The relevant PDEs can be derived from an action principle. We discuss symmetries of the action principle, associated conserved quantities, and other properties, such as a heretofore unexpected Hamiltonian structure in the case of 1D time-independent quantum mechanics (TIQM), which also serves as the basis of an accurate many-D numerical scheme. We single out a specific choice of Lagrangian or gauge, and show why this choice may be regarded as physically preferred. Likewise, we single out a specific choice of trajectory-labeling coordinate(s), in terms of which the resultant PDE exhibits no explicit coordinate dependences. We present simple analytical solutions, and discuss some initial numerical results that appear very promising for molecular and chemical physics applications. The reformulation of quantum mechanics in terms of trajectory ensembles, in addition to shedding light on complex theoretical issues, evidently also provides important practical benefits.

The aim of this paper is to show that for non-relativistic spin-freethe abovebe realized. (A different approach is described in Ref..) In particular, a quantum state can be represented as an ensemble of real-valued quantum trajectories, satisfying a self-containedThis was shown in a recent paperby one of the authors (Poirier) for both the one-dimensional (1D) TISE and TDSE. It transpired that similar work had already been done for the TISE in 1D,the TDSE in 3D,and in greater generality.In the current paper, we simplify, unify, and generalize these previous constructions, presenting quantum trajectoryfor arbitrary configuration spaces and system dimensionalities. The goal here is to present an alternative, standaloneofthat neither relies on the TDSE nor makes any mention of any external constructs such as Ψ, and which, in addition, is likely to provide far-reaching benefits for numerical calculations, e.g., of accurate quantum scatteringfor chemically reactive molecular systems.

For nearly a century,has presented philosophical and interpretational conundrums that remain as controversial as ever. Far from disappearing into the realm of esoteric academic debate, recent experimental advances, e.g., indecoherence, and quantum computing, have brought such questions to the forefront of topical interest. The various competing viewpoints—Copenhagen,Bohmian,many worlds,etc.—differ substantially in terms of their ontological interpretation of Ψ and its collapse, yet are all alike in their reliance upon a complex-valuedand its propagation via the time-independent or time-dependent Schrödinger(TI/DSE). Evidently, a completely self-contained, classical-like, and real-valued formulation ofwavefunctions—if such a thing were possible—would necessarily present novel and potentially very important interpretational and computational ramifications.

The above Hamiltonian approach is proving extremely useful in numerical calculations of quantum reactive scattering phenomena.For any 1D TIQM application,results are obtained simply by propagating a single) trajectory using theHamilton'sof Eq. (6) , until) flattens asymptotically to the desired level of numerical accuracy. The finalvalue then provides a direct measure of the quantum reaction probability. Note that Eq. (6) is amenable to efficient, symplectic numerical integrators as are used in classical simulations, and the conservation ofcan be monitored as an on-the-fly measure of computed accuracy. This approach is extremely robust, accurate, and efficient, leading to 15 digits of accuracy, even in the extremely deep tunneling regime (where absorbing potentials render conventional exact quantum scattering methods intractible). For many-D TIQM applications, classical-like sampling over quantum trajectory initial conditions leads to an approximate simulation scheme that has also proven to be remarkably accurate (i.e., to two or three digits).

In terms of the time derivatives of),Substitution of Eq. (8) into Eq. (7) then revealsto be the conserved Noether energy of Eq. (2) . For the free particle case [∂/∂= 0],is a second conserved quantity, in involution with(and also derivable from Noether's theorem). The importance ofis difficult to overstate; it represents the “particle momentum,” analogous to the well-known “particle energy,”Yet remarkably,has barely been consideredin the previous literature, which generally considersas particle momentum. Note thatis conserved forfree particle TIQM states, including those exhibiting interference, whereasis conservedfor plane wave states, i.e., the classical special case for which

