Exact dynamics of a Gaussian wave-packet in two potential curves coupled at a point

TL;DR

This paper introduces an exact time-domain method for calculating the dynamics of a Gaussian wave-packet in a two-state quantum system with point coupling, providing explicit analytic solutions for the wave functions.

Contribution

The paper develops a novel analytical approach to solve the time-dependent Schrödinger equation for a two-state system with Dirac delta coupling, directly in the time domain.

Findings

Derived exact analytic wave functions for both states.

Demonstrated the method's effectiveness in time-domain calculations.

Provided explicit solutions avoiding Laplace or Fourier domain inversions.

Abstract

We present a method to calculate exact dynamics of a wave-packet in a quantum two-state problem with Dirac delta coupling. The advantage of our method is that the calculations are done in the time domain. Hence inverting the solutions from other domains (Laplace or Fourier domain) do not intervene us from presenting the solution in the time domain. The initial wave packet is considered to be a Gaussian function. The wave propagation in the quantum state is governed by time-dependent Schr\"odinger equation and the coupling is given by non-diagonal element in the matrix representation. We present the exact analytic forms of the wave function for both the states in the time domain.