mirror of
https://asciireactor.com/otho/phy-521.git
synced 2024-11-24 15:25:05 +00:00
350 lines
31 KiB
Plaintext
350 lines
31 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
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"outputs": [],
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"source": [
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"# EPR Paradox Example\n",
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"\n",
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"# A neutral pi meson (pi0) decays into an electron/positron (e-/e+) \n",
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"# pair.\n",
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"\n",
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"# pi0 --> e+ + e- (electron-positron pair)\n",
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"# pi0 has angular momentum l = s = 0\n",
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"\n",
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"# Align electron and positron detectors in opposite directions.\n",
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"\n",
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"# | pi0 |\n",
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"# |e- <------ ------> e+|\n",
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"\n",
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"# Where hbar = 1, the measurement of the spin in some direction is \n",
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"# +/-1 with spin state [1 0] (up) or [0 1] (down).\n",
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"\n",
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"# The Pauli exclusion principle with conserved angular momentum 0\n",
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"# dictates this system must be in the singlet state \n",
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"# chi = [1/sqrt(2) (|up+>|down-> - |down+>|up->)].\n",
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"\n",
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"# In this state, if the positron is measured to have spin [1 0], the \n",
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"# electron must have spin [0 1], or vice versa. There is an equal \n",
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"# probability to find either state during the first measurement.\n",
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"\n",
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"# This view is consistent with the realist view. The realist view could \n",
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"# hold that the electron and position had those angular momenta \n",
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"# from creation.\n",
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"\n",
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"# EPR assumes influences cannot propagate faster than the speed of \n",
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"# light. \"Wave function collapse\" is apparently instantaneous, however.\n",
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"\n",
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"\n",
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"import numpy as np\n",
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"import matplotlib\n",
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"import matplotlib.pyplot as plt\n",
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"import matplotlib.patches as mpatches\n",
