{ "metadata": { "name": "", "signature": "sha256:9e30b4353144ef34efd3c4b9ab05157aa3de69cd80af680c66504804f1dc7010" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "

Utility Functions

\n", "

\n", "This doesn't go with any particular section of the notes. It just\n", "demonstrates the use of the a few of the functions from \"sglib.\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

Contents

\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "

^ Setup Cell

" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%autosave 0\n", "%matplotlib inline\n", "from sglib import *" ], "language": "python", "metadata": {}, "outputs": [ { "javascript": [ "IPython.notebook.set_autosave_interval(0)" ], "metadata": {}, "output_type": "display_data" }, { "output_type": "stream", "stream": "stdout", "text": [ "Autosave disabled\n" ] } ], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "

^ Creating vectors and matrices

\n", "

\n", "You've already been using the col() and row()\n", "functions to create column vectors and row vectors:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "col(1,2,3)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left[\\begin{matrix}1\\\\2\\\\3\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": 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"metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAFIAAAAZBAMAAABQnWQHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iMpl272bNiRBU\nq0QwBS8SAAABFUlEQVQ4EWMQUjJgIAz4lLQYFAgrA6uwJVWlK4rB0za4PkAW4Fq2hIEBaqYisgRD\n/f9vyHzueQyRMJVshcgyDLOuoXiScwGDIEylxUQUlRdQeAz9YH0Q2xfgVSkO1ghWyWGAqvJGWwiy\nqT+9Uh9Abd/GgKpSmsEeKAUD3D8bGGqgKhegqWRgYFSAqWNg4P6/gSF8AziUeBwwVLJ+Rqhk+MTA\n0P8ArHIXA6pK3gkMXP+QVFYCVTaAVZ69e/f7NSQZ9gkMrL+R+EC/Q80EChYiSTCwMjAwTUASOA90\nJ9RHDAwfkSQYmAMY2pEjiXEBN8zvDAv/30BWaro2GpnLsDTWAW4migR2DsnpE7sxKKJDx0wlIvO7\nLgAZp0F1A5wKvwAAAABJRU5ErkJggg==\n", "prompt_number": 5, "text": [ "[4 5 6]" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "I've added one more\n", "\"creation\" function, mat() which will create square\n", "matrices:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "mat(1,0,0,-1)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAyBAMAAAATs7BgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhDNIruZq3bd\nZkRVk0nZAAABRUlEQVQ4EWOQ//+JAR/Q//+RQdjFFZ8ShhAXQwYRvCpAko4QNWwB6CpZJ+10gIpB\n1LD3Y6hhM2A9jqyGtXM9hpqLDAxqyGoYGPZjqOliYLCHCkLdg6nmFwPDewOIQbjUsP4EqtmAXw07\nMGD9FShXwwo0h5BdDED32BNwM8MJBob7hPy+kYFhGsTJsPjCDB8eA1ZYbEPCZ5b8qQtQTTCKVXuX\nA6o5MBmsNDScscrBBEfVwEICO018+ERqT8FuRCpIGGJOMkNMATZFFV/hangFGJgSsKiJmIVQw7GA\ngRnMQ1fHjVDDv4CB7zO6PIiPpCZ+AgPfbwJq6hUYWLAWZkjmEKMmXgHVrrAzIJCC4h7+CQzMhNzM\n08DAS8jv3AIMnAkE/MVwmCHQgZCaoN1LsCnhXPhnLVCc+LSBzRCE2OAzh5i6iYg6DgCd12cy9umd\nxwAAAABJRU5ErkJggg==\n", "prompt_number": 6, "text": [ "\u23a11 0 \u23a4\n", "\u23a2 \u23a5\n", "\u23a30 -1\u23a6" ] } ], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "mat(1,2,3,4,5,6,7,8,9)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\\\7 & 8 & 9\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 7, "text": [ "\u23a11 2 3\u23a4\n", "\u23a2 \u23a5\n", "\u23a24 5 6\u23a5\n", "\u23a2 \u23a5\n", "\u23a37 8 9\u23a6" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "I've also \"predefined\" a number of vectors and matrices that we care about:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sigma_x, sigma_y, sigma_z" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\begin{pmatrix}\\left[\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right], & \\left[\\begin{matrix}0 & - i\\\\i & 0\\end{matrix}\\right], & \\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right]\\end{pmatrix}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 8, "text": [ "\u239b\u23a10 1\u23a4, \u23a10 -\u2148\u23a4, \u23a11 0 \u23a4\u239e\n", "\u239c\u23a2 \u23a5 \u23a2 \u23a5 \u23a2 \u23a5\u239f\n", "\u239d\u23a31 0\u23a6 \u23a3\u2148 0 \u23a6 \u23a30 -1\u23a6\u23a0" ] } ], "prompt_number": 8 }, { "cell_type": "code", "collapsed": false, "input": [ "pX, mX, pY, mY, pZ, mZ" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\begin{pmatrix}\\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right], & \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2}}{2}\\end{matrix}\\right], & \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], & \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], & \\left[\\begin{matrix}1\\\\0\\end{matrix}\\right], & \\left[\\begin{matrix}0\\\\1\\end{matrix}\\right]\\end{pmatrix}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 9, "text": [ "\u239b\u23a1 ___\u23a4 \u23a1 ___ \u23a4 \u23a1 ___ \u23a4 \u23a1 ___ \u23a4 \u239e\n", "\u239c\u23a2\u2572\u2571 2 \u23a5 \u23a2 \u2572\u2571 2 \u23a5 \u23a2 \u2572\u2571 2 \u23a5 \u23a2 \u2572\u2571 2 \u23a5 \u239f\n", "\u239c\u23a2\u2500\u2500\u2500\u2500\u2500\u23a5, \u23a2 \u2500\u2500\u2500\u2500\u2500 \u23a5, \u23a2 \u2500\u2500\u2500\u2500\u2500 \u23a5, \u23a2 \u2500\u2500\u2500\u2500\u2500 \u23a5, \u23a11\u23a4, \u23a10\u23a4\u239f\n", "\u239c\u23a2 2 \u23a5 \u23a2 2 \u23a5 \u23a2 2 \u23a5 \u23a2 2 \u23a5 \u23a2 \u23a5 \u23a2 \u23a5\u239f\n", "\u239c\u23a2 \u23a5 \u23a2 \u23a5 \u23a2 \u23a5 \u23a2 \u23a5 \u23a30\u23a6 \u23a31\u23a6\u239f\n", "\u239c\u23a2 ___\u23a5 \u23a2 ___ \u23a5 \u23a2 ___ \u23a5 \u23a2 ___ \u23a5 \u239f\n", "\u239c\u23a2\u2572\u2571 2 \u23a5 \u23a2-\u2572\u2571 2 \u23a5 \u23a2\u2572\u2571 2 \u22c5\u2148\u23a5 \u23a2-\u2572\u2571 2 \u22c5\u2148 \u23a5 \u239f\n", "\u239c\u23a2\u2500\u2500\u2500\u2500\u2500\u23a5 \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5 \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5 \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5 \u239f\n", "\u239d\u23a3 2 \u23a6 \u23a3 2 \u23a6 \u23a3 2 \u23a6 \u23a3 2 \u23a6 \u23a0" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", "\n", "

