Nodal structure of bound-state wave functions for systems with quartic dispersion

TL;DR

This paper investigates the nodal structure of bound-state wave functions in one-dimensional quantum systems with quartic dispersion, using semiclassical and variational methods, revealing breakdown of classical oscillation theorem in forbidden regions.

Contribution

It introduces a detailed analysis of wave function nodes for quartic dispersion systems, extending classical theorems and comparing analytical and variational approaches.

Findings

Classical oscillation theorem breaks down in forbidden regions with quartic dispersion.

Derived quantization condition using complex Wentzel method with higher-order corrections.

Validated results with exactly solvable fourth-order Schrödinger equation.

Abstract

The nodal structure of bound-state wave functions for one-dimensional quantum systems with quartic energy-momentum dispersion and polynomial potentials is analysed by using the semiclassical approximation and variational approach. For energies of bound states, we derive the quantization condition, obtained by using the complex Wentzel method, where we take into account perturbative (up to the fourth order) and nonperturbative in the Planck constant corrections. The bound-state energies and wave functions for the harmonic and quartic potentials are compared with those found by applying the variational approach utilizing the universal Gaussian basis. It is shown that the classical oscillation theorem, valid for systems with quadratic energy-momentum dispersion, breaks down in the classically forbidden region where wave functions also have nodes, while it still remains valid in the classically allowed region. These results are confirmed in addition via the solutions of the exactly solvable problem of the fourth-order Schrodinger equation with a square well potential.