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2 Bohmian trajectories

Consider a time-dependent Hamiltonian function

n p2 H(x,p,t)= j + U(x,t) (1) =1 2mj Xj and the corresponding quantum operator

n ~2 2 ~ ∂ H(x, i x,t)= − 2 + U(x,t). (2) − ∇ =1 2mj ∂xj Xj b The associated Schr¨odinger equation is ∂Ψ i~ = H(x, i~ ,t)Ψ , Ψ(x, 0)=Ψ0(x). (3) ∂t − ∇x Let us write Ψ in polarb form ReiΦ/~; here R = R(x,t) 0 and Φ= S(x,t) ~ ≥ are real functions. On inserting ReiS/ into Schr¨odinger’s equation and separating real and imaginary parts, one sees that the functions R and S satisfy, at the points (x,t) where R(x,t) > 0, the coupled system of non- linear partial differential equations

n 2 n ∂S 1 ∂S ~2 ∂2R + + U(x,t) 2 = 0 (4) ∂t =1 2mj ∂xj − =1 2mjR ∂xj Xj   Xj 2 n ∂R 1 ∂ 2 ∂Φ + R = 0. (5) ∂t =1 mj ∂xj ∂xj Xj  

2 The crucial step now consists in recognizing the first equation as a Hamilton– Jacobi equation, and the second as a continuity equation. In fact, introduc- ing the quantum potential n 2 2 Ψ ~ ∂ R Q = 2 (6) − =1 2mjR ∂xj Xj (Bohm and Hiley [1]) and the velocity field

Ψ 1 ∂Φ 1 ∂Φ v (x,t)= , ..., (7) m1 ∂x1 m ∂x  n n  the equations (4) and (5) become

∂Φ Ψ + H(x, Φ,t)+ Q (x,t) = 0 (8) ∂t ∇x ∂ρ Ψ 2 + div(ρv )=0 , ρ = R (9) ∂t The main postulate of the Bohmian theory of motion is that particles follow quantum trajectories, and that these trajectories are the solutions of the differential equations Ψ ~ 1 ∂Ψ x˙ j = Im . (10) mj Ψ ∂xj The phase space interpretation is that the Bohmian trajectories are de- termined by the equations Ψ Ψ 1 Ψ Ψ ∂U Ψ ∂Q Ψ x˙ j = pj ,p ˙j = (x ,t) (x ,t). (11) mj −∂xj − ∂xj It is immediate to check that these are just Hamilton’s equations for the Hamiltonian function n 2 Ψ p Ψ H (x,p,t)= j + U(x,t)+ Q (x,t) (12) =1 2mj Xj which can be viewed as a perturbation of the original Hamiltonian H by the quantum potential QΨ (see Holland [13, 14] for a detailed study of quantum trajectories in the context of Hamiltonian mechanics). The Bohmian equations of motion are a priori only defined when R = 0 6 (that is, outside the nodes of the wavefunction); this will be the case in our constructions since for sufficiently small times this condition will be satisfied by continuity if we assume that it is case at the initial time. An important feature of the quantum trajectories is that they cannot cross; thus there will be no conjugate points like those that complicate the usual Hamiltonian dynamics.

3 3 The Short-Time Propagator

The solution Ψ of Schr¨odinger’s equation (3) can be written

Ψ(x,t)= K(x,x0; t)Ψ0(x0)dx0 Z where the kernel K is the “quantum propagator”:

K(x,x0; t)= x exp( iHt/~) x0 . h | − | i Schr¨odinger’s equation (3) is then equivalentb to ∂K i~ = H(x, i~ ,t)K , K(x,x0;0) = δ(x x0) (13) ∂t − ∇x − where δ is the Diracb distribution. Physically this equation describes an isotropic source of point-like particle emanating from the point x0 at initial time t0 = 0. We want to find an asymptotic formula for K for short time intervals ∆t. Referring to the usual literature, such approximations are given by expressions of the type 2 1 n/ i K(x,x0;∆t)= ρ(x,x0;∆t)exp S(x,x0;∆t) 2πi~ ~     p where S(x,x0;∆t) is the action along the classical trajectory from x0 to x in time ∆t and 2 ∂ S(x,x0;∆t) ρ(x,x0;∆t) = det − ∂x ∂x  j k 1≤j,k≤n is the corresponding Van Vleck determinant. It is then common practice (especially in the Feynman path integral literature) to use the following “midpoint approximation” for the generating function S:

