{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Numerov's Algorithm & Shooting Method: Finite Square Well\n", "\n", "### Time indepedent Schrodinger Equation\n", "\n", "In units where $\\hbar = 1$, the 1D TISE can be expressed in the form:\n", "\n", "\\begin{equation}\n", " \\frac{d^2 \\psi}{dx^2} = -2m\\left(E - V(x) \\right) \\psi = -g(x) \\psi\n", "\\end{equation}\n", "\n", "This differential equation can be solved numerically via Numerov's method ([see page 10 - 11](http://www.fisica.uniud.it/~giannozz/Corsi/MQ/LectureNotes/mq-cap1.pdf)). For a 1D spatial grid, the wavefunction at the $n+1$th point along can be approximated by:\n", "\n", "\\begin{equation}\n", " \\psi_{n+1} = \\frac{(12-10f_n) \\ \\psi_n-f_{n-1}\\psi_{n-1}}{f_{n+1}}\n", "\\end{equation}\n", "\n", "where:\n", "\\begin{equation}\n", " f_n \\equiv \\left( 1 + \\frac{\\delta x^2}{12}g_n \\right), \\ \\ \\ \\ \\ \\ \\ g_n = 2m(E-V(x_n))\n", "\\end{equation}\n", "\n", "Hence to start Numerov's method we require $\\psi_0$ and $\\psi_1$, in other words $\\psi(x = x_{min})$ and $\\psi(x = x_{min} + \\delta x)$, where $\\delta x$ is the step size." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Shooting Method:\n", "\n", "When investigating bound states we require $\\psi( x \\to \\pm \\infty) = 0$. However we cannot consider a infinite domain, instead we must choose a big enough domain that setting $\\psi( x = x_{min}) = 0$ is a good approximation (and similarly for $x_{max}$).\n", "\n", "With Numerov's Method in place, the shooting method can be used to find the energy eigenstates. It goes as follows:\n", "\n", "1. Setting $\\psi_0 = 0$ \"satisfies\" the boundary condition that the wavefunctions must vanish at the boundary\n", "2. Since the Schrodinger Equation is linear and homogeneous we are free to set $\\psi_1$ to any non-zero constant as multiplying by a constant does not affect the solution. In this case we shall set $\\psi_1 = \\delta x$.\n", "3. Using the Numerov algorithm, $\\psi(x)$ can be found. Exponential growth near $x_{max}$ can be observed if the input energy is not near a energy eigenvalue\n", "4. A bisection search can be performed to calculate the energy eigenvalue and eigenstate" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Bisection Search:\n", "\n", "We wish to find a function $f(E) = 0$. First we must find values of $E$ which bracket the solution, that is:\n", "