p-Adic quantum mechanics, infinite potential wells, and continuous-time quantum walks

TL;DR

This paper introduces a p-adic quantum mechanics model for infinite potential wells, solves the Schrödinger equation in this setting, and constructs a continuous-time quantum walk on a graph derived from p-adic fractal structures, linking p-adic QM to quantum computing.

Contribution

It develops a rigorous p-adic quantum mechanical model for infinite potential wells and connects it to quantum walks on graphs, bridging p-adic analysis and quantum computing.

Findings

Solved the p-adic Schrödinger equation for infinite potential wells.

Constructed a quantum walk on a graph based on p-adic fractals.

Established a link between p-adic QM and quantum computing.

Abstract

This article discusses a p-adic version of the infinite potential well in quantum mechanics (QM). This model describes the confinement of a particle in a p-adic ball. We rigorously solve the Cauchy problem for the Schr\"odinger equation and determine the stationary solutions. The p-adic balls are fractal objects. By dividing a p-adic ball into a finite number of sub-balls and using the wavefunctions of the infinite potential well, we construct a continuous-time quantum walk (CTQW) on a fully connected graph, where each vertex corresponds to a sub-ball in the partition of the original ball. In this way, we establish a connection between p-adic QM and quantum computing.