{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Perturbation Theory: Energy Shifts and Perturbed Wavefunctions \n", "\n", "#### Advanced Quantum Physics content\n", "\n", "In this exercise we will be investigating first order perturbation theory on a harmonic potential and a square well. Detailed below are the results derived within the AQP course which we shall be using.\n", "\n", "Total Hamiltonian:\n", "\\begin{equation}\n", " \\hat{H} = \\hat{H}^{(0)}+\\hat{H}'\n", "\\end{equation}\n", "\n", "First order energy shift:\n", "\\begin{equation}\n", " \\Delta E_n^{(1)} = \\langle n^{(0)} | \\hat{H}' | n^{(0)} \\rangle\n", "\\end{equation}\n", "\n", "First order perturbed wavefunctions:\n", "\\begin{equation}\n", " |n^{(1)}\\rangle \\approx | n^{(0)} \\rangle + \\sum_{m \\neq n} | m^{(0)} \\rangle \\frac{\\langle m^{(0)} | \\hat{H}' | n^{(0)} \\rangle}{E_n^{(0)}-E_m^{0}}\n", "\\end{equation}\n", "\n", "In the following we are working in units where $\\hbar = 1$\n", "\n", "Importing libraries which we will be using in this exercise, pycav.quantum contains functions specifically designed for this program. The following will outline the basic use of this program so that it can be used with ease in the exercise at the end." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#NAME: Perturbation Theory\n", "#DESCRIPTION: Investigating energy shifts and perturbed wavefunctions.\n", "\n", "import numpy as np\n", "from scipy.special import hermite\n", "from scipy.misc import factorial\n", "from scipy.integrate import quad\n", "\n", "import matplotlib.pyplot as plt\n", "\n", "import pycav.quantum as pm\n", "\n", "%matplotlib notebook" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Define a potential for use in plotting. The function should take an array argument, params, containing the necessary parameters which define the potential. The function should then return a function of position, x.\n", "\n", "E.g. Harmonic potential: params[0] is the mass and params[1] is the angular frequency $V(x) = \\frac{1}{2} m \\omega^2 x^2$" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def harmonic_potential(params):\n", " def V(x):\n", " return 0.5*params[0]*params[1]**2*x**2\n", " return V" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Define the form of the unperturbed wavefunctions. The function should take the params argument as well as the primary quantum number, n. The function should then return the wavefunction as a function of position, like with the potential. The harmonic oscillator wavefunctions are of the form given here\n", "\n", "Also define the form of the unperturbed energies. The function just needs to return the value of the energy of state $n$ by using the params array." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def harmonic_unperturb_wf(params,n):\n", " alpha = params[0]*params[1]\n", " def psi_n(x):\n", " n_fact = factorial(n)\n", " y = np.sqrt(alpha)*x\n", " n_herm = hermite(n)\n", " return (alpha/np.pi)**(0.25)*(1./(2**n*n_fact)**0.5*n_herm(y)*np.exp(-y**2/2.))\n", " return psi_n\n", " \n", "def harmonic_unperturb_erg(params,n):\n", " return (n+0.5)*params[1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With these functions in place we can plot the unperturbed system. This is done by the plot_levels function in the perturb_module. It will plot the first 4 wavefunctions, with their origins placed at their energy level. The function requires the arguments: unperturb_wf, unperturb_erg, potential and params with an additional optional arguments, x_lims which takes a list with the $x$ axis limits e.g. x_lims = [-0.1,0.1] and plot_zoom which takes a list which contains one element which is the factor by which wavefunctions are multiplied on the plot so they are visible." