Ground-state energy eigenvalue calculation of the quantum mechanical well $V(x)={1/2}kx^{2}+\lambda {x^{4}}$ via analytical transfer matrix method

TL;DR

This paper applies the analytical transfer matrix method to compute the ground-state energy of a symmetric quartic potential in quantum mechanics, providing a new analytical approach and validating results against established numerical methods.

Contribution

The paper introduces an analytical transfer matrix technique for calculating ground-state energies in quartic potentials, offering an alternative to traditional numerical and perturbative methods.

Findings

The method yields accurate ground-state energy eigenvalues.

Results are consistent with numerical shooting, perturbation theory, and WKB methods.

Analytical approach simplifies calculations for complex potentials.

Abstract

The analytical transfer matrix technique is applied to the Schr\"{o}dinger equation of symmetric quartic-well potential problem in the form $V(x)={1/2}kx^{2}+\lambda{x^{4}}.$ This gives quantization condition from which we can calculate the ground-state energy eigenvalues numerically. We also compare the results with those obtained from numerical shooting method, perturbation theory, and WKB method.