Monte Carlo Hamiltonian: the Linear Potentials

TL;DR

This paper evaluates the Monte Carlo Hamiltonian method's effectiveness in studying excited states in quantum mechanics, demonstrating its accuracy through models with linear potentials and comparing results with analytical solutions.

Contribution

It extends the Monte Carlo Hamiltonian approach to linear potential models, validating its ability to accurately compute spectra and wave functions for excited states.

Findings

Results agree with analytical and Runge-Kutta calculations.

The method effectively computes spectra and wave functions.

It demonstrates validity for symmetric and asymmetric linear potentials.

Abstract

We further study the validity of the Monte Carlo Hamiltonian method. The advantage of the method, in comparison with the standard Monte Carlo Lagrangian approach, is its capability to study the excited states. We consider two quantum mechanical models: a symmetric one $V(x) = |x|/2$; and an asymmetric one $V(x)=\infty$, for $x < 0$ and $V(x)=x$, for $x \ge 0$. The results for the spectrum, wave functions and thermodynamical observables are in agreement with the analytical or Runge-Kutta calculations.