n mj 2 1 S(x,x0;∆t) (xj x0) (U(x,t0)+ U(x0,t0))∆t (14) ≈ =1 2∆t − − 2 Xj or, worse,

n mj 2 1 S(x,x0; t,t0) (xj x0) (U( 2 (x + x0),t0)∆t (15) ≈ =1 2∆t − − Xj However, as already pointed out by Makri and Miller [15, 16], these “approx- imations” are wrong; they fail to be correct even to first orderin∆t! In fact,

4 Makri and Miller, and one of us [3] have shown independently, and using different methods, that the correct asymptotic expression for the generating function is given by

n mj 2 2 S(x,x0;∆t)= (xj x0) U(x,x0)∆t + O(∆t ) (16) =1 2∆t − − Xj e where U(x,x0, 0) is the average value of the potential over the straight line joining x0 at time t0 to x at time t with constant velocity:

e 1 U(x,x0)= U(λx + (1 λ)x0, 0)dλ. (17) 0 − Z For instance whene

1 2 2 2 2 H(x,p)= (p + m ω x ) 2m is the one-dimensional harmonic oscillator formula (16) yields the correct expansion

2 m 2 mω 2 2 S(x,x0; t)= (x x0) (x + xx0 + x0)∆t + O(∆t ); (18) 2∆t − − 6 the latter can of course be deduced directly from the exact value

mω 2 S(x,x0; t,t0)= ((x + x0) cos ω∆t 2xx0) (19) 2sin ω∆t − by expanding sin ω∆t and cos ω∆t for ∆t 0. This correct expression is → of course totally different from the erroneous approximations obtained by using the “rules” (14) or (15). Introducing the following notation,

n 2 (xj x0) S(x,x0;∆t)= mj − U(x,x0)∆t, (20) =1 2∆t − Xj e e leads us to the Makri and Miller approximation (formula (17c) in [15]) for the short-time propagator:

n/2 1 i 2 K(x,x0;∆t)= ρ(x,x0;∆t)exp S(x,x0;∆t) + O(∆t ) 2πi~ ~     p (21) e

5 where 2 ∂ S(x,x0;∆t) ρ(x,x0;∆t) = det . ∂x ∂x − j k !1 e ≤j,k≤n It turns out that this formula can be somewhat improved. The Van Vleck determinant ρ(x,x0;∆t) is explicitly given, taking formula (20) into account, by 1 ′′ ρ(x,x0;∆t) = det M U (x,x0)∆t −∆t − x,x0   where M is the mass matrix (the diagonal matrixe with positive entries the masses mj) and 2 ∂ U(x,x0) U ′′ = . x,x0 ∂x ∂x − j k !1 e ≤j,k≤n Writing e

1 ′′ 1 −1 ′′ 2 M U (x,x0)∆t = M[I × M U (x,x0)∆t ] −∆t − x,x0 −∆t n n − x,x0   1 2 e = M[I × + O(∆te)], −∆t n n we have by taking the determinant of both sides

m1 mn 2 ρ(x,x0;∆t)= · · · det I × + O(∆t ) . (∆t)n n n 2 2
Noting that det In×n + O(∆t ) =1+ O(∆t ), we thus have

m1 mn 2 ρ(x,x0;∆t)= · · · (1 + O(∆t )). (22) (∆t)n Writing m1 m ρ(∆t)= · · · n (23) (∆t)n which is just the Van Vleck density for the free particle Hamiltonian. We e thus have 2 ρ(x,x0;∆t)= ρ(∆t)+ O(∆t ) (24) and hence we can rewrite formula (21) as e n/2 1 i 2 K(x,x0;∆t)= ρ(∆t)exp S(x,x0;∆t) + O(∆t ). (25) 2πi~ ~     p We will see below that this formulae allows ane easy study of the quantum potential for K.

