{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Finite Difference Method for Solving Schrödinger Equation\n", "### Infinite square well potential\n", "\n", "Schrödinger equation:\n", "\n", "$$ \n", "-\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2}\\psi + U(x)\\psi = E\\psi$\n", "$$\n", "\n", "Use a dimensionless length parameter\n", "\n", "$$\n", "x^\\prime = \\frac{x}{a},\n", "$$\n", "\n", "where $a$ is a characteristic length scale of system. In the case of the infinite\n", "square well it's reasonable to let $a$ be the width of the potential. Using this\n", "dimensionless $x^\\prime$ the Schrödinger equation becomes\n", "\n", "$$\n", "-\\frac{d^2}{(dx^\\prime)^2}\\psi(x^\\prime) + \\frac{2ma^2}{\\hbar^2}U(x^\\prime) = \n", "\\frac{2ma^2}{\\hbar^2} E\\psi(x^\\prime)\\quad \\longrightarrow \\quad \n", "-\\frac{d^2}{(dx^\\prime)^2}\\psi(x^\\prime) + \\pi^2 U^\\prime(x^\\prime) \n", "= \\pi^2 E^\\prime\\psi(x^\\prime),\n", "$$\n", "\n", "where dimesionless energy parameters are defined as\n", "\n", "$$=\n", "U^\\prime \\equiv \\frac{U}{\\frac{h^2}{8ma^2}}\\quad \\mbox{and}\\quad \n", "E^\\prime \\equiv \\frac{E}{\\frac{h^2}{8ma^2}}\n", "$$\n", "\n", "Using these dimensionless parameters, the energies in an infinite square-box\n", "potential are \n", "\n", "$$\n", "E_m^\\prime = m^2.\n", "$$\n", "\n", "I use a finite-difference method to turn the solving of Schrödinger's into an eigenvalue problem. Briefly, after discretizing $x$, ($x^\\prime_j = x^\\prime_0 + j\\Delta$), an approximate version of Schrödinger's equation can be written as \n", "\n", "$$\n", "\\frac{-\\psi_{j+1} + 2\\psi_j - \\psi_{j-1}}{\\pi^2\\Delta^2} + \\pi^2 U^\\prime_j \\psi_j = \\pi^2 E^\\prime u_j.\n", "$$\n", "\n", "This is an eigenvalue problem:\n", "\n", "$$ \n", "H_{ji}\\psi_i = E^\\prime \\psi_j,\n", "$$\n", "\n", "where \n", "\n", "\n", "$$\n", "H_{ji} = \\left\\{\\begin{array}{cl}\n", " \\frac{2}{\\pi^2\\Delta^2} & \\mbox{for $i=j$} \\\\\n", " -\\frac{1}{\\pi^2\\Delta^2} & \\mbox{for $i = j\\pm 1$}\\\\\n", " 0 & \\mbox{otherwise}\n", " \\end{array}\\right.\n", "$$\n", "\n", "The eigenvalues give the energy of the states, and the eigenvectors are numerical\n", "approximations of the wavefunctions.\n", "\n", "[The method can be extended to more than one dimension and to situations with more than one particle. I have used this technique for a variety of one-dimensional potentials, and I have extended it to treat the two-dimensional harmonic oscillator, the hydrogen atom radial equation, and excited states of helium.) For a recent pedagogical discussion of the method, \n", "see Matrix Numerov method for solving Schrödinger's equation,\n", "Mohandas Pillai, Joshua Goglio, and Thad G. Walker, Am. J. Phys. 80, 1017 (2012)]\n", "\n", "\n", "\n", "Marty Ligare" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "from scipy import linalg\n", "\n", "import matplotlib as mpl\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Following is an Ipython magic command that puts figures in the notebook.\n", "%matplotlib notebook\n", "\n", "# M.L. modification of matplotlib defaults\n", "# Changes can also be put in matplotlibrc file, \n", "# or effected using mpl.rcParams[]\n", "mpl.style.use('classic')\n", "plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures\n", "plt.rc('axes', labelsize=16, titlesize=14)\n", "plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new size" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "def u2(x):\n", " '''Infinite square well potential'''\n", " return 0" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [], "source": [ "xl = 0 # Left endpoint\n", "xr = 1 # Right endpoint\n", "n = 1000 # Number of intervals between xl and xr\n", "dim = n - 1 # Number of internal points \n", "delta = (xr-xl)/n\n", "x = np.linspace(xl,xr,n+1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Fill Hamiltonian" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "h = np.zeros((dim,dim),float)\n", "\n", "for i in range(len(h)-1):\n", " h[i,i+1] = h[i+1,i] = -1/(np.pi*delta)**2 # Off-diagonal elements\n", "\n", "for i in range(len(h)):\n", " h[i,i] = 2./(np.pi*delta)**2 + u2(x[i+1]) # Diagonal elements\n", "\n", "vals, vecs = linalg.eigh(h) # Note: eigenvectors in columns of vecs\n", "#psi = vecs.T # \n", "ends = np.zeros(dim)\n", "vecs = np.append([ends],np.append(vecs,[ends],axis=0), axis=0)\n", "psi = vecs.T " ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. 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