brianjp93/Lab 3 (Brownian Motion).ipynb

## Lab 3 (Brownian Motion).ipynb
{
 "metadata": {
  "name": "Lab 3 (Brownian Motion)"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "Brownian Motion Lab"
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "IPython Code written by Brian Perrett on 2/16/14 for PHYS 391"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Import some necessary modules"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "%matplotlib qt\n%pylab inline\nimport csv\nnp = numpy\npl = pylab",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Populating the interactive namespace from numpy and matplotlib\n"
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": "WARNING: pylab import has clobbered these variables: ['f']\n`%pylab --no-import-all` prevents importing * from pylab and numpy\n"
      }
     ],
     "prompt_number": 124
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Set path to my Brownian Motion data.  This is a CSV file with columns (time,x1,y1,x2,y2)"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "datapath = \"C:/users/brian/google drive/sophomore/physics 391 lab/lab_3_brownian_motion/data.csv\"",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 125
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "\"results\" is a list of lists containing time,x1,x2,y1, and y2 data"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "results = []\nwith open(datapath) as f:\n    reader = csv.reader(f)\n    for row in reader:\n        results.append(row)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 126
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "show what is assigned to 'results'.  It is a list of lists containing data for time, x1, y1, x2, and y2"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "results",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 127,
       "text": "[['time(s)', 'x1(mm)', 'y1', 'x2', 'y2'],\n ['0', '0.203960396', '0.022574257', '0.296237624', '0.034059406'],\n ['10.00166667', '0.203960396', '0.021782178', '0.295841584', '0.032475248'],\n ['20.00166667', '0.205940594', '0.02019802', '0.295841584', '0.031683168'],\n ['30.00166667', '0.205148515', '0.018217822', '0.294257426', '0.031683168'],\n ['40.00166667', '0.203564356', '0.019405941', '0.296237624', '0.027722772'],\n ['50.00166667', '0.202772277', '0.019405941', '0.296633663', '0.029306931'],\n ['60.00166667', '0.202772277', '0.018217822', '0.297821782', '0.028118812'],\n ['70.00166667', '0.201980198', '0.018217822', '0.297029703', '0.027326733'],\n ['80.00166667', '0.201188119', '0.017029703', '0.297425743', '0.027326733'],\n ['90.00166667', '0.201584158', '0.018217822', '0.297029703', '0.026138614'],\n ['100.0016667', '0.201188119', '0.01980198', '0.299009901', '0.026930693'],\n ['110.0016667', '0.19960396', '0.021386139', '0.296237624', '0.026930693'],\n ['120.0016667', '0.199207921', '0.022970297', '0.297029703', '0.026138614'],\n ['130.0016667', '0.2', '0.023762376', '0.298613861', '0.025742574'],\n ['140.0016667', '0.201584158', '0.024158416', '0.29980198', '0.025742574'],\n ['150.0016667', '0.200792079', '0.024950495', '0.299009901', '0.026138614'],\n ['160.0016667', '0.201584158', '0.026930693', '0.299009901', '0.025742574'],\n ['170.0016667', '0.202376238', '0.028514851', '0.299009901', '0.025346535'],\n ['180.0016667', '0.20039604', '0.029306931', '0.300990099', '0.026138614'],\n ['190.0033333', '0.201188119', '0.028514851', '0.301386139', '0.025346535'],\n ['200.0033333', '0.201584158', '0.028118812', '0.299405941', '0.025346535'],\n ['210.0033333', '0.201188119', '0.028118812', '0.297425743', '0.025742574'],\n ['220.0033333', '0.201188119', '0.026930693', '0.297029703', '0.026930693'],\n ['230.0033333', '0.201188119', '0.027326733', '0.297029703', '0.025346535'],\n ['240.0033333', '0.2', '0.026930693', '0.298217822', '0.024950495'],\n ['250.0033333', '0.200792079', '0.025742574', '0.296237624', '0.024554455'],\n ['260.0033333', '0.2', '0.026138614', '0.299009901', '0.024554455'],\n ['270.0033333', '0.201584158', '0.024554455', '0.298613861', '0.022178218'],\n ['280.0033333', '0.201188119', '0.023366337', '0.299009901', '0.021782178'],\n ['290.0033333', '0.20039604', '0.024158416', '0.297821782', '0.020990099'],\n ['300.0033333', '0.2', '0.024158416', '0.295841584', '0.021386139'],\n ['310.0033333', '0.20039604', '0.024554455', '0.295049505', '0.02019802'],\n ['320.0033333', '0.2', '0.025346535', '0.293861386', '0.019009901'],\n ['330.0033333', '0.201188119', '0.026138614', '0.294257426', '0.018613861'],\n ['340.0033333', '0.202376238', '0.027326733', '0.290693069', '0.018217822'],\n ['350.0033333', '0.202376238', '0.027722772', '0.290693069', '0.020990099'],\n ['360.0033333', '0.201584158', '0.028118812', '0.291881188', '0.024554455'],\n ['370.0033333', '0.2', '0.028910891', '0.292277228', '0.025742574'],\n ['380.005', '0.201188119', '0.027722772', '0.293069307', '0.024158416'],\n ['390.005', '0.202376238', '0.028118812', '0.293069307', '0.022574257'],\n ['400.005', '0.201188119', '0.028514851', '0.293069307', '0.02019802'],\n ['410.005', '0.201980198', '0.028514851', '0.294257426', '0.021386139'],\n ['420.005', '0.201584158', '0.029306931', '0.295841584', '0.019009901'],\n ['430.005', '0.201980198', '0.03049505', '0.294653465', '0.02019802'],\n ['440.005', '0.201584158', '0.030891089', '0.295049505', '0.020990099'],\n ['450.005', '0.202376238', '0.02970297', '0.295445545', '0.022178218'],\n ['460.005', '0.202376238', '0.03009901', '0.291881188', '0.021782178'],\n ['470.005', '0.202376238', '0.031683168', '0.28990099', '0.022178218'],\n ['480.005', '0.203960396', '0.032079208', '0.29029703', '0.023366337'],\n ['490.005', '0.201980198', '0.032871287', '0.29029703', '0.024950495'],\n ['500.005', '0.201584158', '0.033663366', '0.28950495', '0.024950495'],\n ['510.005', '0.201980198', '0.031683168', '0.28990099', '0.024158416'],\n ['520.005', '0.201584158', '0.032079208', '0.29029703', '0.024554455'],\n ['530.005', '0.202376238', '0.033663366', '0.291089109', '0.026930693'],\n ['540.005', '0.202772277', '0.032475248', '0.29029703', '0.024950495'],\n ['550.005', '0.201980198', '0.031287129', '0.29029703', '0.025346535'],\n ['560.0066667', '0.200792079', '0.03049505', '0.288712871', '0.022970297'],\n ['570.0066667', '0.201584158', '0.028910891', '0.289108911', '0.021782178'],\n ['580.0066667', '0.2', '0.027722772', '0.289108911', '0.021386139'],\n ['590.0066667', '0.19960396', '0.029306931', '0.289108911', '0.019405941'],\n ['600.0066667', '0.2', '0.02970297', '0.288316832', '0.020990099'],\n ['610.0066667', '0.200792079', '0.029306931', '0.287524752', '0.022970297'],\n ['620.0066667', '0.200792079', '0.02970297', '0.287920792', '0.022970297'],\n ['630.0066667', '0.2', '0.027326733', '0.29029703', '0.024554455'],\n ['640.0066667', '0.20039604', '0.028118812', '0.289108911', '0.021386139'],\n ['650.0066667', '0.200792079', '0.028118812', '0.290693069', '0.021782178'],\n ['660.0066667', '0.202376238', '0.03049505', '0.28990099', '0.020990099'],\n ['670.0066667', '0.202772277', '0.030891089', '0.29029703', '0.024554455'],\n ['680.0066667', '0.202772277', '0.032079208', '0.291485149', '0.024158416'],\n ['690.0066667', '0.202376238', '0.030891089', '0.291485149', '0.023762376'],\n ['700.0066667', '0.202772277', '0.032079208', '0.291881188', '0.023762376'],\n ['710.0066667', '0.202376238', '0.032079208', '0.292277228', '0.023762376'],\n ['720.0066667', '0.202376238', '0.02970297', '0.291485149', '0.023762376'],\n ['730.0066667', '0.202772277', '0.030891089', '0.291881188', '0.023762376'],\n ['740.0066667', '0.201980198', '0.033267327', '0.291485149', '0.023366337'],\n ['750.0083333', '0.203960396', '0.032475248', '0.291881188', '0.023762376'],\n ['760.0083333', '0.202772277', '0.033267327', '0.291881188', '0.023762376'],\n ['770.0083333', '0.202376238', '0.033663366', '0.291881188', '0.024158416'],\n ['780.0083333', '0.201584158', '0.033267327', '0.291881188', '0.024158416'],\n ['790.0083333', '0.201980198', '0.034455446', '0.291485149', '0.024158416'],\n ['800.0083333', '0.202772277', '0.034455446', '0.291881188', '0.023762376'],\n ['810.0083333', '0.200792079', '0.032079208', '0.292277228', '0.024158416'],\n ['820.0083333', '0.200792079', '0.032475248', '0.292277228', '0.023366337'],\n ['830.0083333', '0.20039604', '0.032871287', '0.291485149', '0.024554455'],\n ['840.0083333', '0.201584158', '0.03049505', '0.291485149', '0.024554455'],\n ['850.0083333', '0.201584158', '0.03049505', '0.291485149', '0.023366337'],\n ['860.0083333', '0.200792079', '0.029306931', '0.291485149', '0.023762376'],\n ['870.0083333', '0.201188119', '0.028118812', '0.291485149', '0.024158416'],\n ['880.0083333', '0.200792079', '0.028910891', '0.291881188', '0.023762376'],\n ['890.0083333', '0.201188119', '0.028514851', '0.291881188', '0.023762376'],\n ['900.0083333', '0.201584158', '0.027326733', '0.291881188', '0.023762376'],\n ['910.0083333', '0.201584158', '0.028910891', '0.291881188', '0.023762376'],\n ['920.0083333', '0.20039604', '0.028910891', '0.291881188', '0.023762376'],\n ['930.01', '0.20039604', '0.029306931', '0.291881188', '0.023762376'],\n ['940.01', '0.198811881', '0.03009901', '0.291485149', '0.024158416'],\n ['950.01', '0.19960396', '0.03049505', '0.291485149', '0.022970297'],\n ['960.01', '0.198811881', '0.028910891', '0.291485149', '0.022970297'],\n ['970.01', '0.198811881', '0.028514851', '0.291485149', '0.023366337'],\n ['980.01', '0.198811881', '0.027722772', '0.291485149', '0.023762376'],\n ['990.01', '0.197623762', '0.026930693', '0.291881188', '0.024158416']]"
