{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Finite Difference Method for Schroedinger Equation\n", "### Harmonic Oscillator\n", "\n", "Calculate wavefunctions and energies for given potential $u(x)$.\n", "\n", "Use finite-difference method to turn this into an eigenvalue problem.\n", "See ~/research/notes/finiteDifference/finiteDifference_notes.pdf, \n", "along with Mathematica notebooks that do similar calculations.\n", "\n", "Schrödinger equation:\n", "\n", "$$\n", "-\\frac{d^2\\psi}{dx^2} - \\frac{2m}{\\hbar^2}\\, \\frac{1}{2}kx^2\\psi \n", " = \\frac{2m}{\\hbar^2} E\\psi\n", "$$\n", "\n", "Define a dimensionless length parameter\n", "\n", "$$\n", "x^\\prime \\equiv \\frac{x}{a},\n", "$$\n", "\n", "where $a$ will be determined shortly. Rewriting the Schrödinger\n", "equation gives\n", "\n", "$$\n", "-\\frac{d^2\\psi}{d{x^\\prime}^2} + \\left(\\frac{kma^4}{\\hbar^2}\\right)\n", " \\, {x^\\prime}^2\\psi \n", " = \\frac{2ma^2}{\\hbar^2} E\\psi\n", "$$\n", "Now choose $a$ so that the factor in parentheses is unity:\n", "$$\n", "a^4 = \\frac{\\hbar^2}{km} = \\frac{\\hbar^2}{m^2\\omega^2}\n", "\\quad\\longrightarrow \\quad a^2 = \\frac{\\hbar}{m\\omega}.\n", "$$\n", "\n", "The Schrödinger equation can now be written as\n", "\n", "$$\n", "-\\frac{d^2\\psi}{d{x^\\prime}^2} + {x^\\prime}^2 \\psi\n", " = \\frac{E}{\\hbar\\omega} \\psi\n", "$$\n", "\n", "which sugggests working with a dimensionless energy\n", "\n", "$$ \n", "E^\\prime \\equiv \\frac{2E}{\\hbar\\omega}\n", "$$\n", "\n", "giving a dimensionless version of the Schrödinger equation\n", "\n", "$$\n", "-\\frac{d^2\\psi}{d{x^\\prime}^2} + {x^\\prime}^2 \\psi\n", " = E^\\prime \\psi.\n", "$$\n", "\n", "This is good for computational work because there are no messy \n", "physical constants to worry about.\n", "\n", "I use a finite-difference method to turn the solving of Schrödinger's into an eigenvalue problem. Briefly, after discretizing $x^\\prime$, (i.e., $^\\prime_j = \n", "x_0 + j\\Delta$), an approximate version of Schrödinger's equation can be written as \n", "\n", "$$\n", "\\frac{-\\psi_{j+1} + 2\\psi_j - \\psi_{j-1}}{\\Delta^2} + x_j^2\\psi_j = \n", " E^\\prime \\psi_j.\n", "$$\n", "\n", "This is an eigenvalue problem:\n", "\n", "$$ \n", "H_{ji}\\psi_i = E^\\prime \\psi_j,\n", "$$\n", "\n", "where \n", "\n", "$$\n", "H_{ji} = \\left\\{\\begin{array}{cl}\n", " \\frac{2}{\\Delta^2} + x_i^2 & \\mbox{for $i=j$} \\\\\n", " -\\frac{1}{\\Delta^2} & \\mbox{for $i = j\\pm 1$}\\\\\n", " 0 & \\mbox{otherwise}\n", " \\end{array}\\right.\n", "$$\n", "Used as supplementary exercise in PHYS 212E, Spring 2016\n", "\n", "Marty Ligare" ] }, { "cell_type": "code", "execution_count": 1, "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 u(x): # Potential energy function\n", " return x**2" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "n = 4000 # Number of intervals (J=1 in my notes)\n", "dim = n - 1 # Number of internal points\n", "xl = -20. # xl is set far enough to left of well such that \\psi=0\n", "xr = 20. # xr is set far enough to right of well such that \\psi=0\n", " # For HO, results should be good up to energies of \n", " # about (xr/2)**2 in units of \\hbar/\\omega\n", "\n", "delta = (xr-xl)/n\n", " \n", "x = np.linspace(xl+delta,xr-delta,dim)\n", "\n", "#Fill Hamiltonian\n", "\n", "h = np.zeros((dim,dim),float)\n", "for i in range(len(h)-1):\n", " h[i,i+1] = h[i+1,i] = -1/delta**2 \n", "\n", "for i in range(len(h)):\n", " h[i,i] = 2./delta**2 + u(x[i]) \n", "\n", "# Calculate eigenvalues and eigenvectors\n", "vals, vecs = linalg.eigh(h) #Note: eigenvectors are in columns of vecs" ] }, { "cell_type": "code", "execution_count": 5, "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. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
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');\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('