Solving the time-dependent Schr\"odinger equation via Laplace transform

TL;DR

This paper introduces a Laplace transform method to solve the time-dependent Schrödinger equation for arbitrary initial states, avoiding eigenstate summation and path integrals, and demonstrates it on piecewise constant potentials.

Contribution

It presents a novel Laplace transform approach for solving the time-dependent Schrödinger equation directly from stationary solutions, simplifying analysis of wave packet dynamics.

Findings

Successfully solves the initial value problem for specific potentials.

Provides explicit wave packet dynamics including reflection and transmission.

Analyzes classical and quantum properties of wave evolution.

Abstract

We show how the Laplace transform can be used to give a solution of the time-dependent Schr\"odinger equation for an arbitrary initial wave packet if the solution of the stationary equation is known. The solution is obtained without summing up eigenstates nor do we need the path integral. We solve the initial value problem for three characteristic piecewise constant potentials. The results give an intuitive picture of the wave packet dynamics, reproducing explicitly all possible reflection and transmission processes. We investigate classical and quantum properties of the evolution and determine the reflection and transmission probabilities.