^ Printing things

\n", "

\n", "As you can see above, when you simply put the name of an object\n", "in a cell and execute it, python displays the object. But the way\n", "it's displayed, is not always the way we would like to see\n", "it. For example, in the vectors above the denominators are \"rationalized\"\n", "instead of leaving the $\\sqrt{2}$ in the denominator.\n", "

\n", "

\n", "Here's a function called sg_print() which will attempt\n", "to display vectors and matrices in a way that is similar to what we do\n", "on the board. It works pretty well, but it is not always successful.\n", "

" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sg_print(pX)\n", "sg_print(mX)\n", "sg_print(mY)" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "$\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] }, { "html": [ "$\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix}1\\\\-1\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] }, { "html": [ "$\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix}1\\\\- i\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] } ], "prompt_number": 10 }, { "cell_type": "code", "collapsed": false, "input": [ "M = mat(1/sqrt(2), 1/sqrt(2), 1/sqrt(2), -1/sqrt(2))\n", "M" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAA/BAMAAAARe/vjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARM1UEHaJu2Yimasy\n3e/JfTDcAAACX0lEQVRIDZ2XsW4TQRRFnx1hEcU2kZDSgOSVkFJRRNqCMhYUtPmEtKSBIhUNLaKy\nlA9IIiHR8wWIH6CioKDgCwApDdVmht157955OyPMFN43c++Zsdb7dL2y6rpGthlvuu5IVm27vw0k\n99vjQBlyrwvDplShtiTqgHw8QY2p89mH9+y1GWpE7Zy8kI/mo4o0oqbyTb5syKwT0oh6Kz/ksnA/\nSSOqCTsfr3V7KpowU02pC5E7R0H5St5+4rRETV6u5XnwzK485bVEyfRUrgPwzkPiNaUmN7uvRCan\nk0+ec5pScrgX7vnn9snaU04zavUg2H+OP4a5ZtT0z8ghw1KuGTU/KVO5ZlSZ8UqiYmMNw5mSEK6D\nlihnrS4gVWkuOXu9hn2QKjeXzDZ7TYEqN5csNnP8YfCscnPJcn/3d+EsaCBwDOXODazhWePNNZgX\n+BAQNdZc6YSzVMQrUaPN1bvvXvXX/hOp8ebqfQfyEDCkCs0V3fNH7XmBKjRXdC+77leBEsEEAJOW\nScdvKIIJoFYoks4UJgCYtUw6UZQAarVCdaIoAcyslepEUQKo1wrViWqCQRPAzFqprpRLAPX2BemJ\n8gnAFOuJ8gnAFOtKuQTIKNKVcgmQUaQblSdATqFuVJ4AOYW6UXkC5BTqRuWu2jxRPgH+Uk+/x/FY\nhPVE1Xb2GlK1dIikJQRSlXSIECQEUpV0iBQkBFKVdIgUJARS9eYKnCYEU2N/veIxw1ikhCCqlg4R\n1IQgqpIOEbKEQKqWDpGyhECqkg4RgoRAqpIOkYKEQCpK/zYi9V9vN8+2f7tpN7emE+53mL7w/gAA\nAABJRU5ErkJggg==\n", "prompt_number": 11, "text": [ "\u23a1 ___ ___ \u23a4\n", "\u23a2\u2572\u2571 2 \u2572\u2571 2 \u23a5\n", "\u23a2\u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500 \u23a5\n", "\u23a2 2 2 \u23a5\n", "\u23a2 \u23a5\n", "\u23a2 ___ ___ \u23a5\n", "\u23a2\u2572\u2571 2 -\u2572\u2571 2 \u23a5\n", "\u23a2\u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n", "\u23a3 2 2 \u23a6" ] } ], "prompt_number": 11 }, { "cell_type": "code", "collapsed": false, "input": [ "sg_print(M)" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "$\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix}1 & 1\\\\1 & -1\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "sg_print() can also do decimal approximations, which can\n", "be handy when you're deailing with an object too complicated in its\n", "exact format:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "M2 = M/(1+sqrt(3))\n", "M2" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left[\\begin{matrix}\\frac{\\sqrt{2}}{2 + 2 \\sqrt{3}} & \\frac{\\sqrt{2}}{2 + 2 \\sqrt{3}}\\\\\\frac{\\sqrt{2}}{2 + 2 \\sqrt{3}} & - \\frac{\\sqrt{2}}{2 + 2 \\sqrt{3}}\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAKMAAABCBAMAAADJWnuUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 13, "text": [ "\u23a1 ___ ___ \u23a4\n", "\u23a2 \u2572\u2571 2 \u2572\u2571 2 \u23a5\n", "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n", "\u23a2 \u239b ___\u239e \u239b ___\u239e\u23a5\n", "\u23a22\u22c5\u239d1 + \u2572\u2571 3 \u23a0 2\u22c5\u239d1 + \u2572\u2571 3 \u23a0\u23a5\n", "\u23a2 \u23a5\n", "\u23a2 ___ ___ \u23a5\n", "\u23a2 \u2572\u2571 2 -\u2572\u2571 2 \u23a5\n", "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n", "\u23a2 \u239b ___\u239e \u239b ___\u239e\u23a5\n", "\u23a32\u22c5\u239d1 + \u2572\u2571 3 \u23a0 2\u22c5\u239d1 + \u2572\u2571 3 \u23a0\u23a6" ] } ], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [ "sg_print(M, exact=False, ndigs=3)" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "$\\left[\\begin{matrix}0.707 & 0.707\\\\0.707 & -0.707\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", "If you set the \"exact\" parameter to False, then you will get a decimal\n", "approximation. Note that python requires \"True\" and \"False\" to begin with\n", "a capital \"T\" and \"F.\" The \"ndigs\" parameter tells how many \"digits\n", "of accuracy\" you want. In many cases, this will end up meaning the number\n", "of digits after the decimal point. There is a limit to how many digits\n", "you can get, but we are mainly interested in making the number small, so\n", "that things print out in a readable way.\n", "

\n", "

\n", "The reason I used the name sg_print() is because python\n", "already has a print() function:\n", "

" ] }, { "cell_type": "code", "collapsed": false, "input": [ "print(M)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Matrix([[sqrt(2)/2, sqrt(2)/2], [sqrt(2)/2, -sqrt(2)/2]])\n" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Python's print tends to prints thing out the way you would enter them\n", "into a program. This can be useful if you're trying to figure out\n", "how to enter something." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "

^ Finding Eigenvectors

\n", "