Any fourth-order ODE can be rewritten as a set of four coupled, first-order ODE's. Remarkably, Eq. (4) can be rewritten aswhich arefor the Hamiltonian,In the above) are the “classical” dimension phase space variables, and () correspond to an additional, “quantum” dimension, essentially describing quantum interference. Note that Eq. (7) reduces to the classical Hamiltonian when= 0 and

Action extremization applied to Eqs. (1) and (3) yields the following fourth-order autonomous ordinary differential(ODE), describing 1D TIQM quantum trajectories:Two of the four constants of integration correspond to time-translation invariance and energy, as in the classical case. The other two constants determine which particular TIQM state the trajectory is associated with. A one-to-one correspondence thus exists between trajectory solutions [of Eq. (4) ] and (scattering) TIQM states.

As in classicalthe quantum trajectories,), are obtained via extremization of the action,= ∫. Sinceis autonomous, the resultant) solutions exhibit time-translation invariance and energy conservation for any choice of. However, for general, the conserved energy for Eq. (1) obtained via Noether's theorem does not take the form of Eq. (2) . Requiring this equivalence imposes rather special conditions on, e.g., thatmust be invariant under time rescaling. The simplest, well-behaved, nontrivialgiving the form of Eq. (2) isHere ℏ is, in principle, an arbitrary positive constant. However, with the usual identification of ℏ as Planck's constant, Eq. (3) is equivalent to the quantum potential of Bohmian mechanicsfor the 1D Cartesian TISE.This is surprising, given that the expression Eq. (3) , being universal and kinematic, is determined entirely by the trajectory,). In particular, no reference to theΨ, or the TISE itself, is used in this derivation. Our quantum trajectories are nevertheless Bohmian trajectories, although our formulation and interpretation are not at all that of Bohmian mechanics, because no Ψ is involved.

As shown in Ref., 1D TIQM states can be represented uniquely with a single trajectory,). Theis one of a broad class of dynamical laws, for which theand the energy are of the form (1) and (2) are natural generalizations of the well-known classical forms. The quantum correction,, is similar to the potential, in that it appears with opposite signs inand, but is actually related to the TISE kinetic energy operator.It has a universal “kinematic” form, i.e., no explicitdependence.

III. THE 1D TIME-DEPENDENT CASE Section: Choose Top of page ABSTRACT I.INTRODUCTION II.THE 1D TIME-INDEPENDEN... III.THE 1D TIME-DEPENDENT... << IV.THE MANY-D TIME-DEPEND... V.CONCLUDING REMARKS REFERENCES CITING ARTICLES

For the case of 1D time-dependent quantum mechanics (TDQM), any self-contained formulation must involve a PDE, rather than an ODE. It is no longer possible to exactly represent a quantum state as a single trajectory, x(t), but rather as a one-parameter ensemble of trajectories, x(C, t), where the real-valued, space-like coordinate C labels individual trajectories. The equation of motion should be a PDE involving C and t partial derivatives, preferably derived from a field-theoretic action principle.

7 and 9 370, 4 (2010). 7. B. Poirier, Chem. Phys., 4 (2010). https://doi.org/10.1016/j.chemphys.2009.12.024 315, 505 (2005). 9. P. Holland, Ann. Phys., 505 (2005). https://doi.org/10.1016/j.aop.2004.09.008 10 461, 3659 (2005). 10. P. Holland, Proc. R. Soc. London, Ser. A, 3659 (2005). https://doi.org/10.1098/rspa.2005.1525 C was chosen as the initial trajectory value [x(C, 0) = x 0 = C]. The resultant PDE is complicated, exhibits explicit x 0 dependence through the initial probability density, ρ 0 (x 0 ) = ρ(x 0 , 0), and bears little resemblance to Eq. Q is expressed in terms of C rather than t derivatives of x. The PDE can be simplified by a better choice of the trajectory parameter, C, which in general can be taken to be any monotonic function of x 0 (regardless of the initial wavefunction). A crucial idea of the current paper is that C should be chosen so as to uniformize the probability density . In particular, since ρ C (C) dC = ρ(x, t) dx, if we choose C = ∫ − ∞ x 0 ρ 0 ( x 0 ′ ) d x 0 ′ , (9) then C takes values from 0 to 1 (for normalized wavepackets), and ρ C (C) = 1. In Refs., andwas chosen as the initial trajectory value [, 0) =]. The resultantis complicated, exhibits explicitdependence through the initial) = ρ(, 0), and bears little resemblance to Eq. (4) . In addition,is expressed in terms ofrather thanderivatives of. Thecan be simplified by a better choice of the trajectory parameter,, which in general can be taken to be any monotonic function of(regardless of the initialA crucial idea of the current paper is that. In particular, since ρ= ρ(, if we choosethentakes values from 0 to 1 (for normalized wavepackets), and ρ) = 1.