      }
     ],
     "prompt_number": 127
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Initialize new variables"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "time_data = []\nx1_data = []\nx2_data = []\ny1_data = []\ny2_data = []",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 128
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "fill time_data list, delete first entry (Becase it is a string).   This also converts the distances from mm to m."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for item in results:\n    time_data.append(item[0])\ndel time_data[0]\n\nfor item in results:\n    x1_data.append(item[1])\ndel x1_data[0]\nfor i in range(len(x1_data)):\n    x1_data[i] = float(x1_data[i])/1000\n\nfor item in results:\n    y1_data.append(item[2])\ndel y1_data[0]\nfor i in range(len(y1_data)):\n    y1_data[i] = float(y1_data[i])/1000\n\nfor item in results:\n    x2_data.append(item[3])\ndel x2_data[0]\nfor i in range(len(x2_data)):\n    x2_data[i] = float(x2_data[i])/1000\n\nfor item in results:\n    y2_data.append(item[4])\ndel y2_data[0]\nfor i in range(len(y2_data)):\n    y2_data[i] = float(y2_data[i])/1000",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 129
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "plots a time vs x graph (unneeded, but you can run this if you'd like the graph)"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "ignore the error.  use of 'f' is not longer necessary"
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "The plot below is a plot of (time, position).  Time is in seconds.  Position is in Meters"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(len(x1_data)):\n    matplotlib.pyplot.plot(time_data[i],x1_data[i],'bo')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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gQ0REqmOwISIi1THYEBGR6hhsiIhIdQw2RESkOgYbIiJSHYMNERGpjsGGiIhU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\nAAAASUVORK5CYII=\n",
       "text": "<matplotlib.figure.Figure at 0x7cf4048>"
      }
     ],
     "prompt_number": 130
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Initialize Change in Position"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "deltax1=[]\ndeltay1=[]\ndeltax2=[]\ndeltay2=[]",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 131
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "initialize delta values (from above)"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(len(x1_data)-1):\n    change = float(x1_data[i+1]) - float(x1_data[i])\n    deltax1.append(change)\n    \nfor i in range(len(y1_data)-1):\n    change = float(y1_data[i+1]) - float(y1_data[i])\n    deltay1.append(change)\n    \nfor i in range(len(x2_data)-1):\n    change = float(x2_data[i+1]) - float(x2_data[i])\n    deltax2.append(change)\n    \nfor i in range(len(y2_data)-1):\n    change = float(y2_data[i+1]) - float(y2_data[i])\n    deltay2.append(change)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 132
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Find DeltaR1 and DeltaR2 using our just found delta values.  Use distance formula."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "deltar1 = []\ndeltar2 = []",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 133
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(len(deltax1)):   #I chose deltax1, but it doesn't really matter which one you choose.  They are all the same length.\n    r = sqrt((deltax1[i]**2)+(deltay1[i]**2))\n    deltar1.append(r)\n    \nfor i in range(len(deltax1)):   #I chose deltax1, but it doesn't really matter which one you choose.  They are all the same length.\n    r = sqrt((deltax2[i]**2)+(deltay2[i]**2))\n    deltar2.append(r)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 134
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Import Math to use laTex."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "from IPython.display import Math",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 135
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$$mean(x)^2$$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "meanx_squared = mean(deltax1)**2",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 136
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(len(deltax1)):\n    deltax1[i] = float(deltax1[i])",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 137