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"%matplotlib inline "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": "iVBORw0KGgoAAAANSUhEUgAAAXwAAAEXCAYAAACu1P9TAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAHHpJREFUeJzt3XmcXGWd7/FPZZlAJsYEZgaDBvFCCIsOizDIIjQgAqMEZIIRGIcEr5gL6Au3GTalEwcUGZhhE4aLGRhFL6g4kHEQBGmILMFhuyJkuQoGEtaYQDbI1veP3yn69Mmp7qpOVyrN83m/XvWqruc8p+qp6lPfc85znnMKJEmSJEmSJEmSJEmSJEmSJKkubcB64PwWt6NRzwLPtLoRBe3EZ3lwi9uhPhjU6gaobm3EF+2eHupsn9XZ3EJic9G5kfNPJj7fkze+KXVrtM3HAz8HXgZWA68CvwW+D/xdP7WnetMAM6TVDVDD6vmi+WVsrs31870W+J/ASuBnxIq/AuwCfBw4CPj3jXyNK4EfAs9t5POoBQx8qXGVVjegxIFE2D8H7AcsKkwfQv90wyzObhqA7NJJQztd/a4TgYeBFcQX94fAtjXm2wq4AHgyq78UeBz4JjC8UHccsfW4EHgzu78B2LGX9pwIzAaW09UV1dt0stc/O2vPcmAZ8ADwqRrvpcwHgcuAJ4jPYhUwD/gnYFShbgcwI/v737L2VW/b5eoNAU4DHgJeJz63R4HTqb2iOIPodlkFPA9cAbyzgfcBsH92/xM2DHuAtcDdhbI2uo5t7AfcRfyPXye6hT5Y8jzt2TwHFcqr3Y1bE3saLwBvEMvO5BptPpn4n71CvPcF2et+skZ9bSS38NNyGjABuJX4cn4ImATsDuxB9PlWvS+rsx3w38B3iA2E8cCZwNXEFxRgHyIsRmTP/RTRjfC3wDHAR7LnKPoycDhwGxFGxZCrNX0U8MuszY8A383adiTwA2A34Gt1fB6fBY4lwvzO7Dn2Br4EHAXsS6xMIEJ+SfZ+/oNY0VS9lt0PBWYCHwXmEP3mbwCHEiG+Lxv2o18GfJ4I6X8lgvmYrO5QYuVZj1ez+53qrJ+3L7Hy/AXRZTMOOI4I9Y8Cv6rzeUYB9xNtvhkYRoT3DGKFkO9OuhA4C/g98H+Iz3BbYlmamM0vJauN+NL8soc622d1fl8ob8/KlxJhmHdjNu34QvkDWfk/lLzOVsSXGWKr9WlgHXBCod4ns+d4mu5bt9X2LCNWNkW9Tb8+m/6VQvkw4PasLfn52rL6Xy/U347yre5Tsvp/XyifnJXXOvhZbfdlhecdBFyXTZuQK98/K5tH9z2KYXR9/sX/ZS3bEiuk9cRK9wRi76qn7qc2uvZSTitMm5BrW9n/rmwLfz2xdZ+vvwuwhtiDyVtMbDBsUdKurXtos5SENjY+8Kf38LzfzpV9MCt7pI52HZDVrbUVeF82/cMl7bmkxjw9Td+a2AqeXWPe3bN5L8qVtVEe+LVUiC3Ouwrlk6kd+IOIEFtIeVfpKGJFdFOu7H9Te9TPwTQW+BDvcz7du5teJ1aCJ5W0qy2rM7fG893DhuHeXlIGXSvoESXPcy/x3vPdgK8CvwP+pOa7Ub+zSyctZd0qz2f3o3NlH8ru76jjOffK7mutiO4hDijuAcwqTHu4l+cum74PXcHVXjJ9aHa/Sy/PXa37OaLff1dgJN1D8d11PEfVTsRnOJ/aK5Y3Cu3aixjxc29J3fuJEG1ER9aOA4gVxh7Z30dkt5OJ0TqrC/MV/y9V9+ae5746Xn8+XV1gec8RK9HRxAgiiD3LzxPdfzdnr/UQXd1jagIDf+Cofvl7OtBenVYrKJaWlK3N7gfnyqrdCwvraFe1X/2FGtOr5cWDoAAv9vLcZdOru/v7ZLcyncCf9vLcEFvbxxJbmj/NXu9NIpzOpKvbqh7Vdo2jduAX21X97F4qqbuWrn75RnQSe1v5Pa7DiQPoHwH+F9HllFf2+tD1+dd7ALls+YLyZeyLxN7LFKIv/6ys3n8Rx25+V+drqgGO0hk4qls+PfVv/ll2X+uLV6/q/O+po261Xe+qMX1MoV5eb+PZy6ZXn+dSYvktuw0GDuvlufcmwv4XxIHozwDnEt1e02ks7PPtuqWXdu1QMk/ZZzeErv/nxvoFcF729yEl07epMV+1Xc3Y6q4e69gje/2/IVa6E4iROnb1NIGBP3DMIXbFdyIOmpbZL7t/YiNf68Hs/gh6H3P+aHZfFiT58kdrTG/UbMr7kBtVHS56GxvuEe1L+cHEddn94JJpTxMryv2of8/5EeLzLRsffyD9+/2sdrWU/T8/XKO8Lbt/rB/bUeYVIuwnEV2AO7Dh4AL1AwN/4HiTGDM/FLi4ZPp7gK8SW8XXb+RrPUqMEtmD8lE6W9O1BXw/cdDvQGIrLW9iVj6X+of29eYVov93b2KrtWwZ3oE4gN2T6pj+4orqL4CrasxTPeHovSXT1hFDL8cAl1O+whhD9z7867P7c+l+DGUL4lyHRhxJDKUsW9mMILqooLwvfhwbjtI5hlipzqd2H39f/QlxbKFoKLEx00lXX7/6kX34A8uXiX7rKXSdKPM6EUDHEP3DF9E/X9C/JQ4CXkgE+b3EVuA4ok94PF3j8E8mug1uIoYEzs2mH5u1rz+u4ZJ3RtaO6cCniZXOS8TQxF2IlcGniIuP1fLrbL7jsvv7ia6FI4m9qUVsuNX7ABFEZxIrvWrf9+XE+/wGMUpoKnA0sbW6kFiJjCOGYZ5D7A1Un+8K4uDlk8RJU2uI/+Vi4vhHvWf1jgf+mRiaOQv4f0Sf+HuAjxH98A8R4+yLfk6MiDoK+L/E3s9xxMlQp9T5+o0Ynmvjo8AfiJXc4cDOdC1DUvL+lDhJ5mGib3U1EU63EmFV5nxiC7SsG2R7oktjRsm0rYBvEQG4Cvgj8QX9BrBloe5OxIk1i7I2Lcwej2uwPfVMh9gaPJ0I6qXECJhniRXPF+je7dVG+bDM0cTW/DPZ+5sP/GP23p6hfEjkEURQL8uecx3dz7SFWFneRYT2m8QolfuIA5NlI39OJ0arvEHXmbYje2hDma2JDYEfEGPe/0j8H14iTlqbyoYbeG10fS4fIj6717JbrTNta/1vehoy/G90/5yGEHuj/0WE/aqsnQ8Ap5a0U5K0kdpo7PwEDXD19uEPIbZO5hNbIc8RoySKzsmmrSS6AMrOktyV2OJYQWwFTmugHZKkJvs+Ec6fJY7on0Ts+uadTQT9acS1Q35GHGDLD/kaTezy30kMm/scMXrgG01su6RybbiFr4Ijib7AnXuoswXR73dermw48SMM+TA/m+jXzJ9+/VVia/8d/dFYSXVrw8BXwc3EtTh6ciix4BSv1Pddup/Ofx9xUClvu2zej29EGyVJvain7/yviL77K4mt+BXE8LExuTo7E0fh5xfmnUP3PYPxWVneAqIraHzdrZYkNayewB9DXCXwL4kz4aYQw7V+mqszmuiLL54Kv4To2hmSq1d22v8Sup94IknqZ/WMd62e+HEMEcwQJ4TcS/QBdvR7qyRJ/a6ewP8jceW6Jbmy+4kDubsRgb+EOBBboftWfvVyqNWr5S2h/Mp7owvPz+677975xBMbe0kYSUrOE8RlUTZQT5fO0zXq5cN9DnFBqeLvl+5M12nk1XrF65SPJbp9uvXtP/HEE3R2diZ5O//881vehoHcPm9p/+8HQhubeaP8/CeoEeRF/wl8gO6X5T2IOLW9+rueDxDXEsn/+PBw4noi+RE+txOnpueHZU6i60QtSVKT1BP41xJj52cSQydPBL5HXHfjgazOG8Q1V84hTrw6DPhRNu2K3HNdQ1xb5JaszqnEtTkupfyXciRJ/aSePvxlxDj7y4lfl18N/AfxizV53yJWIGcTewO/Jq5+90quzlIi6K8kViBLiLBv7+sbeDtqa2trdRN6tLm3T80zEP73A6GNrVLvpVdboTPrj5Ik1alSqUCNbPeiZZKUCANfkhJh4EtSIgx8SUqEgS9JiTDwJSkRBr4kJcLAl6REGPiSlAgDX5ISYeBLUiIMfElKhIEvSYkw8CUpEQa+JCXCwJekRBj4kpQIA1+SEmHgS1IiDHxJSoSBL0mJMPAlvS0MGjSIKVOmtLoZmzUDPzGzZ89m4sS
