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

What is it about?

Some form of the time-independent logarithmic Schrödinger equation (log SE) arises in almost every branch of physics. Nevertheless, little progress has been made in obtaining analytical or numerical solutions due to the nonlinearity of the logarithmic term in the Hamiltonian. Even for a central potential, the Hamiltonian does not commute with L^2 or Lz; the Hamiltonian is invariant under the parity operation only if the wave function is an eigenstate of the parity operator. We show that the solutions with well-defined parity can be expressed as a linear combination of eigenstates of L^2 and Lz, where the parity restrictions on l and m determine the nodal structure of the wave function. The dominant contribution in the sum is designated as l ̃ and m ̃ ; these serve as approximate quantum numbers. Using an iterative finite element approach, we also carry out fully converged numerical calculations in 1D, 2D and 3D for the special case of a Coulomb potential. The nodal structure of the wave functions and the expectation values <L^2> and are consistent with the analytical predictions. Values for the energy and expectations values are tabulated for the low-lying states. The methods developed for this problem are applicable across many areas of physics.

Featured Image

Photo by G-R Mottez on Unsplash

Photo by G-R Mottez on Unsplash

Why is it important?

This work shows how we can practically do quantum mechanics with the Logarithmic Schrodinger equation (LogSE) which is the equation governing the dilaton field in dilatonic quantum gravity. This work is a computational tour-de-force from the Finite Element Methods (FEM) of Janine Shertzer.

Perspectives

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This is a building block towards a theory of quantum gravity: one which starts with General Relativity (GRT) and leads to the Logarithmic Schrodinger Equation (LogSE). This particular shows the quantum mechanical computational options in accurately solving it.

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