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$$mean(x^2)$$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "mean_xsquared = mean(power(deltax1, 2))",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 138
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "meanx2_squared = mean(deltax2)**2",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 139
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "mean_x2squared = mean(power(deltax2, 2))",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 140
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Compare our mean(x^2) and mean(x)^2"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "our mean(x)^2 should be much less than mean(x^2)"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "meanx_squared",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 141,
       "text": "4.0968197581834192e-15"
      }
     ],
     "prompt_number": 141
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "mean_xsquared",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 142,
       "text": "7.9849566398240025e-13"
      }
     ],
     "prompt_number": 142
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "meanx2_squared",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 143,
       "text": "1.9363875749510819e-15"
      }
     ],
     "prompt_number": 143
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "mean_x2squared",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 144,
       "text": "1.1834847638428302e-12"
      }
     ],
     "prompt_number": 144
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "And it is.  This means that the total movement is basically = 0"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$\\overline{(\\Delta x^2)}$ and $\\overline{(\\Delta x)}^2$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$$\\overline {(\\Delta x)}^2 = 1.94e-15 m$$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$$\\overline {(\\Delta x^2)} = 1.18e-12 m$$"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Define a general formula for $\\overline{(\\Delta x^2)}$ and $\\overline{(\\Delta x)}^2$"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "meanf_squared()    and    mean_fsquared()"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def meanf_squared(f):\n    \"\"\"\n    takes one list input (f), of position-change values\n    returns: mean(f)^2\n    \"\"\"\n    return mean(f)**2",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 145
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def mean_fsquared(f):\n    \"\"\"\n    returns mean(f^2)\n    \"\"\"\n    return mean(power(f, 2))",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 146
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "%who",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Math\t change\t csv\t d_r1\t d_r2\t datapath\t deltar1\t deltar2\t deltax1\t \ndeltax2\t deltay1\t deltay2\t every_fifth_deltar2\t every_fourth_deltar2\t every_other_deltar2\t every_sixth_deltar2\t every_third_deltar2\t first\t \nfirstf\t i\t item\t kb_1\t kb_2\t mean_fsquared\t mean_x2squared\t mean_xsquared\t meanf_squared\t \nmeanx2_squared\t meanx_squared\t pl\t r\t reader\t results\t rms_deltar2_1\t rms_deltar2_2\t rms_deltar2_3\t \nrms_deltar2_4\t rms_deltar2_5\t rms_deltar2_6\t rms_list\t rms_r\t root_deltat_list\t row\t second\t secondf\t \nsig_avg_delta_r1squared\t sig_avg_delta_r2squared\t sig_d\t sig_delta_r1_squared\t sig_delta_r2_squared\t sigf\t thirdf\t time_data\t uncert_delta_r_squared\t \nuncertposition\t x1_data\t x2_data\t y1_data\t y2_data\t \n"
      }
     ],
     "prompt_number": 147
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "(Page 22) - Histograms of each delta array"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hist(multiply(deltax1,1000000), bins=10)\nxlabel(\"Delta x (microns)\")\nylabel(\"occurances\")\ntitle(\"Delta x1\")",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 148,
       "text": "<matplotlib.text.Text at 0x833b4e0>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7Q7EIpCTLC6Da64Ys4yBDBrWXBVlz6P0InI1FICdZXgDVXjdkGQcZMqi9LMiawz4+unTp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\niIg0jkVARKRxLAIiIo1jERARadz/A+y67dIku/gBAAAAAElFTkSuQmCC\n",
       "text": "<matplotlib.figure.Figure at 0x7ffcf28>"
      }
     ],
     "prompt_number": 148
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hist(multiply(deltay1,1000000), bins=10)\nxlabel(\"Delta y (microns)\")\nylabel(\"occurances\")\ntitle(\"Delta y1\")",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 149,