6 4 Short-Time Bohmian Trajectories

Let us determine the quantum potential Q corresponding to the propagator K = K(x,x0; t) using the asymptotic formulas above. Recall that it de- scribes an isotropic source of point-like particle emanating from the point x0 at initial time t0 = 0. We have, by definition,

n 2 2 ~ ∂ √ρ Q = 2 − =1 2mj√ρ ∂xj Xj which we can rewrite 2 −1 ~ M x x√ρ Q = ∇ ·∇ − 2 √ρ where M is the mass matrix defined above. We have, using (24),

2 √ρ = ρ(∆t)(1 + O(∆t )) and hence p 2 e ∂ √ρ 2 2 = O((∆t) )). ∂xj From this it follows that the quantum potential associated with the propa- gator satisfies 2 Q(x,x0;∆t)= O(∆t ). (26) The discussion above suggests that the quantum trajectory of a sharply located particle should be identical with the classical (Hamiltonian) tra- jectory for short times. Let us show this is indeed the case. If we want to monitor the motion of a single, we have of course to specify its initial momentum which gives its direction of propagation at time t0 = 0; we set

p(0) = p0. (27)

In view of formula (10), the trajectory in position space is obtained by solving the system of differential equations

−1 M xK x˙ = ~ Im ∇ , x(0) = x0. (28) K Replacing K with its approximation

2 1 n/ i K(x,x0;∆t)= ρ(∆t)exp S(x,x0;∆t) 2πi~ ~     p e e 7 e we have, since K K = O(∆t2) in view of (25), − −1 M K 2 e x˙ = ~ Im ∇x + O(∆t ). K e A straightforward calculation, using the expression (20) for the approximate e action S(x,x0;∆t), leads to (cf. the proof of Lemma 248 in [3]) the relation

x(∆t) x0 −1 2 e x˙(∆t)= − M U(x(∆t),x0)∆t + O(∆t ). (29) ∆t − ∇x This equation is singular at time t = 0 hencee the initial condition x(0) = x0 is not sufficient for finding a unique solution; this is of course consistent with the fact that (29) describes an arbitrary particle emanating from x0; to single out one quantum trajectory we have to use the additional condition (27) giving the direction of the particle at time t = 0 (see the discussion in Holland [12], 6.9). We thus have § −1 2 x(∆t)= x0 + M p0∆t + O(∆t ); in particular x(∆t)= x0 + O(∆t) and hence, by continuity,

U(x(∆t),x0)= U(x0,x0)+ O(∆t). ∇x ∇x Let us calculate U(x0,x0). We have, taking definition (17) into account, ∇x e e 1 xU(ex,x0)= λ xU(λx + (1 λ)x0, 0)dλ ∇ 0 ∇ − Z and hence e 1 1 xU(x0,x0)= λ xU(x0, 0)dλ = xU(x0, 0). ∇ 0 ∇ 2∇ Z We can thus rewritee equation (29) as

x(∆t) x0 1 −1 2 x˙(∆t)= − M U(x0, 0)∆t + O(∆t ). ∆t − 2 ∇x Let us now differentiate both sides of this equation with respect to ∆t:

x(t) x0 x˙(t) 1 −1 x¨(t)= − + M U(x0, 0) + O(∆t) (30) (∆t)2 ∆t − 2 ∇x that is, replacingx ˙(∆t) by the value given by (29),

p˙(t)= Mx¨(t)= U(x0, 0) + O(∆t). (31) −∇x

8 Solving this equation we get 2 p(t)= p0 U(x0, 0)∆t + O(∆t ). (32) −∇x Summarizing, the solutions of the Hamilton equations are given by p0 2 x(∆t)= x0 + ∆t + O(∆t ) (33) m 2 p(∆t)= p0 U(x0, 0)∆t + O(∆t ). (34) −∇x These equations are, up to the error terms O(∆t2) the equations of motion of a classical particle moving under the influence of the potential U; there is no trace of the quantum potential, which is being absorbed by the terms O(∆t2). The motion is thus identical with the classical motion on time scales of order O(∆t2).