\n", "The print_eigenvectors() function finds the eigenvalues and\n", "eigenvectors of an operator (a matrix) and prints them out in a readable\n", "format:\n", "

" ] }, { "cell_type": "code", "collapsed": false, "input": [ "print_eigenvectors(sigma_y)" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "The operator is: $\\left[\\begin{matrix}0 & - i\\\\i & 0\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] }, { "html": [ "eigenvalue: $-1$, eigenvector: $\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix}1\\\\- i\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] }, { "html": [ "eigenvalue: $1$, eigenvector: $\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix}1\\\\i\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] } ], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "But if you want to get the actual eigenvectors and assign them to\n", "variables for further calculation, then use the \n", "find_eigenvectors() function:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "eval, evec = find_eigenvectors(sigma_y)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", "\"eval\" and \"evec\" are just variables. You can name them anything you\n", "want. The first variable will end up with a list of all the eigenvalues,\n", "and the second one will end up with a list of all the eigenvectors.\n", "You can get a specific one by supplying an index in square \n", "brackets. Note that python, like most computer languages, uses zero for\n", "the first index, rather than one.\n", "

" ] }, { "cell_type": "code", "collapsed": false, "input": [ "eval" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\begin{bmatrix}-1, & 1\\end{bmatrix}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAZBAMAAACP0x4YAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAu90iEM0ymat272ZEieIx\nuvsAAAB6SURBVCgVYxBSMmDADw4pXWBQwK8EJMvcQI6qEAyDWR0wzDJ8gq6KWQ5DlWMFuioW8T4M\nVQyc6KoYGOYNelWuSiAQxjCoXe8XgBJLmKHK3vWom4EJJfwr1mlswJpyDqOYBeJgS18biFHFQpQq\nbgyjsNpIc1VKROTHBgCD+TIQNJw1owAAAABJRU5ErkJggg==\n", "prompt_number": 18, "text": [ "[-1, 1]" ] } ], "prompt_number": 18 }, { "cell_type": "code", "collapsed": false, "input": [ "eval[0]" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$-1$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJ1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAilU6eAAAADHRSTlMAEM3dMiK7mat272a9noTYAAAALklE\nQVQIHWNggANFOIuBIewggiNWgcRh4CSTI2QMAioMZBsAcxB71+FuGBtMAwDSnw2AS5zPhQAAAABJ\nRU5ErkJggg==\n", "prompt_number": 19, "text": [ "-1" ] } ], "prompt_number": 19 }, { "cell_type": "code", "collapsed": false, "input": [ "eval[1]" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$1$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAjSURBVAgd\nY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCHcAAAAABJRU5ErkJggg==\n", "prompt_number": 20, "text": [ "1" ] } ], "prompt_number": 20 }, { "cell_type": "code", "collapsed": false, "input": [ "evec" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\begin{bmatrix}\\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\- \\frac{\\sqrt{2} i}{2}\\end{matrix}\\right], & \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2} i}{2}\\end{matrix}\\right]\\end{bmatrix}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAK4AAAA/BAMAAACLGO47AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1QAAAABJRU5ErkJggg==\n", "prompt_number": 21, "text": [ "\u23a1\u23a1 ___ \u23a4 \u23a1 ___ \u23a4\u23a4\n", "\u23a2\u23a2 \u2572\u2571 2 \u23a5 \u23a2 \u2572\u2571 2 \u23a5\u23a5\n", "\u23a2\u23a2 \u2500\u2500\u2500\u2500\u2500 \u23a5, \u23a2 \u2500\u2500\u2500\u2500\u2500 \u23a5\u23a5\n", "\u23a2\u23a2 2 \u23a5 \u23a2 2 \u23a5\u23a5\n", "\u23a2\u23a2 \u23a5 \u23a2 \u23a5\u23a5\n", "\u23a2\u23a2 ___ \u23a5 \u23a2 ___ \u23a5\u23a5\n", "\u23a2\u23a2-\u2572\u2571 2 \u22c5\u2148 \u23a5 \u23a2\u2572\u2571 2 \u22c5\u2148\u23a5\u23a5\n", "\u23a2\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5 \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\u23a5\n", "\u23a3\u23a3 2 \u23a6 \u23a3 2 \u23a6\u23a6" ] } ], "prompt_number": 21 }, { "cell_type": "code", "collapsed": false, "input": [ "sg_print(evec[0])\n", "sg_print(evec[1])" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "$\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix}1\\\\- i\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] }, { "html": [ "$\\frac{1}{\\sqrt{2}}\\left[\\begin{matrix}1\\\\i\\end{matrix}\\right]$" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] } ], "prompt_number": 22 } ], "metadata": {} } ] }