       "text": "<matplotlib.text.Text at 0x81e52e8>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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4/fbbxcGDB4UQHRfxaGtrE9nZ2cLf319cuHBBCCHE448/LlatWiWEEGLPnj1C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\nYII=\n",
       "text": "<matplotlib.figure.Figure at 0x80030f0>"
      }
     ],
     "prompt_number": 149
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hist(multiply(deltax2,1000000), bins=10)\nxlabel(\"Delta x (microns)\")\nylabel(\"occurances\")\ntitle(\"Delta x2\")",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 150,
       "text": "<matplotlib.text.Text at 0x81a8630>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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w9ImIFIShTw7D1dUVOp0OERER0Gq1eOONNyx+sKqwsNC8nO+3336Lzz//XJLa\nzp8/j/j4+Ha9ZunSpbJ8mnnSpEkoKCiw+7jkHBj65DDc3d2Rm5uL7777Drt27cLnn3+O1FTrr7Wa\nm5uLzz77TJLaMjMz8fTTT7frNampqRg9erRV2zY1NXWgqmubPn063nzzTZvtj24y9vrEGJElnp6e\nLe7/9NNPokePHkKIK+vNz5s3TwwdOlRERUWJNWvWCCGEKCgoEBEREaKhoUH069dP9OzZU2i1WrFx\n40Zx8OBBcffddwudTieGDx8uTpw40WbMrVu3itGjRwshhPj5559FcHDwNT89HRoaal6vfN26dWL8\n+PEiNjZWaDQakZGRIVatWiV0Op0YNmyYqKioEEIIkZSUJLZs2SKEEOLgwYNi+PDhIjo6WsTExIia\nmhqxbt06kZCQIO677z6h1+tFRUWFGD9+vIiKihLDhg0TR48eFUIIsXTpUjF16lSh1+vFHXfcIdLT\n04UQQtTW1oqHHnpIREdHi4iICLFx40YhhBANDQ0iKCioc/8ZdNOSdcE1ohvp378/TCYTzp8/j+zs\nbPj4+ODgwYO4fPkyRo4cibi4OPO2bm5ueOWVV3D48GHzVY9qamqwb98+uLq64osvvsCiRYuwZcuW\nFmNMnDgRW7duRWZmJv75z3/i5ZdfbrPGU0lJCVxdXc3rlQPAsWPHkJeXh/r6egQFBWHVqlU4cuQI\n5syZgw8//BAvvPACVCoVVCoVGhoa8Pjjj2PTpk0YPHgwamtr0b17dwBX/jrJz8+Hj48PZs6cicGD\nByM7Oxt79uxBYmIicnNzAQAnT57Enj17UF1djYEDB2LGjBnIyclB37598emnnwIAqqurzd+Lvn37\n4vvvv+f1p6kNhj45hZ07dyI/P98c2tXV1Th16hQGDBhg3kYI0eIYQFVVFRITE3Hq1CmoVCo0NjZe\nc98ZGRkIDw/H8OHD8dhjj7V5/r///S969+5tvq9SqWAwGODh4QEPDw/4+PiYl3KOjIzE0aNHW9R0\n4sQJ9O7dG4MHDwYA85K9KpUKsbGx8PHxAQAcOHAAW7duBQAYDAb88ssvqKmpgUqlQnx8PNzc3NCj\nRw/06tUL58+fR1RUFObNm4cFCxZg7NixGDlypHncPn36oLCwkKFPbbCnTw7rp59+gqurq3nmnZmZ\nidzcXOTm5uL06dO4//77b/j6JUuWYPTo0cjPz8c//vEPXLp06ZrbnT17Fq6urigtLb3ugePWj99y\nyy3m2y4zNy6EAAACFklEQVQuLub7Li4ubfrzN1or38PD44bjXNW1a1fzbVdXVzQ1NeHOO+9Ebm4u\nIiMj8dJLL+GVV15psZ+rVz4j+i3+VJBDKisrw3PPPYeZM2cCAMaMGYO3337bHKgnT55EXV1di9d4\ne3ujpqbGfL+6uhp9+vQBgOtewKapqQnJycn4+9//jpCQELzxxhtttrn99ttbXKzlesF8redUKhUG\nDhyI4uJiHDp0CMCVtpPJZGqz7ahRo7B+/XoAgNFoRM+ePeHl5XXd8YqLi9GtWzdMnjwZ8+bNa3EB\n+OLiYtx+++3XrZOUi+0dchj19fXQ6XRobGxEly5dkJiYiNmzZwMAnnnmGRQWFmLQoEEQQqBXr17I\nzs4G8OtM2mAwIC0tDTqdDgsXLsSLL76IpKQk/PnPf0Z8fPw1Z9zLly/HPffcg+HDhyMqKgpDhw7F\n2LFjMXDgQPM2/v7+aGpqQl1dHdzd3c29+qta3249jpubGzZu3IiZM2eivr4e7u7u2LVrV5ttU1JS\nMG3aNERHR8PDwwNZWVnX3ScA5OfnY/78+XBxcUHXrl3xzjvvALhyMZBz584hJCSkff8BpAhcWpnI\nCikpKQgNDb1mz9/R7Ny5E59++ilWr14tdynkgNjeIbLCH/7wB/PM29G9//775r+QiFrjTJ+ISEE4\n0yciUhCGPhGRgjD0iYgUhKFPRKQgDH0iIgVh6BMRKcj/Axdgg2i/Jkk9AAAAAElFTkSuQmCC\n",
       "text": "<matplotlib.figure.Figure at 0x81dc710>"
      }
     ],
     "prompt_number": 150
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hist(multiply(deltay2,1000000), bins=10)\nxlabel(\"Delta y (microns)\")\nylabel(\"occurances\")\ntitle(\"Delta y2\")",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 151,
       "text": "<matplotlib.text.Text at 0x85c1c50>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ee++9ht/QLFsc7lBL3rh75swZ6+309HTx9NNPK5jm1j7//HMRHBwsCgoKlI7i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\n+ImIVIbFT0SkMv8Pha0FDLVFN1AAAAAASUVORK5CYII=\n",
       "text": "<matplotlib.figure.Figure at 0x80721d0>"
      }
     ],
     "prompt_number": 151
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "Find Standard Deviations"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "std(deltax1)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 152,
       "text": "8.9129054983446167e-07"
      }
     ],
     "prompt_number": 152
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "std(deltay1)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 153,
       "text": "1.1157602757294859e-06"
      }
     ],
     "prompt_number": 153
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "std(deltax2)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 154,
       "text": "1.0869905134212889e-06"
      }
     ],
     "prompt_number": 154
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "std(deltay2)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 155,
       "text": "1.228559188267289e-06"
      }
     ],
     "prompt_number": 155
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "(Page 24) - Dispersion Constant (D)"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$$\\overline {(\\Delta r)^2} = 4D \\Delta t$$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "so.."
