside the bulk of the packet, in which case the theory seems no good in the firstplace, it seems that one can ignore such initial conditions.) In the case of the hydrogenatom, a couple of trajectories were simulated and it was found that the more the initialvelocity deviates from (5), the more the particle seems to escape from the packet.Here we provide further evidence, in the form of analytical results, that relaxing theconstraint on the velocities is untenable. We consider some potentials V and quantumstates for a single particle for which the QPD can be easily analysed. In particular,we consider the case of central potentials and bound energy eigenstates (for which | ψ | is appreciably different from zero only in a certain region of space) and show that theparticle motion is often unbounded, so that particles escape to infinity. This implies thatthe QPD is empirically inadequate. Perhaps more importantly, this work also seems toreveal a potential source of instabilities in particular ways of simulating wave functionevolution. We will explain this further near the end of the paper.Let us first consider some properties of energy eigenstates. For a single particle, theSchr¨odinger equation implies ∂ t S + 12 m ∇ S · ∇ S + V + Q = 0 . (6)Hence for an energy eigenstate ψ = φ ( x )e − i Et/ ~ we have E = 12 m ∇ S · ∇ S + V + Q (7)and the total force in the QPD is given by − ∇ ( V + Q ) = − m ∇ ( ∇ S · ∇ S ) . (8)In the special case that the phase S does not depend on x , for example for non-degenerateenergy eigenstates, the total force is zero, so that the particle is free. In that case, par-ticles will move to infinity unless their velocity is zero, which is the case for a trajectoryof the de Broglie-Bohm theory.We can define the energy of a particle as e E = m | ˙X | V + Q , (9)where we have used the tilde to distinguish it from E , the energy of the wave function.We have that d e E/dt = ∂Q/∂t , so that in general d e E/dt = 0. However, in the case ofan energy eigenstate, for which Q does not depend on t , we have that d e E/dt = 0. For atrajectory given by the de Broglie-Bohm theory (i.e., corresponding to an initial velocitygiven by (5)), we have moreover that e E = E . (But e E = E does not guarantee that thetrajectory is given by the de Broglie-Bohm theory.)From (7), it follows that the total potential V + Q E . Hence, no matter how muchthe classical potential V would confine the particles, the total potential is bounded fromabove. As such, it would seem that if the energy e E of the particle were large enough, it3ould escape to infinity. Indeed, in the case of one spatial dimension, we have that when e E > E ( > V + Q ), then the particle will escape to infinity, because there are no turningpoints. Actually, for a bound state in one dimension, we even have that ∂ x S = 0, sothat E = V + Q and the particle is free.Let us now turn to explicit examples. We start with one spatial dimension. Firstconsider a free particle. For a Gaussian wave function centered around the origin (weassume ~ = 2 m = σ = 1, where σ is the width of the packet) ψ ( x, t ) = (cid:18) π (1 + i t ) (cid:19) / e − x / t ) , (10)the QPD reads ¨ X = X/ (1 + t ) , so that the possible trajectories are X ( t ) = √ t ( X + V arctan t ) , (11)where X and V are respectively the initial position and velocity. The velocity constraint(5) corresponds to V = 0. The standard deviation of the density | ψ | is given by σ ( t ) = √ t . Hence, if the initial speed | V | of the particle is large enough, it willescape the bulk of the packet. For example, a particle with 0 X < σ (0) = 1will have a position X ( t ) > σ ( t ) for t > tan((1 − X ) /V ) if V > − X ) /π . Inthe case of an ensemble of particles with initial position and (independent) velocitydistribution respectively given by | ψ ( x, | = e − x / / √ π and e − ( v − v o ) / σ / √ π ˜ σ (i.e.,the initial velocity distribution is Gaussian with mean v and standard deviation ˜ σ ), theposition distribution ρ v ( x, t ) at later times t is Gaussian with mean v √ t arctan t and standard deviation p (1 + t )(1 + ˜ σ arctan t ). Hence, if v = 0 the center of ρ v ( x, t ) moves away from that of the distribution | ψ ( x, t ) | = ρ v =0 ( x, t ), up to a distance v π/ /σ ( t )). ρ v also spreads more than | ψ | , by up to a factor p σ π / V for x > ~ = 2 m = 1), theenergy eigenstates ψ ( x, t ) = φ ( x ) exp( − i Et ) for E < V are given by φ ( x ) = (cid:26) cos( kx − α/ x < α/ − κx x > , (12)where k = √ E , κ = √ V − E and α = 2 tan − ( − κ/k ). Since the phase of the wavefunction does not depend on the spatial coordinate, we have (as mentioned above)that the particle is free; it is not confined by the potential. (Actually, in this case,also the de Broglie-Bohm motion is unphysical, since particles will just stand still. Amore realistic treatment should consider an approximately localized packet that movestowards the potential step.)Consider now the harmonic oscillator, with V = x / ~ = m = ω = 1).The energy eigenstates are bound states and hence, as mentioned before, the particle isfree. Other states of interest are coherent states ψ ( x, t ) = π − / exp (cid:20) −
12 ( x − a cos t ) − i2 (cid:18) t + +2 xa sin t − a t (cid:19)(cid:21) . (13) For the de Broglie-Bohm treatment of all the systems considered here, see [2]. t de Broglie-Bohmtrajectoryde Broglie-Bohmtrajectoryde Broglie-Bohmtrajectoryde Broglie-Bohmtrajectoryde Broglie-Bohmtrajectory Figure 1: Tractories for the QPD in the case of the coherent state (13) (with a = 1).The density | ψ | is represented by the shaded area. One de Broglie-Bohm trajectoryis plotted, which follows the center of the density | ψ | . It is a trajectory with initialvelocity V = 0. The trajectories respectively moving to the right and left have initialvelocities V = ± .
25 and escape the wave packet.The corresponding density | ψ ( x, t ) | = π − / exp (cid:2) − ( x − a cos t ) (cid:3) (14)is a Gaussian function whose center oscillates between the points x = ± a . The QPDreads ¨ X = − a cos t , so that the possible trajectories are X ( t ) = X + V t + a (cos t − . (15)Since ∂S ( x, /∂x = 0, the constraint (5) implies that V = 0, so that the particle inthe de Broglie-Bohm theory performs a harmonic oscillation around the point X − a .However, if V = 0 then the particle will escape to infinity, oscillating around X + V t − a .Some trajectories are plotted in figure 1.We now turn to three spatial dimensions and a central potential V = V ( r ). En-ergy eigenstates with energy E , orbital angular momentum number l and the magneticquantum number m (here we denote the mass by m ) are of the form ψ Elm ( r, θ, φ, t ) = R El ( r ) Y lm ( θ, φ )e − i Et/ ~ , (16)where ( r, θ, φ ) are spherical coordinates determined by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos φ . (17)5he functions Y lm are the spherical harmonics and R El are real functions that are so-lutions to the radial Schr¨odinger equation. Due to the spherical symmetry there is a(2 l + 1)-fold degeneracy: for given E and l all values of m obeying − l m l arepossible. For bound states there is usually no further degeneracy so that general energyeigenstates are of the form ψ ( r, θ, φ, t ) = l X m = − l c m ψ Elm ( r, θ, φ, t ) = R El ( r ) l X m = − l c m Y lm ( θ, φ )e − i Et/ ~ , (18)where the c m are arbitrary complex coefficients. In the case of the harmonic potentialand the Coulomb potential additional symmetries imply further degeneracy. In thefollowing, we restrict our attention to states of the form (18). Using (7) and the factthat the phase of ψ does not depend on r , we find that for such a state the total potentialis of the form V + Q = E − m ∇ S · ∇ S = E + 1 r f ( θ, φ ) , (19)with f = − r ∇ S · ∇ S/ m a function of the angular variables only. So, droppingthe constant E , the effective potential is of the form f ( θ, φ ) /r . This potential wasconsidered in detail in [8]. The corresponding Lagrangian is L = m r + r ˙ θ + r sin θ ˙ φ ) − fr , (20)with corresponding equations of motion m ¨ r = m r ( ˙ θ + sin θ ˙ φ ) + 2 fr , (21) ddt ( m r ˙ θ ) = m r sin θ cos θ ˙ φ − r ∂f∂θ , (22) ddt ( m r sin θ ˙ φ ) = − r ∂f∂φ . (23)The energy e E is a constant of the motion and reads (dropping again the constant E ): e E = m r + r ˙ θ + r sin θ ˙ φ ) + fr . (24)In the case of a trajectory given by the de Broglie-Bohm theory we have that e E = 0. (Wethen also have that ˙ r = 0, so that the particle moves on a sphere.) Another constant ofthe motion is C = 12 m | L | + f , (25)where | L | = m r ( ˙ θ + sin θ ˙ φ ) with L the angular momentum. We can use it to writethe energy as e E = m r + Cr . (26)6 V r rC = 0 C > 0C < 0 Figure 2: The shape of the potential V r = C/r for C = ± , r moves under an effective potential V r = C/r , whichis plotted in figure 2.We can now qualitatively analyze the QPD. In the case C = 0, we have that e E = m ˙ r /
