Chapter 7 - Angular Momentum and Rotation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chapter 7 - Angular Momentum and Rotation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from numpy import sqrt, pi, exp, angle\n",
    "from qutip import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our usual spin operators (spin-1/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pz = basis(2,0)\n",
    "mz = basis(2,1)\n",
    "px = 1/sqrt(2)*(pz + mz)\n",
    "mx = 1/sqrt(2)*(pz - mz)\n",
    "py = 1/sqrt(2)*(pz + 1j*mz)\n",
    "my = 1/sqrt(2)*(pz - 1j*mz)\n",
    "Sx = 1/2.0*sigmax()\n",
    "Sy = 1/2.0*sigmay()\n",
    "Sz = 1/2.0*sigmaz()\n",
    "Splus = Sx + 1j*Sy\n",
    "Sminus = Sx - 1j*Sy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 0.]\n",
       " [ 0.]]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Splus*pz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0\\\\0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 1.]\n",
       " [ 0.]]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Splus*mz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the spin-1 operators. We use J just to keep them apart. Could be S instead."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Jx = 1/sqrt(2)*Qobj([[0,1,0],[1,0,1],[0,1,0]])\n",
    "Jy = 1/sqrt(2)*Qobj([[0,-1j,0],[1j,0,-1j],[0,1j,0]])\n",
    "Jz = Qobj([[1,0,0],[0,0,0],[0,0,-1]])\n",
    "Jplus = Jx + 1j*Jy\n",
    "Jminus = Jx - 1j*Jy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rotations of spin-1/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Rz90 = (-1j*pi/2*Sz).expm()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}(0.707-0.707j) & 0.0\\\\0.0 & (0.707+0.707j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[ 0.70710678-0.70710678j  0.00000000+0.j        ]\n",
       " [ 0.00000000+0.j          0.70710678+0.70710678j]]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rz90"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}(0.500-0.500j)\\\\(0.500+0.500j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 0.5-0.5j]\n",
       " [ 0.5+0.5j]]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rz90*px"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Doesn't look like example 7.4 because it's not simplified. How to check:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}(0.500-0.500j)\\\\(0.500+0.500j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 0.5-0.5j]\n",
       " [ 0.5+0.5j]]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(-1j*pi/4)*py"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Yes, this agrees."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Can also use inner product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}-0.707j\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[ 0.-0.70710678j]]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mx.dag()*Rz90*px"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angle(0.707 - 0.707j) == -pi/4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Find the spin-1 states (the eigenstates of the corresponding matrix)\n",
    "According to the postulates, the allowed values of an observable are the eigenvalues with corresponding eigenstates.\n",
    "\n",
    "We know the matrix representation of Jx, Jy, Jz in the Z-basis so we can find all 9 states (in the Z basis). Why nine? There are three possible values $\\hbar$, 0, $-\\hbar$ for each of three directions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "yevals, (yd,y0,yu) = Jy.eigenstates()\n",
    "zevals, (zd,z0,zu) = Jz.eigenstates()\n",
    "xevals, (xd,x0,xu) = Jx.eigenstates()\n",
    "xd = -xd  # fix the signs to match book\n",
    "yu = -yu  # fix the signs to match book"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.707\\\\0.0\\\\0.707\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 0.70710678]\n",
       " [ 0.        ]\n",
       " [ 0.70710678]]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Rz90 = (-1j*pi/2*Jz).expm()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}-0.707j\\\\0.0\\\\-0.707j\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 0.-0.70710678j]\n",
       " [ 0.+0.j        ]\n",
       " [ 0.-0.70710678j]]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rz90*x0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.707\\\\0.0\\\\0.707\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 0.70710678]\n",
       " [ 0.        ]\n",
       " [ 0.70710678]]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.500\\\\0.707\\\\0.500\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 0.5       ]\n",
       " [ 0.70710678]\n",
       " [ 0.5       ]]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xu"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 7.10 A spin-1 particle is measured to have $S_y=\\hbar$. What is the probability that a subsequent measurement will yield $S_z=0$? $S_z=\\hbar$? $S_z=-\\hbar$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25000000000000011"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(z0.dag()*yu).norm()**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25000000000000011"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(zu.dag()*yu).norm()**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(zd.dag()*yu).norm()**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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