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arXiv:1601.03993 · WORLD SCIENTIFIC (EUROPE) eBooks

The Quantum State as Spatial Displacement

Peter Holland

Published Nov 14, 2016• 5 citations
arXivHTMLPDF

Abstract

We give a simple demonstration that the Schrödinger equation may be recast as a self-contained second-order Newtonian law for a congruence of spacetime trajectories. This provides a pictorial representation of the quantum state as the displacement function of the collective whereby quantum evolution is represented as the deterministic unfolding of a continuous coordinate transformation. Introducing gauge potentials for the density and current density it is shown that the wave-mechanical and trajectory pictures are connected by a canonical transformation. The canonical trajectory theory is shown to provide an alternative basis for the quantum operator calculus and the issue of the observability of the quantum state is examined within this context. The construction illuminates some of the problems involved in connecting the quantum and classical descriptions.

DOI
10.1142/9781786341419_0010

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We give a simple demonstration that the Schrödinger equation may be recast as a self-contained second-order Newtonian law for a congruence of spacetime trajectories. This provides a pictorial representation of the quantum state as the displacement function of the collective whereby quantum evolution is represented as the deterministic unfolding of a continuous coordinate transformation. Introducing gauge potentials for the density and current density it is shown that the wave-mechanical and trajectory pictures are connected by a canonical transformation. The canonical trajectory theory is shown to provide an alternative basis for the quantum operator calculus and the issue of the observability of the quantum state is examined within this context. The construction illuminates some of the problems involved in connecting the quantum and classical descriptions.
Abstract We observe that Schrödinger’s equation may be written as two real coupled Hamilton–Jacobi (HJ)-like equations, each involving a quantum potential. Developing our established programme of representing the quantum state through exact free-standing deterministic trajectory models, it is shown how quantum evolution may be treated as the autonomous propagation of two coupled congruences. The wavefunction at a point is derived from two action functions, each generated by a single trajectory. The model shows that conservation as expressed through a continuity equation is not a necessary component of a trajectory theory of state. Probability is determined by the difference in the action functions, not by the congruence densities. The theory also illustrates how time-reversal symmetry may be implemented through the collective behaviour of elements that individually disobey the conventional transformation ( T\mathrm{T}T T ) of displacement (scalar) and velocity (reversal). We prove that an integral curve of the linear superposition of two vectors can be derived algebraically from the integral curves of one of the constituent vectors labelled by integral curves associated with the other constituent. A corollary establishes relations between displacement functions in diverse trajectory models, including where the functions obey different symmetry transformations. This is illustrated by showing that a ( T\mathrm{T}T T -obeying) de Broglie-Bohm trajectory is a sequence of points on the (non- T\mathrm{T}T T ) HJ trajectories, and vice versa.
We observe that Schrodinger's equation may be written as two real coupled Hamilton-Jacobi (HJ)-like equations, each involving a quantum potential. Developing our established programme of representing the state through exact free-standing deterministic trajectory models, it is shown how quantum evolution may be treated as the autonomous propagation of two coupled congruences. The wavefunction at a point is derived from two action functions, each generated by a single trajectory. The model shows that conservation as expressed through a continuity equation is not a necessary component of a trajectory theory of state; probability is determined by the difference in the actions, and we prove that it is not generated by the densities of the two HJ congruences. The theory also illustrates how time-reversal symmetry may be implemented by transformations that disobey the conventional transformation (T) of displacement components (scalars) and velocity (reversal); rather, the time-reversal transform of one flow is the T-reversal of the other. We prove a theorem whereby an integral curve of the linear superposition of two vectors is derived algebraically from the integral curves of one of the constituent vectors labelled by integral curves associated with the other constituent. A corollary establishes relations between displacement functions in diverse trajectory models, in particular where the functions obey different symmetry transformations. This is illustrated by showing that a (T-obeying) de Broglie-Bohm trajectory is a sequence of points on the (non-T) HJ trajectories, and vice versa.
Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.
By re-expressing the generalized Schrodinger equation in its coordinate representation in terms of amplitude and phase of the state vector, a quantum-mechanical change of a system can be made to correspond to an ensemble of classical motions of the system to which is added some internal potential. This ensemble may be taken as the statistical. This enables an alternative formulation of quantum mechanics with which is associated that classical picture. Such a formulation leads, however, if developed straightforwardly according to the picture, to a theoretical scheme quite different from that of ordinary quantum mechanics concerning the problems other than the equation of motion. Furthermore, this formulation is not applicable to Fermions when spin, or exclusion principle is taken into account. Bohm's recent renewed form of the statistical interpretation of quantum mechanics referring to this formulation is criticized in several points. On this formulation we can construct a singular ensemble which is shown to correspond to the transformation kernel of the wave function to infinitesimally later instant, and thus a connection between this formulation and that of Feynman's is taken out. Hydrodynamical picture formally equivalent to the statistical picture in one-body problems is also considered, and from such analogy certain formal generalization of Schrodinger equation is suggested.
An adapted representation of quantum mechanics sheds new light on the relationship between quantum states and classical states. In this approach the space of quantum states splits into a product of the state space of classical mechanics and a Hilbert space, and expectation values of observables decompose into their classical value plus a quantum correction. The splitting is preserved under time evolution of the Schrödinger equation under certain assumptions, and the time evolution of the classical part of a quantum state is governed by Hamilton's equation. The new representation is obtained from the usual Hilbert space representation of quantum mechanics by introducing a gauge degree of freedom in a time-dependent unitary transformation, followed by a non-conventional gauge fixing condition.
We present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum-classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time-dependent Schrödinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase-space Schrödinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (ℏ→0) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantum-classical correspondence, and quasiclassical trajectory simulations of quantum dynamics.