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arXiv:quant-ph/9710008

Relativistically Extended Modification of the Schroedinger Equation

Waldemar Puszkarz

Published Oct 2, 1997• 1 citation
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Abstract

We propose a nonlinear modification of the Schrödinger equation that possesses the main properties of this equation such as the Galilean invariance, the weak separability of composite systems, and the homogeneity in the wave function. The modification is derived from the relativistic relation between the energy and momentum of free particle and, as such, it is the best relativistic extension of the Schrödinger equation that preserves the properties in question. The only change it effectively entails in the Schrödinger equation involves the conserved probability current. It is pointed out that it partially retains the linear superposition principle and that it can be used to model the process of decoherence.

DOI
10.48550/arxiv.quant-ph/9710008

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We propose a nonlinear modification of the Schrödinger equation that possesses the main properties of this equation such as the Galilean invariance, the weak separability of composite systems, and the homogeneity in the wave function. The modification is derived from the relativistic relation between the energy and momentum of free particle and, as such, it is the best relativistic extension of the Schrödinger equation that preserves the properties in question. The only change it effectively entails in the Schrödinger equation involves the conserved probability current. It is pointed out that it partially retains the linear superposition principle and that it can be used to model the process of decoherence.
A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in Δmln⁡(Ψ∗/Ψ)Δ^{m}\ln (Ψ^{*}/Ψ)Δmln(Ψ∗/Ψ) multiplied by Ψ∗ΨΨ^{*}ΨΨ∗Ψ. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase SSS rather than the amplitude RRR of the wave function Ψ=Rexp⁡(iS)Ψ=R\exp (iS)Ψ=Rexp(iS). From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to nnn particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.
A Schrödinger-like equation for a single free quantum particle is presented. It is argued that this equation can be considered a natural relativistic extension of the Schrödinger equation for energies smaller than the energy associated to the particle’s mass. Some basic properties of this equation: Galilean invariance, probability density, and relation to the Klein-Gordon equation are discussed. The scholastic value of the proposed Grave de Peralta equation is illustrated by finding precise quasi-relativistic solutions for the infinite rectangular well and the quantum rotor problems. Consequences of the non-linearity of the proposed equation for the quantum superposition principle are discussed.
We present an extension of Staruszkiewicz's modification of the Schrödinger equation which preserves its main and unique feature: in the natural system of units the modification terms do not contain any dimensional constants. The extension, similarly as the original, is formulated in a three-dimensional space and derives from a Galilean invariant Lagrangian. It is pointed out that this model of nonlinearity violates the separability of compound systems in the fundamentalist approach to this issue. In its general form, this modification does not admit stationary states for all potentials for which such states exist in linear quantum mechanics. This is, however, possible for a suitable choice of its free parameters. It is only in the original Staruszkiewicz modification that the energy of these states remains unchanged, which marks the uniqueness of this variant of the modification.
The free Schrodinger equation is shown to be a consequence of space-time homogeneity in the non-relativistic domain. First, we show that space-time translation invariance of natural laws, which is a conseqiuence of space-time homogeneity, entails that the state of a free particle with definite momentum and energy assumes the plane wave form when the time evolution of the state is linear. Next, we show that the conservation of energy and momentum, which is a consequence of space-time translation invariance, may further determine the energy-momentum relation In the non-relativistic domain. These results then lead to the free Schrodinger equation naturally. The new integrated analysis may help understand the origin of the wave equations in quantum theory.
We modify the Schrödinger equation in a way that preserves its main properties but makes use of higher order derivative terms. Although the modification represents an analogy to the Doebner-Goldin modification, it can differ from it quite distinctively. A particular model of this modification including derivatives up to the fourth order is examined in greater detail. We observe that a special variant of this model partially retains the linear superposition principle for the wave packets of standard quantum mechanics remain solutions to it. It is a peculiarity of this variant that a periodic structure emerges naturally from its equations. As a result, a free particle, in addition to a plane wave solution, can possess band solutions. It is argued that this can give rise to well-focused particle trajectories. Owing to this peculiarity, when interpreted outside quantum theory, the equations of this modification could also be used to model pattern formation phenomena.
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.