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"text/plain": [
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"<matplotlib.figure.Figure at 0x7f7490162da0>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"# If information about the measurement of the wave function propagated\n",
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"# at a finite speed, the particles could conceivably be measured such\n",
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"# that both are equally likely to hold either spin up or spin down.\n",
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"\n",
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"# What would happen if the measurements were uncorrelated?\n",
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"\n",
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"plt.hist(np.random.randint(0,2,1000),bins=4)\n",
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"plt.hist(np.random.randint(3,5,1000),bins=4)\n",
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"elepatch = mpatches.Patch(color='blue', label='e-')\n",
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"pospatch = mpatches.Patch(color='orange', label='e+')\n",
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"#plt.legend(handles=[elepatch,pospatch])\n",
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"plt.text(0.5,565,\"e-\",size=20)\n",
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"plt.text(3.5,565,\"e+\",size=20)\n",
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"\n",
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"plt.suptitle(\"Uncorrelated Spins\",fontsize=20)\n",
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"plt.ylim([400,600])\n",
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"plt.xlim([-1,5])\n",
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"plt.xticks([0.125,0.85,3.125,3.85],[\"down\",\"up\",\"down\",\"up\"])\n",
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"plt.tick_params(axis='both',labelsize=15)\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<matplotlib.figure.Figure at 0x7f7490130940>"
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]
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},
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"execution_count": 16,
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"metadata": {},
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"output_type": "execute_result"
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},
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{
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"data": {
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"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<matplotlib.figure.Figure at 0x7f74897bf128>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"<matplotlib.figure.Figure at 0x7f7490130940>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"# How many violations of angular momentum would be measured?\n",
|
||
|
|
"\n",
|
||
|
|
"# From running several simulations, it's evident a violation in the \n",
|
||
|
|
"# conservation of angular momentum would be measured half of the time.\n",
