Schrodinger Equation

CHAPTER 3 – Theoretical and Numerical Computational Solution of the Schrodinger Equation

3.1 Theoretical Solution

The theoretical solution of the time independent and the time dependent Schrodinger equation is analysed.

Solution to Time Dependent Schrodinger Equation: method of separation of variables [6]

TDSE: EΨ(t,x)= (〖-ħ〗^2/2m d^2/(dx^2 ) + U(x))Ψ(t,x)-→ EΨ(t,x) = ĤΨ

The potential energy in the Hamiltonian is time independent: U = U(x).

Assuming: Ψ(t,x) = Ψ(x)f(t)

So TDSE is re-written as: iħΨ(x) df(t)/dt = 〖-ħ〗^2/2m (d^2 Ψ(x))/(dx^2 ) f(t)+ UΨ(x)f(t)

Multiply through by (1/Ψ(x)f(t)) to make the R.H.S exclusively t-dependent and the L.H.S exclusively x-dependent

TDSE is re-written as: iħ 1/(f(t))

The particle is a box analysis allows the prediction of the possible stationary state wave functions (Ψ(x) ), and the corresponding Energies (E), for a given potential energy function (U(x) ) [6]. Figure 1: A particle with mass m, moves along a straight line at constant speed, within two rigid walls which are distance L apart

Figure 1, describes a simple model of a particle bounded so that it cannot escape to infinity, the particle is confined to a restricted region of space between two rigid walls separated by a distance (L). The motion of the particle is purely one dimensional along the x-axis. Particle moves from x=0 to x =L. the potential energy corresponding to the rigid walls is infinite, so the particle cannot escape the box, between the walls, the potential energy is zero.

In order to solve the Schrödinger equation for the particle in a box system, some assumptions will be made regarding the wave function: Since the particle is confined to the space 0≤x≤L: so Ψ(x) is expected to be zero outside this region: so if the term U(x)Ψ(x) in equation 12 is to be finite, then Ψ(x) must be zero and U(x) is infinite

For the n x n Hamiltonian matrix, there will be n number of Eigen values and each Eigen value corresponds to a particular wave function (Ψ_1:Ψ_n) [11]

3.6.1.1 Critical Review of Finite Difference solutions

A brief look at some of the works that have been done on the solution of the Schrödinger equation using the Crank-Nicholson method and their conclusion about the finite difference approach to the Schrödinger equation.

The reason for discretization in the finite difference solution of the Schrödinger equation is to obtain a problem which can be solved through a finite procedure [24]. During the discretization, the derivative terms are re-written in terms of finite differences which enable an eventual formulation of the original problem (Schrödinger equation) in the form of a matrix equation that is solved using inverse iteration