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$$D = \\frac {\\overline {(\\Delta r)^2}}{(4\\Delta t)}$$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "where delta(t) = 10 seconds."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "print(\" the mean of (delta r)^2 = \" + str(mean_fsquared(deltar1)))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": " the mean of (delta r)^2 = 2.04535304445e-12\n"
      }
     ],
     "prompt_number": 156
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "d_r1 = mean_fsquared(deltar1)/(4*10)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 157
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "d_r1",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 158,
       "text": "5.1133826111332259e-14"
      }
     ],
     "prompt_number": 158
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": "Find delta(r2)"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "str(mean_fsquared(deltar2))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 159,
       "text": "'2.70284444302e-12'"
      }
     ],
     "prompt_number": 159
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "d_r2 = mean_fsquared(deltar2)/(4*10)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 160
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "d_r2",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 161,
       "text": "6.7571111075470226e-14"
      }
     ],
     "prompt_number": 161
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$$D_1 = 5.1133826111332259e-14$$"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$$D_2 = 6.7571111075470226e-14$$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The above two numbers are fairly similar (same magnitude).  They are measuring the same value, so that's good."
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "(Page 25) - Measuring the Boltzmann Constant (Kb)"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$$Df = K_bT$$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "so"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$$K_b = \\frac {Df}{T}$$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Where T is approximately room temperature (298K).  "
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$f = 6(pi)*(1e-3)*R = 6.03185795e-8$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "f = 6*pi*1e-3*3.2e-6\nf",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 162,
       "text": "6.031857894892403e-08"
      }
     ],
     "prompt_number": 162
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$f = 6 \\pi \\eta R = 6.0318795e-8$"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Find $K_b$ using our $D_{r1}$ and $D_{r2}$ values"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "kb_1 = (d_r1*f/298)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 163
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "kb_2 = (d_r2*f/298)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 164
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "(Page 25) - Boltzmann constant values"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "kb_1",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 165,
       "text": "1.0350066198848819e-23"
      }
     ],
     "prompt_number": 165
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "kb_2",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 166,
       "text": "1.3677159053933777e-23"
      }
     ],
     "prompt_number": 166
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "(Page 26) - Error Propagation For Measurement 2"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "I am only using Measurement 2 for the error propagation because that gave me the better data"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "For each measurement in position I have x+-1.6 microns"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "so."
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$ \\delta x = \\sqrt{1.6^2 + 1.6^2}$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "uncertposition = sqrt(1.6**2+1.6**2)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 167
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "uncertposition = uncertposition * 1e-6\nuncertposition",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 168,
       "text": "2.2627416997969522e-06"
      }
     ],
     "prompt_number": 168
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The uncertainty in position will be 2.263 microns for every measurement of position"
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "(Page 27 and 31) Find Error in delta (r^2)"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$( \\Delta r_i^2) = \\Delta x_i^2 + \\Delta y_i^2$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "$\\delta \\Delta(r^2)$ can be found in quadrature"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "for each $\\Delta(r^2)$, the uncertainty will be"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$   \\delta \\Delta (r_i^2) = \\sqrt{ \\left( \\frac {\\partial (\\Delta r_i^2)}{\\partial x_i} \\delta \\Delta x \\right)^2  +  \\left( \\frac {\\partial (\\Delta r_i^2)}{\\partial y_i} \\delta \\Delta y \\right)^2 }   $"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "After taking the partials, you get..."