|
||
|
|
"# We can conclude that the information that the entangled particles are in\n",
|
||
|
|
"# orthogonal spin states is instantaneously agreed once a measurement\n",
|
||
|
|
"# is made.\n",
|
||
|
|
"\n",
|
||
|
|
"violations = 0\n",
|
||
|
|
"\n",
|
||
|
|
"for trial in range(0,1000):\n",
|
||
|
|
" elespin = np.random.randint(0,2)\n",
|
||
|
|
" posspin = np.random.randint(0,2)\n",
|
||
|
|
" if elespin == 0:\n",
|
||
|
|
" elespin = np.matrix('0 1')\n",
|
||
|
|
" else:\n",
|
||
|
|
" elespin = np.matrix('1 0')\n",
|
||
|
|
" if posspin == 0:\n",
|
||
|
|
" posspin = np.matrix('0 1')\n",
|
||
|
|
" else:\n",
|
||
|
|
" posspin = np.matrix('1 0')\n",
|
||
|
|
" \n",
|
||
|
|
" elespin.transpose()\n",
|
||
|
|
" posspin.transpose()\n",
|
||
|
|
" chi_squared = elespin*np.matrix('1; 0')*posspin*np.matrix('0; 1') - posspin*np.matrix('1; 0')*elespin*np.matrix('0; 1')\n",
|
||
|
|
" \n",
|
||
|
|
" if chi_squared == 0:\n",
|
||
|
|
" violations = violations + 1\n",
|
||
|
|
"\n",
|
||
|
|
"zeroes = np.zeros(violations,dtype=int)\n",
|
||
|
|
"ones = np.full((1000-violations),1,dtype=int)\n",
|
||
|
|
"result = np.concatenate((ones,zeroes))\n",
|
||
|
|
"\n",
|
||
|
|
"plt.ylim([400,600])\n",
|
||
|
|
"plt.xlim([-1,2])\n",
|
||
|
|
"plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n",
|
||
|
|
"plt.tick_params(axis='both',labelsize=15)\n",
|
||
|
|
"plt.suptitle(\"Conservation Violations\",fontsize=20)\n",
|
||
|
|
"plt.hist([result],bins=4)\n",
|
||
|
|
"plt.figure()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 12,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"<matplotlib.figure.Figure at 0x7fb39afb7128>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 12,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"# Bell's Experiment took this a step further, to rule out locality \n",
|
||
|
|
"# completely. Establish the detectors to \"float\" such that they \n",
|
||
|
|
"# measure the components of the spins of the electron and positron \n",
|
||
|
|
"# along a unit vector a or b, with angles phi_a and phi_b, \n",
|
||
|
|
"# respectively. Compute a product P of the spins in units of hbar/2. \n",
|
||
|
|
"# This will give +/-1. \n",
|
||
|
|
"\n",
|
||
|
|
"# / pi0 \\\n",
|
||
|
|
"# /e- <------ ------> e+\\\n",
|
||
|
|
"\n",
|
||
|
|
"# QM predicts P(a,b) = -a dot b, the expectation value of the product \n",
|
||
|
|
"# of the spins.\n",
|
||
|
|
"\n",
|
||
|
|
"# In 1964, Bell derived the Bell inequality for a local hidden variable\n",
|
||
|
|
"# theory: abs(P(a,b) - P(a,c)) <= 1 + P(b,c)\n",
|
||
|
|
"\n",
|
||
|
|
"# For any local hidden variable theory, the Bell inequality must hold. \n",
|
||
|
|
"# It introduces a third unit vector c, which is any other unit vector \n",
|
||
|
|
"# than a or b.\n",
|
||
|
|
"\n",
|
||
|
|
"# Does the quantum mechanical prediction violate the Bell inequality?\n",
|
||
|
|
"# Testing several detector configurations in a plane, systematically \n",
|
||
|
|
"# from 0 to pi, we determine whether the QM prediction is consistent \n",
|
||
|
|
"# with a local hidden variable theory.\n",
|
||
|
|
"\n",
|
||
|
|
"violations = 0\n",
|
||
|
|
"trials = 0\n",
|
||
|
|
"for step_a in range(0,6):\n",
|
||
|
|
" for step_b in range (0,6):\n",
|
||
|
|
" phi_a = step_a/6*np.pi\n",
|
||
|
|
" phi_b = step_b/6*np.pi\n",
|
||
|
|
" phi_c = phi_a - 0.5*phi_b\n",
|
||
|
|
" P_ab = -1 * np.cos(phi_a - phi_b)\n",
|
||
|
|
" P_ac = -1 * np.cos(phi_a - phi_c)\n",
|
||
|
|
" P_bc = -1 * np.cos(phi_b - phi_c)\n",
|
||
|
|
" \n",
|
||
|
|
" bell_lhs = np.abs(P_ab - P_ac)\n",
|
||
|
|
" bell_rhs = 1 + P_bc\n",
|
||
|