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\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", " 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", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\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": [ "pm.plot_perturb(perturbation,harmonic_unperturb_wf,harmonic_unperturb_erg,harmonic_potential,\n", " h_params,tolerance = 0.01,plot_zoom = [200,10000])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercises\n", "\n", "1. For the harmonic potential, investigate linear and quadratic perturbations and higher order $x^n$. What do you notice about the first order energy shift as a function of $(n+1/2)$?\n", "(Theoretical Exercise: Why do the shifts have the shown function form? Hint: use ladder operators)\n", "\n", "2. For a square well with infintely high walls, write down the unperturbed wavefunctions and the unperturbed energies and create the required functions\n", "\n", "3. Investigate a small step in potential near the centre of the well such that perturbation has the form:\n", "\\begin{align}\n", " V'(x) &= \\epsilon \\ \\mbox{for} \\ |x| < b/2\n", "\\end{align}\n", "where $b \\ll a$ where $a$ is the width of the well. Give the first order energy shift to the ground state and first excited state\n", "\n", "4. Investigate a linear perturbation, $V'(x) \\propto -x$, and give the first order energy shift to the ground state and first excited state. Now write a function to calculate the second order energy shift:\n", "\\begin{equation}\n", " \\Delta E_n^{(2)} = \\sum_{m \\neq n} \\frac{\\langle m^{(0)} | \\hat{H}' | n^{(0)} \\rangle}{E_n^{(0)}-E_m^{0}}\n", "\\end{equation}\n", "Use [SciPy Quad](http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html) to calculate the matrix elements. You should truncate the sum when the terms become negligibly small. Why should you perform seperate sums for even and odd $m$? Find the second order energy shift for the ground and first excited states.\n", "\n", "5. Use the optional argument return_list = True in first_order_wf to return a tuple containing (perturbed_wf, k_list, I_list). k_list, I_list contain a list of the principle quantum numbers and prefactors in the sum of the unperturbed states respectively. i.e.\n", "\\begin{equation}\n", " |n^{(1)}\\rangle \\approx |n^{(0)}\\rangle + \\sum_{k \\in k_{list}} I_{list}[k] \\ |k^{(0)}\\rangle\n", "\\end{equation}\n", "Use this to show the linear combination of states which creates the perturbed state (increasing the perturbation size makes this clearer). Also plot the perturbed wavefunction for the ground state, what is the interpretation of this?\n", "\n", "6. With the system in the ground state of the perturbed system ($V'(x) \\propto -x$), write down the time dependence once the perturbation is switched off (only consider the above expansion to first order in the sum). \n", "\n", "7. (Extra) Plot this evolution." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.1" }, "widgets": { "state": { "1412be1b5bf0405bb9a946c35ea63657": { "views": [] }, "1715affd35b14d39b7af3786eb1949c3": { "views": [] }, "2a4e2c0e430c4a4184c1b91e187bf53c": { "views": [] }, "506debe03127468da114ab8278fab643": { "views": [] }, "569436effbb04248a2bb736967e1d719": { "views": [] }, "788cdab7a49b4be999cd642f4624a24d": { "views": [] }, "858a79b17a7f420b8904670c97a2641a": { "views": [] }, "888dfe39ae994808a5f8d8b0eebadafa": { "views": [] }, "8ac7b2dfc10541778f95d221cdf8180e": { "views": [] }, "974f8774d83c4fd18308414a1d08c9f5": { "views": [] }, "9d792c9ac97c46268bca1c65174ef356": { "views": [] }, "a3aea97c247a497fa3edcbdc2bf7cbd3": { "views": [] }, "da0a835f7c9542fb98ef55f4a39ae2f5": { "views": [] }, "da85ba8c4b484ab6bbcf04480732f43a": { "views": [] }, "dc4de2ba4f364368bf984a826dc7f1fc": { "views": [] }, "e09b6235dd2c45439f6a3c06b0ba37fb": { "views": [] }, "e3efb4216c4b4f12a7917b3b1e9bab59": { "views": [] }, "eb621ec16a26478388d8b116a4dd1f0d": { "views": [] } }, "version": "1.1.2" } }, "nbformat": 4, "nbformat_minor": 0 }