C), and writing x′ = ∂x/∂C, x ̇ = ∂ x / ∂ t etc., the PDE of Ref. 7 370, 4 (2010). 7. B. Poirier, Chem. Phys., 4 (2010). https://doi.org/10.1016/j.chemphys.2009.12.024 m x ̈ + ∂ V ( x ) ∂ x + ℏ 2 4 m x ′ ′ ′ ′ x ′ 4 − 8 x ′ ′ ′ x ′ ′ x ′ 5 + 10 x ′ ′ 3 x ′ 6 = 0 . (10) Equation equation for the 1D TDQM case; it has no explicit coordinate dependences. It also bears an extremely close resemblance to Eq. C derivatives with t derivatives in the last term on the left-hand side (representing the quantum force). More formally, Eq. x(C, t) = x(t − λC), where λ is a constant. In the 1D TIQM context, t thus serves as an effective uniformizing coordinate. Equation t-independent solutions that correspond to the bound (fluxless) 1D TIQM quantum states (Sec. Working with Eq. (9) (or any uniformizing choice of), and writing= ∂/∂etc., theof Ref.simplifies very substantially to (10) is the perturbed Newtonfor the 1D TDQM case; it has no explicit coordinate dependences. It also bears an extremely close resemblance to Eq. (4) , obtained by replacingderivatives withderivatives in the last term on the left-hand side (representing the quantum force). More formally, Eq. (4) is obtained on looking for travelling wave solutions of Eq. (10) , i.e., solutions of the form) =− λ), where λ is a constant. In the 1D TIQM context,thus serves as an effective uniformizing coordinate. (10) also admits-independent solutions that correspond to the(fluxless) 1D TIQM quantum states (Sec. II concerns only the scattering states).

equation, obtained by extremizing the action, ∫ ∫ d C d t 1 2 m x ̇ 2 − V ( x ) − ℏ 2 4 m x ′ ′ ′ x ′ 3 − 5 2 x ′ ′ 2 x ′ 4 , (11) cf. Eqs. t and C. By Noether's theorem, this gives rise to two conservation laws, which are easily found to be, respectively, ∂ ∂ t 1 2 m x ̇ 2 + V ( x ) + ℏ 2 4 m x ′ ′ ′ x ′ 3 − 5 2 x ′ ′ 2 x ′ 4 + ℏ 2 4 m ∂ ∂ C x ′ ′ ′ x ′ 4 − 2 x ′ ′ 2 x ′ 5 x ̇ + 2 x ′ ′ x ̇ ′ x ′ 4 − x ̇ ′ ′ x ′ 3 = 0 , (12) ∂ ∂ t [ m x ̇ x ′ ] + ∂ ∂ C − 1 2 m x ̇ 2 + V ( x ) + ℏ 2 4 m x ′ ′ ′ x ′ 3 − 5 2 x ′ ′ 2 x ′ 4 = 0 . (13) The first corresponds to conservation of energy. In the free particle case, there is also a momentum conservation law, arising from x-translation symmetry, ∂ ∂ t m x ̇ + ℏ 2 4 m ∂ ∂ C x ′ ′ ′ x ′ 4 − 2 x ′ ′ 2 x ′ 5 = 0 . (14) As in the 1D TIQM case, Eq. (10) is a variationalobtained by extremizing the action,cf. Eqs. (1) and (3) . This action is invariant under translations of both coordinatesand. By Noether's theorem, this gives rise to twowhich are easily found to be, respectively,The first corresponds to conservation of energy. In the free particle case, there is also alaw, arising from-translation symmetry,

