Phase aspect of nonlinear modifications of the Schroedinger equation

We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The solutions include a coherent state, a phase-modified Gaussian wave packet in the potential of harmonic oscillator whose strength varies in time, a free Gaussian soliton, and a similar soliton in the potential of harmonic oscillator comoving with the soliton. The last of these solutions implies that there exist an energy level in the spectrum of harmonic oscillator which is not predicted by the linear theory. The free solitonic solution can be considered a model for a particle aspect of the wave-particle duality embodied in the quantum theory. The physical size of this particle is naturally rendered equal to its Compton wavelength in the subrelativistic framework in which the self-energy of the soliton is assumed to be equal to its rest-mass energy. The solitonic solutions exist only for the negative coupling constant for which the Gaussian wave packets must be larger than some critical finite size if their energy is to be bounded, i.e., they cannot be point-like objects.We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The solutions include a coherent state, a phase-modified Gaussian wave packet in the potential of harmonic oscillator whose strength varies in time, a free Gaussian soliton, and a similar soliton in the potential of harmonic oscillator comoving with the soliton. The last of these solutions implies that there exist an energy level in the spectrum of harmonic oscillator which is not predicted by the linear theory. The free solitonic solution can be considered a model for a particle aspect of the wave-particle duality embodied in the quantum theory. The physical size of this particle is naturally rendered equal to its Compton wavelength in the subrelativistic framework in which the self-energy of the soliton is assumed to be equal to its rest-mass energy. The solitonic solutions exist only for the negative coupling constant for which the Gaussian wave packets must be larger than some critical finite size if their energy is to be bounded, i.e., they cannot be point-like objects.

arXiv:quant-ph/9903010Mar 4, 19991 citation

Nonlinear Phase Modification of the Schroedinger Equation

Waldemar Puszkarz

A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in $Δ^{m}\ln (Ψ^{*}/Ψ)$ multiplied by $Ψ^{*}Ψ$ . This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase $S$ rather than the amplitude $R$ of the wave function $Ψ=R\exp (iS)$ . 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 $n$ 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 nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in $Δ^{m}\ln (Ψ^{*}/Ψ)$ multiplied by $Ψ^{*}Ψ$. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase $S$ rather than the amplitude $R$ of the wave function $Ψ=R\exp (iS)$. 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 $n$ 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.

arXiv:quant-ph/9710010Oct 2, 19971 citation

ASPECTS OF NONLINEAR QUANTUM MECHANICS

Marek Czachor

Jan 1, 19934 citations

On the Staruszkiewicz Modification of the Schroedinger Equation

Waldemar Puszkarz

We discuss Staruszkiewicz's nonlinear modification of the Schrödinger equation. It is pointed out that the expression for the energy functional for this modification is not unique as the field-theoretical definition of energy does not coincide with the quantum-mechanical one. As a result, this modification can be formulated in three different ways depending on which physically relevant properties one aims to preserve. Some nonstationary one-dimensional solutions for suitably chosen potentials, including a KdV soliton, are presented, and the question of finding stationary solutions is also discussed. The analysis of physical and mathematical features of the modification leads to the conclusion that the Staruszkiewicz modification is a very subtle modification of the fundamental equation of quantum mechanics.We discuss Staruszkiewicz's nonlinear modification of the Schrödinger equation. It is pointed out that the expression for the energy functional for this modification is not unique as the field-theoretical definition of energy does not coincide with the quantum-mechanical one. As a result, this modification can be formulated in three different ways depending on which physically relevant properties one aims to preserve. Some nonstationary one-dimensional solutions for suitably chosen potentials, including a KdV soliton, are presented, and the question of finding stationary solutions is also discussed. The analysis of physical and mathematical features of the modification leads to the conclusion that the Staruszkiewicz modification is a very subtle modification of the fundamental equation of quantum mechanics.

arXiv:quant-ph/9912006Dec 2, 19992 citations

A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in

Δ^{m}\ln (Ψ^{*}/Ψ)

multiplied by

Ψ^{*}Ψ

. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase

S

rather than the amplitude

R

of the wave function

Ψ=R\exp (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

n

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.