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$ \\delta \\Delta (r_i^2) = \\sqrt{(2 \\Delta x_i * \\delta x)^2 + (2 \\Delta y_i* \\delta y)^2} $"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Finding $\\delta \\Delta(r^2)$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def uncert_delta_r_squared(deltax, deltay, sigdeltaposition):\n    first = ((2*deltax)*sigdeltaposition)\n    second = ((2*deltay)*sigdeltaposition)\n    return sqrt(first**2 + second**2)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 169
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sig_delta_r2_squared = []",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 170
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Finding error in $ \\Delta (r^2)$ for our second measurements"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(len(deltax2)):\n    first = deltax2[i]\n    second = deltay2[i]\n    sig_delta_r2_squared.append(uncert_delta_r_squared(first,second,uncertposition))\n\nsig_delta_r2_squared",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 171,
       "text": "[7.3897198222632879e-12,\n 3.5845448911503125e-12,\n 7.1690807313337393e-12,\n 2.003819487395229e-11,\n 7.3897231150365327e-12,\n 7.6039616000000541e-12,\n 5.0693056000000418e-12,\n 1.7922724455754323e-12,\n 5.6676588223274318e-12,\n 9.6516724143743937e-12,\n 1.2545893542576163e-11,\n 5.06930560000002e-12,\n 7.3897198222631361e-12,\n 5.3768128112424245e-12,\n 4.0076389747910253e-12,\n 1.7922724455751716e-12,\n 1.7922679200917748e-12,\n 9.6516724143741756e-12,\n 4.0076389747909914e-12,\n 8.9613531769089267e-12,\n 9.1388224109392671e-12,\n 5.6676588223274318e-12,\n 7.1690807313338621e-12,\n 5.6676588223274374e-12,\n 9.1388232984596066e-12,\n 1.2545893542576163e-11,\n 1.090195428430289e-11,\n 2.5346560000001867e-12,\n 6.4621239264063672e-12,\n 9.1388232984596034e-12,\n 6.4621239264062008e-12,\n 7.6039616000000541e-12,\n 2.5346560000000135e-12,\n 1.6229703273985769e-11,\n 1.2545893542575995e-11,\n 1.7002970742647628e-11,\n 5.6676588223273058e-12,\n 8.0152759257232655e-12,\n 7.1690852568172742e-12,\n 1.0753621097000824e-11,\n 7.6039616000000428e-12,\n 1.2924247852812552e-11,\n 7.6039615999998812e-12,\n 4.0076389747909914e-12,\n 5.6676588223272912e-12,\n 1.6229703773741357e-11,\n 9.1388232984596066e-12,\n 5.6676588223273833e-12,\n 7.1690807313338774e-12,\n 3.5845448911503739e-12,\n 4.0076389747909914e-12,\n 2.5346528000020355e-12,\n 1.1335316213570966e-11,\n 9.6516724143742919e-12,\n 1.7922724455751716e-12,\n 1.2924250363099024e-11,\n 5.6676588223274609e-12,\n 1.7922679200917748e-12,\n 8.9613531769090494e-12,\n 8.0152759257232655e-12,\n 9.6516740950972801e-12,\n 1.7922724455751869e-12,\n 1.2924247852812742e-11,\n 1.531316394925185e-11,\n 7.3897187246721923e-12,\n 5.0693056000000305e-12,\n 1.6229699275937348e-11,\n 5.667657391244474e-12,\n 1.7922724455751562e-12,\n 1.7922679200918053e-12,\n 1.7922724455749416e-12,\n 3.5845403656667469e-12,\n 1.7922679200918053e-12,\n 2.5346496000000284e-12,\n 2.5346496000000284e-12,\n 0.0,\n 1.7922724455751562e-12,\n 0.0,\n 1.7922679200918053e-12,\n 2.5346528000020351e-12,\n 2.5346559999998289e-12,\n 3.5845403656669311e-12,\n 6.4621201609764085e-12,\n 0.0,\n 5.3768082857587055e-12,\n 1.7922679200917748e-12,\n 1.7922724455751562e-12,\n 2.5346528000020351e-12,\n 0.0,\n 0.0,\n 0.0,\n 0.0,\n 0.0,\n 2.5346528000020351e-12,\n 5.3768128112421022e-12,\n 0.0,\n 1.7922724455751716e-12,\n 1.7922679200917748e-12,\n 2.5346528000020351e-12]"
      }
     ],
     "prompt_number": 171
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "rms_r = sqrt(sum(power(sig_delta_r2_squared, 2)))/(len(sig_delta_r2_squared))",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 172
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Uncertainty of $ \\overline{\\Delta r^2} = 7.48e-13$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "rms_r",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 173,
       "text": "7.4775255314684716e-13"
      }
     ],
     "prompt_number": 173
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": "Now we can find uncertainty in D."
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "We know..."
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$D=\\frac {\\overline {\\Delta r^2}}{4\\Delta t}$"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "So... The uncertainty in D is..."
