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Project Overview

In problem one of this project the goal was to solve the one-dimensional time-dependent Schrodinger equation using the Crank-Nicolson discretization method. The Schrodinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. This wave function which we will call ψ is a function that assigns a complex number to each point x at each time t. The parameter V(x,t) is the potential that represents the environment in which the particle exists. The Crank–Nicolson method is a finite difference method used for numerically solving partial differential equations and is second-order in time. This method is based on the trapezoidal rule which gives second-order convergence in time. We can also note that for linear equations, the trapezoidal rule is equivalent to the implicit midpoint method.

In problem two of this project our goal was to solve the two-dimensional Schrodinger equation using the alternating direction implicit (ADI) discretization technique. More detail of this method can be found in the report. This problem results in interesting wave functions heavely dependent on initial conditions.

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