Solution of the logarithmic Schrödinger equation with a Coulomb potential

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 10 10 . The excited states (n>1) are converged to better than one part in 10 8. 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.

Solution of the 3D logarithmic Schrödinger equation with a central potential

“…where n and n′ represent the collective quantum numbers of two different physical states. (See appendix A of [20] for a derivation.) The state y n is orthogonal (in the normal sense) to y k which are solutions of…”

“…In this special case, = l 0 and = m 0 are good quantum numbers. The ground state (which we label s 1 ) is exactly solvable [20], and the solution is…”

“…The exactly solvable s 1 ground state serves as an important check on the accuracy of the method. (Plots of the radial functions are given in [20]).…”

“…We recently considered the special case of the log SE with a Coulomb potential and arbitrary S [20]. We obtained an exact analytical solution for the ground state.…”

Temperature-driven dynamics of quantum liquids: Logarithmic nonlinearity, phase structure and rising force

“…Secondly, the presented analysis is based on a simplified setup of the irrotational superfluid being free of any external forces and containers (logarithmic fluids in certain types of external potential traps, such as harmonic or Coulomb potentials, were discussed in, e.g., refs. [19,22,26]). If one considers other physical setups then both the value and direction of this additional acceleration can be different from formula (21), but the fundamental reasons for the effect remain intact.…”