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$\\delta D = \\frac {1}{4\\Delta t} \\delta {\\overline {\\Delta r^2}}$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sig_d = (1/40)*rms_r",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 174
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Now we know our sigma D"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sig_d",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 175,
       "text": "1.869381382867118e-14"
      }
     ],
     "prompt_number": 175
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "(Page 26) Now we have all the terms to fill in the following equation"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$\\delta k_b = \\sqrt{\\left(\\frac{\\partial k}{\\partial D} \\delta D \\right)^2   +   \\left( \\frac{\\partial k}{\\partial f} \\delta f \\right)^2    +   \\left( \\frac{\\partial k}{\\partial T} \\delta T \\right)^2 } $"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "or"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "$\\delta k_b = \\sqrt{\\left(\\frac{f}{T} \\delta D \\right)^2   +   \\left( \\frac{D}{T} \\delta f \\right)^2    +   \\left( \\frac{-Df}{T^2} \\delta T \\right)^2 }$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sigf = 6*pi*(1e-3)*(.3e-6)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 192
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sigf",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 193,
       "text": "5.6548667764616275e-09"
      }
     ],
     "prompt_number": 193
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "firstf = (f/298)*sig_d\nfirstf",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 194,
       "text": "3.783839883493927e-24"
      }
     ],
     "prompt_number": 194
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "secondf = ((d_r2*sigf)/298)\nsecondf",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 195,
       "text": "1.2822336613062917e-24"
      }
     ],
     "prompt_number": 195
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "thirdf = ((d_r2*6.032e-8*10)/298**2)\nthirdf",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 196,
       "text": "4.5897588172518858e-25"
      }
     ],
     "prompt_number": 196
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt((firstf**2)+(secondf**2)+(thirdf**2))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 197,
       "text": "4.0214706620975975e-24"
      }
     ],
     "prompt_number": 197
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "(Page 28) So now we have found that $\\delta K_b = 4.02e-24$ "
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "Time Evolution"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "every_other_deltar2 = []\nevery_third_deltar2 = []\nevery_fourth_deltar2 = []\nevery_fifth_deltar2 = []\nevery_sixth_deltar2 = []",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 182
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(len(deltar2)):\n    if i%2 == 0:\n        every_other_deltar2.append(deltar2[i])\n        \nfor i in range(len(deltar2)):\n    if i % 3 == 0:\n        every_third_deltar2.append(deltar2[i])\n\nfor i in range(len(deltar2)):\n    if i % 4 == 0:\n        every_fourth_deltar2.append(deltar2[i])\n        \nfor i in range(len(deltar2)):\n    if i % 5 == 0:\n        every_fifth_deltar2.append(deltar2[i])\n        \nfor i in range(len(deltar2)):\n    if i % 6 == 0:\n        every_sixth_deltar2.append(deltar2[i])\n        \n",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 183
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Now we have \"every_other_deltar2\" filled with every other delta r value from my second data set"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "We can solve for D, with delta t is equal to 20 seconds"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "mean(power(every_other_deltar2, 2))/(4*20)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 184,
       "text": "2.9252032024314538e-14"
      }
     ],
     "prompt_number": 184
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "rms_deltar2_1 = mean(power(deltar2,2))\nrms_deltar2_2 = mean(power(every_other_deltar2, 2))\nrms_deltar2_3 = mean(power(every_third_deltar2, 2))\nrms_deltar2_4 = mean(power(every_fourth_deltar2, 2))\nrms_deltar2_5 = mean(power(every_fifth_deltar2, 2))\nrms_deltar2_6 = mean(power(every_sixth_deltar2, 2))\nprint(rms_deltar2_2)\nprint(rms_deltar2_3)\nprint(rms_deltar2_4)\nprint(rms_deltar2_5)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "2.34016256195e-12\n3.35558288154e-12\n1.84452512373e-12\n3.61533205098e-12\n"
      }
     ],
     "prompt_number": 206
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "rms_list = [rms_deltar2_1, rms_deltar2_2, rms_deltar2_3, rms_deltar2_4, rms_deltar2_5, rms_deltar2_6]",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 186
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "root_deltat_list = [(4*10),(4*20),(4*30),(4*40),(4*50),(4*60)]",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 207
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "plot(root_deltat_list, rms_list, 'bo')\nxlabel('Time * 4')\nylabel('RMS')\ntitle('Time Evolution')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 211,
       "text": "<matplotlib.text.Text at 0x8d404e0>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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WLl1qTJ482ejZs6cRGRlpjBw50jh8+LB9+obcFhs2bDAsFosRFRVlREdHG9HR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\nAABMCAcAgMn/AaNbli4T+gSQAAAAAElFTkSuQmCC\n",
       "text": "<matplotlib.figure.Figure at 0x9189940>"
      }
     ],
     "prompt_number": 211
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "That is an ugly scatter plot"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "It is prettier as sqrt(t) vs Dispersion"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "time_list = [10,20,30,40,50,60]",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 220
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Dispersion_list = divide(rms_list,(power(time_list, .5)))",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 222
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "plot(time_list, Dispersion_list, 'bo')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 229,
       "text": "[<matplotlib.lines.Line2D at 0x9132a20>]"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": "<matplotlib.figure.Figure at 0x951b7b8>"
      }
     ],
     "prompt_number": 229
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "This looks a bit more linear"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}