2. Hence, for strictly positive energy the particle flies off to infinity with a constantradial speed. For zero energy (as in the case of a de Broglie-Bohm-trajectory), theparticle’s trajectory is confined to the surface of a sphere. If
C >
0, we have e E > r > ( C/ e E ) / , and the particle eventuallyflies off to infinity. If C <
0, the particle motion is unbounded for e E >
0. For e E < r ( C/ e E ) / . Examples of radial trajectories arepresented in figure 3.The situation is summarized in figure 4. If the energy of the particle is greater thanthe energy in the de Broglie-Bohm theory, i.e., e E >
0, then its motion is unbounded andit flies off to infinity, even for initial conditions arbitrarily close to those of a trajectoryof the de Broglie-Bohm theory. If e E <
0, then the particle motion will be bounded withradial motion oscillating between 0 and the turning point r = ( C/ e E ) / , which may stillbe far from the center of the packet.In conclusion, we have seen that according to the QPD particles often escape from thebulk of the wave packet, even for bound states. In the case of the hydrogen atom (whichis described by the Coulomb potential), this means that the electron will not be boundto its nucleus. More generally we expect that molecules and atoms will not be stable butwill tend to disintegrate. Of course we have only considered energy eigenstates here, butthere seems to be no hope that the total potential will bind the particles in the case of7 t rE~ = -1, C = -1 E~ = 0, C = 0 E~ = 1, C = 1E~ = 1, C = -1 Figure 3: Examples of radial trajectories with the same initial radial coordinate r (0) = 1(and m = 2). If e E > no trajectoriesbounded trajectoriesunbounded trajectories0E~ C
Figure 4: Types of trajectories in terms of the constants of motion e E and C . For pointson the coordinate axes we have the following situation. For the origin, the motion isbounded. For C > e E = 0) or e E < C = 0) we have no trajectories. For C < e E = 0) the motion is unbounded.8 superposition (as was illustrated in [7]). Neither is there any hope that a relativisticor quantum field theoretical treatment will help. And, of course, even if this were nottrue, there would still seem to be no reason whatsoever to expect the predictions ofQPD to be governed by Born probabilities and hence no reason whatsoever that theyshould have anything to do with those of quantum mechanics.This work might also be of relevance for particular techniques for simulating thewave function evolution using de Broglie-Bohm trajectories. For example, one techniqueis roughly as follows [9–12]. The Newtonian equation (3) is considered with | ψ | replacedby √ ρ , where ρ an actual density of a large but finite number of configurations. Given aninitial wave function ψ , the equation is numerically integrated starting with the initialdistribution ρ given by | ψ | (up to some accuracy) and with the initial velocity of eachconfiguration satisfying the constraint (5). From the trajectories one can then obtain atime-dependent wave function ψ . But since ρ equals | ψ | only up to some accuracy, thesimulation will entail small deviations of the velocities from ∇ S/m . As is illustrated byour results, this might cause trajectories to deviate significantly from trajectories in thede Broglie-Bohm theory and hence this may potentially lead to an inaccurate simulationof the wave function. (It may also be that the difference between the force determinedby ρ and the quantum force determined by | ψ | induces corrections to the velocity whichbrings them closer to ∇ S/m .)Recently, this particular way of numerically simulating the wave function formed thebasis of a new approach to quantum mechanics [13, 14]. In this approach, there is a largebut finite number of configurations, each representing a different world, which evolveaccording to a dynamics similar to that used in the wave function simulations. Thereis no wave function on the fundamental level. But given appropriate initial conditionson the velocities, the configurations may (approximately) determine a wave function.However, it is not clear whether such wave functions obey some Schr¨odinger dynamics.One potential source of trouble may be the instabilities caused by deviations of thevelocities from ∇ S/m . Acknowledgments.
