Abstract
The nonlinear logarithmic Schrödinger equation (log SE) appears in many branches of fundamental physics, ranging from macroscopic superfluids to quantum gravity. We consider here a model problem, in which the log SE includes an attractive Coulomb interaction. We derive an analytical solution for the ground state energy and wave function as a function of the strength of the logarithmic interaction. We develop an iterative finite element method to solve the Coulombic log SE for the spherically symmetric states. The ground state results agree with the exact solution to better than one part in 1010. The excited states (n>1) are converged to better than one part in 108. We also construct a remarkably simple variational wave function, consisting of a sum of Gaussons with n free parameters. One can obtain an approximation to the energy and wave function that is in good agreement with the finite element results. Although the Coulomb problem is interesting in its own right, the iterative finite element method and the variational Gausson basis approach can be applied to any central force Hamiltonian.
Highlights
The logarithmic Schrödinger equation is one of the nonlinear modifications of Schrödinger’s equation, with applications in quantum optics [1, 2], nuclear physics [3, 4], diffusion phenomena [5], stochastic quantum mechanics [6], effective quantum gravity [7] and Bose–Einstein condensation [8, 9]
We have obtained an analytical solution to the ground state for the logarithmic Schrödinger equation (log SE) with a Coulomb interaction
A simple variational wave function consisting of a sum of n Gaussons is shown to provide an excellent approximation for the excited states
Summary
Original content from this work may be used under the terms of the Creative Abstract. We consider here a model this work must maintain attribution to the problem, in which the log SE includes an attractive Coulomb interaction. We derive an analytical author(s) and the title of the work, journal citation solution for the ground state energy and wave function as a function of the strength of the logarithmic and DOI. We develop an iterative finite element method to solve the Coulombic log SE for the spherically symmetric states. The ground state results agree with the exact solution to better than one part in 1010. One can obtain an approximation to the energy and wave function that is in good agreement with the finite element results. The Coulomb problem is interesting in its own right, the iterative finite element method and the variational Gausson basis approach can be applied to any central force
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