equations of the general form ∂A/∂t + ∂B/∂C = 0, the first square bracket in Eq. C-derivative to the density A and subtraction of the corresponding t-derivative from the flux B. Consequently, the energy density need not conform to the standard TDQM “field” form, 3 3. P. R. Holland, The Quantum Theory of Motion ( Cambridge University Press , Cambridge, England , 1993). T + V + Q, as in Eq. Lagrangian T − V − Q [cf. Eqs. momentum conservation law Eq. p in Eq. Interpreted as hydrodynamical balanceof the general form ∂/∂+ ∂/∂= 0, the first square bracket in Eq. (12) [Eq. (14) ] represents the energy [momentum] density, and the second term the corresponding flux. This designation is only determined up to addition of a-derivative to the densityand subtraction of the corresponding-derivative from the flux. Consequently, the energy density need not conform to the standard TDQM “field” form,but instead may be chosen to be the particle energy,, as in Eq. (12) . This choice is appropriate for the[cf. Eqs. (1) and (2) ], and has the great advantage of being conserved along individual trajectories in the TIQM limiting case (in general, only the total ensemble energy is conserved). However, in thelaw Eq. (14) , it does not seem to be possible to use a density that reduces to the particle momentumin Eq. (8)

equation, Eq. PDE, the Lagrangian density L is determined only up to the addition of a divergence (i.e., the sum of C- and t-derivatives). We have chosen forms of the action [Eq. conservation laws such that L appears as (minus) the flux in Eq. Lagrangian and energy densities, but not from the flux of Eq. L we have made has the advantage that the trajectory action, S ( C , t ) = ∫ 0 t L ( C , t ′ ) d t ′ , expressed in units of ℏ, can be identified with the change in phase of Ψ. Also of note is the balanceEq. (13) . For any autonomous Euler-Lagrangethedensityis determined only up to the addition of a divergence (i.e., the sum of- and-derivatives). We have chosen forms of the action [Eq. (11) ] and thesuch thatappears as (minus) the flux in Eq. (13) . Through gauge transformations, it is possible to eliminate the third-order derivative from theand energy densities, but not from the flux of Eq. (13) . The choice ofwe have made has the advantage that the trajectory action,, expressed in units of ℏ, can be identified with the change in phase of Ψ.

V = 0) and harmonic oscillator ( V = 1 2 m ω 2 x 2 ) potentials. The respective x(C, t) solutions are x 0 + p 0 ( t − t 0 ) m + a erfinv ( 2 C − 1 ) 1 + ℏ 2 ( t − t 0 ) 2 m 2 a 4 , (15) and x 0 cos ω ( t − t 0 ) + p 0 sin ω ( t − t 0 ) m ω + a erfinv ( 2 C − 1 ) × cos 2 ω ( t − t 0 ) + ℏ 2 sin 2 ω ( t − t 0 ) m 2 a 4 ω 2 , (16) where x 0 , p 0 , t 0 , and a are real wavefunction parameters, and 0 ⩽ C ⩽ 1. Note these solutions diverge as C → 0 or 1. In general, x(C, t) must diverge at the C endpoints; modulo this requirement, any solution of Eq. C generalization) can be used to reconstruct a normalized solution Ψ(x, t) of the 1D TDSE. We now consider Gaussian wavepacket evolution under the free particle (= 0) and harmonic oscillator () potentials. The respective) solutions areandwhere, andare realparameters, and 0 ⩽⩽ 1. Note these solutions diverge as→ 0 or 1. In general,) must diverge at theendpoints; modulo this requirement, any solution of Eq. (10) (or its arbitrary-generalization) can be used to reconstruct a normalized solution Ψ() of the 1D TDSE.