It is a pleasure to thank Roderich Tumulka for discussions.S.G. and W.S. are supported in part by the John Templeton Foundation. The opinionsexpressed in this publication are those of the authors and do not necessarily reflect theviews of the John Templeton Foundation.
References [1] D. Bohm and B.J. Hiley,
The Undivided Universe , Routledge, New York (1993).[2] P.R. Holland,
The Quantum Theory of Motion , Cambridge University Press, Cam-bridge (1993). As such, there is no point in even starting an analysis of positions measurements in QPD, in thehope that such measurements would yield the correct results, since measurement devices or pointerswould not exist in the first place.
93] D. D¨urr and S. Teufel,
Bohmian Mechanics , Springer-Verlag, Berlin (2009).[4] L. de Broglie, in ´Electrons et Photons: Rapports et Discussions du Cinqui`eme Con-seil de Physique , Gauthier-Villars, Paris, 105 (1928); English translation: G. Bac-ciagaluppi and A. Valentini,
Quantum Theory at the Crossroads: Reconsideringthe 1927 Solvay Conference , Cambridge University Press, Cambridge (2009) andarXiv:quant-ph/0609184.[5] D. Bohm, “A Suggested Interpretation of the Quantum Theory in Terms of “Hid-den” Variables. I”,
Phys. Rev. , 166 (1952).[6] D. Bohm, “A Suggested Interpretation of the Quantum Theory in Terms of “Hid-den” Variables. II”, Phys. Rev. , 180 (1952).[7] S. Colin and A. Valentini, “Instability of quantum equilibrium in Bohm’s dynam-ics”, Proc. R. Soc. A , 20140288 (2014) and arXiv:1306.1576v1 [quant-ph].[8] H. Wu and D.W.L. Sprung, “Inverse-square potential and the quantum vortex”,
Phys. Rev. A , 4305 (1994).[9] C.L. Lopreore and R.E. Wyatt, “Quantum Wave Packet Dynamics with Trajecto-ries”, Phys. Rev. Lett. , 5190 (1999).[10] R.E. Wyatt, Quantum Dynamics with Trajectories , Springer, New York (2005).[11] Deckert D.-A., D¨urr D and Pickl P., “Quantum Dynamics with Bohmian Trajecto-ries”,
J. Phys. Chem. A , 10325 (2007) and arXiv:quant-ph/0701190.[12] S. Goldstein, R. Tumulka and N. Zangh`ı,
Bohmian Trajectories as the Foundationof Quantum Mechanics , in
Quantum Trajectories , ed. P.K. Chattaray, CRC Press,Boca Raton (2011) and arXiv:0912.2666 [quant-ph].[13] M.J.W. Hall, D.-A. Deckert and H.M. Wiseman, “Quantum phenomena modelledby interactions between many classical worlds”,
Phys. Rev. X4