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Physical Review A

Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics

Ernst F. Hefter

Published Aug 1, 1985• 144 citations
Publisher

Abstract

Recently a nonlinear Schr\"odinger equation (NLSE) with an inhomogeneous term proportional to $b\mathrm{ln}({|\ensuremath{\psi}|}^{2}|{a}^{3})\ensuremath{\psi}$ has been put forward. It has been proposed to apply it to atomic physics. Subsequent neutron interferometer experiments designed to test the physical reality of such a nonlinearity were not conclusive, thus rejecting it as unphysical. In the present paper it is pointed out that the different length scales $a$ associated with atomic and nuclear physics, for example, lead to different typical energies $b$ for these systems. Guided by the experience with phenomenological NLSE's, the constant $b$ is for the following applications to nuclear physics identified with the compressibility of finite nuclear matter, $C=\frac{K}{9}$, i.e., $b\ensuremath{\equiv}C$. Thus we obtain consistent qualitative and quantitative answers related to the concepts of microworlds and mesoworlds as well as, e.g., the prediction $130<~K<~250$ MeV. However, this necessitates the interpretation of the respective NLSE as an equation for extended objects.

DOI
10.1103/physreva.32.1201

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Recently a nonlinear Schr"odinger equation (NLSE) with an inhomogeneous term proportional to bln(∣\ensuremathψ∣2∣a3)\ensuremathψb\mathrm{ln}({|\ensuremath{\psi}|}^{2}|{a}^{3})\ensuremath{\psi}bln(∣\ensuremathψ∣2∣a3)\ensuremathψ has been put forward. It has been proposed to apply it to atomic physics. Subsequent neutron interferometer experiments designed to test the physical reality of such a nonlinearity were not conclusive, thus rejecting it as unphysical. In the present paper it is pointed out that the different length scales aaa associated with atomic and nuclear physics, for example, lead to different typical energies bbb for these systems. Guided by the experience with phenomenological NLSE's, the constant bbb is for the following applications to nuclear physics identified with the compressibility of finite nuclear matter, C=K9C=\frac{K}{9}C=9K​, i.e., b\ensuremath≡Cb\ensuremath{\equiv}Cb\ensuremath≡C. Thus we obtain consistent qualitative and quantitative answers related to the concepts of microworlds and mesoworlds as well as, e.g., the prediction 130< K< 250130<~K<~250130< K< 250 MeV. However, this necessitates the interpretation of the respective NLSE as an equation for extended objects.
Evolution by the Gross-Pitaevskii equation, which describes Bose-Einstein condensates under certain conditions, solves the unstructured search problem more efficiently than does the Schr\"odinger equation, because it includes a cubic nonlinearity, proportional to ${|\ensuremath{\psi}|}^{2}\ensuremath{\psi}$. This is not the only nonlinearity of the form $f(|\ensuremath{\psi}{|}^{2})\ensuremath{\psi}$ that arises in effective equations for the evolution of real quantum physical systems, however: The cubic-quintic nonlinear Schr\"odinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schr\"odinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a ``database'' when there is a single correct answer may be easier than searching when there are multiple correct answers. In each of these cases the nonlinear equation is an effective approximation to a multiparticle Schr\"odinger equation, for search by which Grover's algorithm is optimal. Thus our results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically.
Via Nelson's stochastic mechanics, a method for the solution of the hydrodynamical version of the logarithmic nonlinear Schr\"odinger equation (LNLSE), with a time-dependent forced-harmonic-oscillator potential, is presented. Based on a new interpretation of the interplay between dispersion and nonlinearity, a revealing general spreading-wave-packet solution is found. The complete stochastic process associated with the LNLSE is also derived and decomposed into underlying processes of independent nature (classical and quantum). Physical consequences and conditions which allow the existence of solitonlike (nonspreading) solutions are described: Among them, it is found that the zero-point energy is given by ${\ensuremath{\epsilon}}_{0}$=\ensuremath{\Elzxh}${\ensuremath{\Omega}}_{0}$/2, where ${\ensuremath{\Omega}}_{0}$=\ensuremath{\lambda}/2+[(\ensuremath{\lambda}/2)+${\ensuremath{\omega}}_{0}^{2}$] 1/2^{1/2}1/2 with ${\ensuremath{\omega}}_{0}$ and \ensuremath{\lambda} being the harmonic frequency and the nonlinearity strength, respectively.
By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schrödinger equation (INLSE). This is possible due to the connection between the INLSE and the nonlinear Schrödinger equation (NLSE), which can be established from a treatment based on Lie point symmetries and is related with Schrödinger equation, under certain conditions. As an illustrative example, we construct exact solutions for the INLSE through a Pösch-Teller potential with a single bound state.
Numerical solution of nonlinear Schrodinger equation (NLSE) can be used to describe an inhomogeneous,weakly interact Bose-Einstein condensate in a small harmonic trap potential at zero temperature. In one dimension situation, we find solutions for the NLSE for ground-state condensate wave function, and obtain the ground-state condensate energy per particle. Furthermore, with the time-dependent NLSE, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering length in one-dimensional (1D) system for a fixed number of atoms, and the negative scattering length solutions have solitonlike properties.
Abstract Inspired by the generalized uncertainty principle, which adds gravitational effects to the standard description of quantum uncertainty, we extend the exact uncertainty principle approach by Hall and Reginatto (2002 J. Phys. A: Math. Gen. 35 3289 ), and obtain a (quasi)nonlinear Schrödinger equation. This quantum evolution equation of unusual form, enjoys several desired properties like separation of non-interacting subsystems or plane-wave solutions for free particles. Starting with the harmonic oscillator example, we show that every solution of this equation respects the gravitationally induced minimal position uncertainty proportional to the Planck length. Quite surprisingly, our result successfully merges the core of classical physics with non-relativistic quantum mechanics in its extremal form. We predict that the commonly accepted phenomenon, namely a modification of a free-particle dispersion relation due to quantum gravity might not occur in reality.
Although quantum mechanics is linear, there are nevertheless quantum systems with multiple interacting particles in which the effective evolution of a single particle is governed by a nonlinear equation. This includes Bose-Einstein condensates, which are governed by the Gross-Pitaevskii equation, which is a cubic nonlinear Schrodinger equation with a term proportional to /[psi]/ ²[psi] scaling of Grover's algorithm. Since the Gross- Pitaevskii equation effectively approximates the multi- particle Schrodinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic bound on the number of particles needed for this approximation to hold, asymptotically. The Gross-Pitaevskii equation is not the only nonlinearity of the form f(/[psi]/²)[psi] that arises in effective equations for the evolution of real quantum physical systems, however : The cubic-quintic nonlinear Schrodinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrodinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a database'' when there is a single correct answer may be easier than searching when there are multiple correct answers. These results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically. Furthermore, strongly regular graphs, which have no global symmetry, are sufficiently complete for quantum search on them to asymptotically behave like unstructured search. Certain sufficiently complete graphs retain the improved runtime and resource scalings for some nonlinearities, so our scheme for nonlinear, analog quantum computation retains its benefits even when some structure is introduced