5 Conclusion

This result puts Bohm’s original perception, which led him to the causal interpretation, on a firm mathematical footing. He writes [11] Indeed it had long been known that when one makes a certain approximation (WKB) Schr¨odinger’s equation becomes equiva- lent to the classical Hamilton–Jacobi theory. At a certain point I asked myself: What would happen, in the demonstration of this equivalence, if we did not make this approximation? I saw imme- diately that there would be an additional potential, representing a kind of force, that would be acting on the particle. The source of this “force” was the quantum potential. In our approach we see that while any classical potential acts immediately, the quantum potential potential does not. From this fact two consequences of our follow. Firstly, it gives a rigorous treatment of the “watched pot” effect. If we keep observing a particle that, if unwatched, would make a transition from one quantum state to another, will no longer make that transition. The un- watched transition occurs when the quantum potential grows to produce the transition. Continuously observing the particle does not allow the quantum potential to develop so the transition does not take place. For details see section 6.9 in Bohm and Hiley [1]. Secondly, in the situation when the quantum potential decreases con- tinuously with time, the quantum trajectory continuously deforms into a classical trajectory [5]. This means that there is no need to appeal to deco- herence to reach the classical domain.

9 References

[1] D. Bohm, B. J. Hiley,The Undivided Universe: An Ontological Inter- pretation of Quantum Theory. London & New York: Routledge (1993) [2] M. de Gosson, The quantum motion of half-densities and the derivation of Schr¨odinger’s equation. J. Phys. A:Math. Gen. 31(2), 158–168, 1998 [3] M. de Gosson, The Principles of Newtonian and quantum mechanics. The need for Planck’s constant ~. With a foreword by Basil Hiley. Im- perial College Press, London, 2001. [4] M. de Gosson, B.J. Hiley, Mathematical and Physical Aspects of Quan- tum Processes, Imperial College Press, London, 2014. [5] B.J. Hiley and A.H. Aziz Mufti, The Ontological Interpretation of Quantum Field Theory applied in a Cosmological Context, Funda- mental Problems in Quantum Physics, ed., M. Ferrero and A. van der Merwe, pp.141-156, Kluwer, Dordrecht. (1995)). [6] B.J. Hiley, Non-Commutative Geometry, the Bohm Interpretation and the Mind-Mater Relationship, in Proc. CASYS’2000, Li`ege, Belgium, Aug. 7–12, 2000 [7] B.J. Hiley, Phase Space Descriptions of Quantum Phenomena, Proc. Int. Conf. Quantum Theory: Reconsideration of Foundations 2, 267- 86, ed. Khrennikov, A., V¨axj¨oUniversity Press, V¨axj¨o, Sweden (2003) [8] B.J. Hiley, On the Relationship between the Wigner-Moyal and Bohm Approaches to Quantum Mechanics: A step to a more General Theory. Found. Phys., 40 (2010) 365-367. DOI 10.1007/s10701-009-9320-y [9] B.J. Hiley, R.E. Callaghan, Delayed-choice experiments and the Bohm approach. Phys. Scr. 74 (2006), 336–348 [10] B.J. Hiley, R.E. Callaghan, O.J.E Maroney, Quantum trajectories, real, surreal or an approximation to a deeper process? quant-ph/0010020 [11] D. Bohm, Hidden Variables and the Implicate Order, in B. J. Hiley and D. F. Peat, Quantum Implications: Essays in Honour of David Bohm, p. 35, Routledge, London, 1987. [12] P.R. Holland, The Quantum Theory of Motion: An account of the de Broglie-Bohm causal interpretation of quantum mechanics (Cambridge University Press, 1993).

10 [13] P. Holland, Hamiltonian theory of wave and particle in quantum me- chanics I: Liouville’s theorem and the interpretation of the de Broglie- Bohm theory. Nuovo Cimento B 116 (2001) 1043–1070.

[14] P. Holland, Hamiltonian theory of wave and particle in quantum me- chanics II: Hamilton-Jacobi theory and particle back-reaction. Nuovo Cimento B 116 1143–1172 (2001).

[15] N. Makri, W.H. Miller, Correct short time propagator for Feynman path integration by power series expansion in ∆t, Chem. Phys. Lett. 151 1–8, 1988.

[16] N. Makri, W.H. Miller, Exponential power series expansion for the quantum time evolution operator. J. Chem. Phys. 90, 904–911, 1989.

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