Ground State Wave Function of the Schr\"odinger Equation in a Time-Periodic Potential

TL;DR

This paper presents an exact solution for the ground state wave function of the Schrödinger equation in a time-periodic potential using a generalized transfer matrix method, revealing soliton-like behavior and wedge-shaped wave fronts.

Contribution

It introduces a novel exact method to solve the Schrödinger equation in time-periodic potentials and connects quantum wave functions with directed polymer models in statistical mechanics.

Findings

Wave function exhibits oscillating soliton-like behavior.

Wave front has a wedge shape.

Solution corresponds to the partition sum of a directed polymer.

Abstract

Using a generalized transfer matrix method we exactly solve the Schr\"odinger equation in a time periodic potential, with discretized Euclidean space-time. The ground state wave function propagates in space and time with an oscillating soliton-like wave packet and the wave front is wedge shaped. In a statistical mechanics framework our solution represents the partition sum of a directed polymer subjected to a potential layer with alternating (attractive and repulsive) pinning centers.