|
" \n",
|
||
|
|
" if bell_lhs > bell_rhs:\n",
|
||
|
|
" violations = violations + 1\n",
|
||
|
|
" \n",
|
||
|
|
" trials = trials + 1\n",
|
||
|
|
" \n",
|
||
|
|
"zeroes = np.zeros(violations,dtype=int)\n",
|
||
|
|
"ones = np.full((trials-violations),1,dtype=int)\n",
|
||
|
|
"result = np.concatenate((ones,zeroes))\n",
|
||
|
|
"\n",
|
||
|
|
"plt.ylim([trials/2-10,trials/2+10])\n",
|
||
|
|
"plt.xlim([-1,2])\n",
|
||
|
|
"plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n",
|
||
|
|
"plt.tick_params(axis='both',labelsize=15)\n",
|
||
|
|
"plt.suptitle(\"Hidden Locality Violations\",fontsize=20)\n",
|
||
|
|
"plt.hist([result],bins=4)\n",
|
||
|
|
"plt.figure()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 13,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"<matplotlib.figure.Figure at 0x7fb39beac240>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 13,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"# It is seen that the QM prediction disagrees with a local hidden \n",
|
||
|
|
"# variable theory in a significant number of configurations.\n",
|
||
|
|
"\n",
|
||
|
|
"# On average, for random orientations between 0 and pi, how often? \n",
|
||
|
|
"# After running several trials, it appears to be about half of the, \n",
|
||
|
|
"# which is what one would expect from the quantum mechanical\n",
|
||
|
|
"# prediction.\n",
|
||
|
|
"\n",
|
||
|
|
"violations = 0\n",
|
||
|
|
"trials = 0\n",
|
||
|
|
"for rand_a in range(0,10):\n",
|
||
|
|
" for rand_b in range (0,10):\n",
|
||
|
|
" phi_a = np.random.rand(1)*np.pi\n",
|
||
|
|
" phi_b = np.random.rand(1)*np.pi\n",
|
||
|
|
" phi_c = phi_a - 0.5*phi_b\n",
|
||
|
|
" P_ab = -1 * np.cos(phi_a - phi_b)\n",
|
||
|
|
" P_ac = -1 * np.cos(phi_a - phi_c)\n",
|
||
|
|
" P_bc = -1 * np.cos(phi_b - phi_c)\n",
|
||
|
|
" \n",
|
||
|
|
" bell_lhs = np.abs(P_ab - P_ac)\n",
|
||
|
|
" bell_rhs = 1 + P_bc\n",
|
||
|
|
" \n",
|
||
|
|
" if bell_lhs > bell_rhs:\n",
|
||
|
|
" violations = violations + 1\n",
|
||
|
|
" \n",
|
||
|
|
" trials = trials + 1\n",
|
||
|
|
"\n",
|
||
|
|
"zeroes = np.zeros(violations,dtype=int)\n",
|
||
|
|
"ones = np.full((trials-violations),1,dtype=int)\n",
|
||
|
|
"result = np.concatenate((ones,zeroes))\n",
|
||
|
|
"\n",
|
||
|
|
"plt.ylim([trials/2-10,trials/2+10])\n",
|
||
|
|
"plt.xlim([-1,2])\n",
|
||
|
|
"plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n",
|
||
|
|
"plt.tick_params(axis='both',labelsize=15)\n",
|
||
|
|
"plt.suptitle(\"Hidden Locality Violations\",fontsize=20)\n",
|
||
|
|
"plt.hist([result],bins=4)\n",
|
||
|
|
"plt.figure()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 14,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"# The prediction (Bell's inequality) made by assuming a local hidden\n",
|
||
|
|
"# variable is violated by a significant number of the possibile \n",
|
||
|
|
"# bborientations. This simulation cannot determine which theory is\n",
|
||
|
|
"# correct, but the QM prediction has been confirmed through experiment.\n",
|
||
|
|
"# No hidden local variable holds actionable information about the \n",
|
||
|
|
"# state. Entangled states retain their entanglement in a non-local \n",
|
||
|
|
"# nature."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 15,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"plt.show()"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": null,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": []
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"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": "3.4.3"
|
||
|
|
}
|
||
|
|
},
|
||
|
|
"nbformat": 4,
|
||
|
|
"nbformat_minor": 2
|
||
|
|
}
|