{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Finite Difference Method for Solving Schrödinger Equation\n", "### Hydrogen radial equation\n", "\n", "The radial equation for hydrogen (in SI units) is \n", "\n", "$$ \n", "-\\frac{\\hbar^2}{2mr}\\frac{d^2}{d r^2}\\, (r\\psi) \n", " + \\frac{\\hbar^2}{2m}\\, \\frac{l(l+1)}{r^2}\\psi \n", " - \\frac{e^2}{4\\pi\\epsilon_0}\\frac{1}{r} \\psi = E\\psi.\n", "$$\n", "\n", "It's standard to rewrite this in terms of a so-called radial wavefunction $u$,\n", "where \n", "\n", "$$\n", "\\psi \\equiv \\frac{u(r)}{r}.\n", "$$\n", "\n", "The radial equation can then be written in the form\n", "\n", "$$\n", "-\\frac{d^2u}{dr^2} + \\frac{l(l+1)}{r^2}u - \\frac{2me^2}{4\\pi\\epsilon_0 \\hbar^2 r}u\n", " = \\frac{2mE}{\\hbar^2}u.\n", "$$\n", "\n", "Experienced physicists might be immediately inclined to rewrite this equation \n", "in terms of the dimensionless length paramater $r^\\prime \\equiv r/a_0$, where\n", "$a_0$ is the Bohr radius, but such previous acquaintance woth the hydrogen atom\n", "isn't essential. Each term in this form of the radial equation has dimensions of (length)$^{-2}$. Examination of the third term on the left side of the equation \n", "suggests that it might useful to use a dimensionless length parameter made up of fundamental constants characterizing the appropriate scale for this system,\n", "\n", "$$\n", "r^\\prime = \\frac{r}{a},\n", "$$\n", "\n", "where \n", "\n", "$$\n", "a \\equiv \\frac{4\\pi\\epsilon_0\\hbar^2}{me^2}.\n", "$$\n", "\n", "This is, in fact the Bohr radius.\n", "\n", "Written terms of $r^\\prime$ the radial equation becomes\n", "\n", "$$\n", "-\\frac{d^2u}{d{r^\\prime}^2} + \\frac{l(l+1)u}{{r^\\prime}^2} - \\frac{2u}{r^\\prime} \n", " = \\frac{2\\hbar^2 (4\\pi\\epsilon_0)^2}{me^4} E u.\n", "$$\n", "\n", "or, defining a dimensionless energy \n", "\n", "$$\n", "E^\\prime \\equiv \\frac{E}{\\frac{me^4}{2\\hbar^2 (4\\pi\\epsilon_0)^2}} = \\frac{E}{13.6\\, \\mbox{eV}},\n", "$$\n", "\n", "the radial equation becomes\n", "\n", "$$\n", "-\\frac{d^2u}{d{r^\\prime}^2} + \\frac{l(l+1)u}{{r^\\prime}^2} - \\frac{2u}{r^\\prime} \n", " = E^\\prime u.\n", "$$\n", "\n", "This is now in a good form for computational work, with no messy physical constants.\n", "\n", "\n", "I use a finite-difference method to turn the solving of Schrödinger's into an eigenvalue problem. Briefly, after discretizing $r$, (i.e., $r^\\prime_j = \n", "j\\Delta$), an approximate version of Schrödinger's equation can be written as \n", "\n", "$$\n", "\\frac{-u_{j+1} + 2u_j - u_{j-1}}{\\Delta^2} + \\left(\\frac{l(l+1)}{{r^\\prime_j}^2} \n", " - \\frac{2}{r^\\prime_j}\\right) u_j = E^\\prime u_j.\n", "$$\n", "\n", "The term in parentheses is sometimes called an effective potential $U_{\\rm eff}$.\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", "$$\n", "H_{ji} = \\left\\{\\begin{array}{cl}\n", " \\frac{2}{\\Delta^2} + U_{\\rm eff}(r^\\prime_i) & \\mbox{for $i=j$} \\\\\n", " -\\frac{1}{\\Delta^2} & \\mbox{for $i = j\\pm 1$}\\\\\n", " 0 & \\mbox{otherwise}\n", " \\end{array}\\right.\n", "$$\n", "\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 \n", "one particle. I have used this technique for a variety of one-dimensional potentials, \n", "and I have extended it to treat the two-dimensional harmonic oscillator, \n", " 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)]" ] }, { "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): # Effective potential energy function\n", " return l*(l+1)/x**2 - 2/x" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "n = 2000 # Number of intervals (J=1 in my notes)\n", "dim = n - 1 # Number of internal points\n", "xl = 0 # xl corresponds to origin\n", "xr = 200. # \n", "delta = (xr-xl)/n\n", " \n", "x = np.linspace(xl+delta,xr-delta,dim)\n", "\n", "l = 1 # orbital quantum number\n", "\n", "#Fill Hamiltonian\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", "vals, vecs = linalg.eigh(h) #Note: eigenvectors 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 = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '
');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
');\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 = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "plt.title(\"Hydrogen Atom, $l = 1$ states\")\n", "plt.xlabel(\"$r$ (units: $a_0$)\")\n", "plt.ylabel(\"$u^2(x)$\")\n", "plt.axhline(0, color='black') #draw x axis\n", "plt.grid(True)\n", "plt.xlim(0,40)\n", "\n", "for m in range(3):\n", " #y = np.append([xl],np.append(np.transpose(vecs)[m],[xr]))\n", " y = np.transpose(vecs)[m]\n", " plt.plot(x, y**2)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "#### Version information\n", "\n", "+ `%version_information` is an IPython magic extension for showing version \n", "information for dependency modules in a notebook;\n", "\n", "+ See https://github.com/jrjohansson/version_information\n", "\n", "+ `%version_information` is available on Bucknell computers on the linux network. \n", "You can easily install it on any computer." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "%load_ext version_information" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "application/json": { "Software versions": [ { "module": "Python", "version": "3.7.7 64bit [GCC 7.3.0]" }, { "module": "IPython", "version": "7.16.1" }, { "module": "OS", "version": "Linux 4.9.0 9 amd64 x86_64 with debian 9.13" }, { "module": "numpy", "version": "1.18.5" }, { "module": "scipy", "version": "1.5.0" }, { "module": "matplotlib", "version": "3.2.2" } ] }, "text/html": [ "
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Fri Jan 01 10:29:55 2021 EST
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