Example-QuantumMechanics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Algebra: Quantum Mechanics\n",
    "\n",
    "Reference: http://qutip.org/docs/latest/index.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from qutip import *\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy.linalg as LA\n",
    "from scipy.linalg import expm\n",
    "%matplotlib inline\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "#!pip install qutip"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Some Facts from Quantum Physics:\n",
    "\n",
    "1. Web function $\\psi(x)$ is projection of abstract quantum state $|\\psi \\rangle $ (in a certain representation) to a position space (representation) basis $|x \\rangle$. Where, $|x\\rangle$ is a continuous basis with orthogonality relation $\\langle x^{'}|x\\rangle = \\delta(x,x^{'})$.\n",
    "\n",
    "$$ \\psi(x) = \\langle x|\\psi \\rangle$$\n",
    "\n",
    "2. Web function $\\phi(p)$ is projection of abstract quantum state $|\\psi \\rangle $ in a certain representation to a momentum space (representation)  basis $|p \\rangle$. Where, $|p\\rangle$ is a continuous basis with orthogonality relation $\\langle p^{'}|p\\rangle = \\delta(p,p^{'})$.\n",
    "\n",
    "$$\\phi(p) = \\langle p|\\phi \\rangle$$\n",
    "\n",
    "\n",
    "\n",
    "3. Spherical Harmonics $Y_{l,m}(\\theta,\\phi)$ is projection of abstract quantum state $| l,m \\rangle$ (in angular momnetum representation) to a position space (representation) basis $|\\theta, \\phi \\rangle$.\n",
    "\n",
    "$$Y_{l,m}(\\theta,\\phi) = \\langle \\theta,\\phi|l,m \\rangle$$\n",
    "\n",
    "4. An unitary operator $U$ can be constructed from exponentiation of Hermitian Operator $H$. \n",
    "$$U = exp(-i\\alpha H)$$\n",
    "One application of this approach is defining Rotation matrix $R$ in Hilbert space by implementation of angular momentum Operator (e.g., $L_x, L_y, L_z$) as generator of rotation in specific irreducible subspace (e.g.,l=0,l=1,l=2...) of the Hilbert space. A general rotation in Hilbert space is infinite dimentional rotation matrix. In eigen basis of anfgular momentum ($L^{2}, L_z$), this matrix appears as block diagonal matrix with block representing rotation in specific irriducible sub-space.\n",
    "\n",
    "\n",
    "5. In hydrogen like system with spinless particle, operator hamiltonian $H$, square of Angular momentum $L^{2}$ and z-omponent of $L$ operator($L_z$) commute with eachother which means these operators are simultaneously diagonalized. The benefit of of this relation is that once we are able to find eigen basis of $L_z$ operator, we get the eigen basis of Hamiltonian as well. The eighen states of hamiltonian represents the energy level of the system."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. Hydrogen Atom: Angular Momentum and Spherical Harmonics\n",
    "\n",
    "The matrix element of general angular momnetum operators $J,J_z,J_+,J_-$ are as follows:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![img](https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcTRMh9_FL09_N_ftKHwlFvsrwyJQv_wgu_hVg&usqp=CAU)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One can write a python function to provide a matrix element for an arbitrary operators $J,J_z,J_+,J_-$ , but we dont have to work hard now python package ```qutip``` provides us these operators as quantum object ```Quobj``` (of kind operators). We will try to play around with some of them."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####   Spin Angular Momentum $s = 1/2, ms = -1/2, 1/2$\n",
    "Matrix size = 2x2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 1.0\\\\1.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 1.]\n",
       " [1. 0.]]"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmax()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[0.+0.j, 1.+0.j],\n",
       "        [1.+0.j, 0.+0.j]]), array([[0.+0.j, 0.-1.j],\n",
       "        [0.+1.j, 0.+0.j]]), array([[ 1.+0.j,  0.+0.j],\n",
       "        [ 0.+0.j, -1.+0.j]]))"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sx = np.array(sigmax())\n",
    "sy = np.array(sigmay())\n",
    "sz = np.array(sigmaz())\n",
    "sx,sy,sz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[1.+0.j, 0.+0.j],\n",
       "        [0.+0.j, 1.+0.j]]), array([[1.+0.j, 0.+0.j],\n",
       "        [0.+0.j, 1.+0.j]]), array([[1.+0.j, 0.+0.j],\n",
       "        [0.+0.j, 1.+0.j]]))"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(sx,sx), np.dot(sy,sy), np.dot(sz,sz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----------\n",
    "\n",
    "#### Experiment 1 :\n",
    "\n",
    "- A generic rotation $ U_n(\\theta) = e^{-i \\theta n.\\sigma} = e^{-i (\\theta_x \\sigma_x + \\theta_y \\sigma_y + \\theta_z \\sigma_z)} $\n",
    "\n",
    "Note: $ U_n(\\theta) = e^{-i \\theta n.\\sigma} \\neq e^{-i (\\theta_x \\sigma_x)} e^{-i (\\theta_y \\sigma_y)} e^{-i(\\theta_z \\sigma_z)} $ since pauli matrices are non-commuting.\n",
    "\n",
    "- Generic quantum state in ($s =1/2$) subspace: $|\\psi \\rangle = \\alpha |\\psi_{1/2}\\rangle + \\beta |\\psi_{-1/2}\\rangle$\n",
    "\n",
    "[Non-commuting Matrix exponentials](https://math.stackexchange.com/questions/588007/non-commuting-matrix-exponentials/590120)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us tast above fact by evaluating generic rotation operator $U_n$ around a arbitrary axix $n$ (i.e., $e^{-i (\\theta_x \\sigma_x + \\theta_y \\sigma_y + \\theta_z \\sigma_z)})$ as ```U_direct``` and the product of individual rotation operator (i.e.,  $ e^{-i (\\theta_x \\sigma_x)} e^{-i (\\theta_y \\sigma_y)} e^{-i(\\theta_z \\sigma_z)}$) as ```U_prod``` in the code cells below. We will clearly see that thes two terms are not equal verifying relation $e^{-i \\theta n.\\sigma} \\neq e^{-i (\\theta_x \\sigma_x)} e^{-i (\\theta_y \\sigma_y)} e^{-i(\\theta_z \\sigma_z)}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Calculate $e^{-i (\\theta_x \\sigma_x + \\theta_y \\sigma_y + \\theta_z \\sigma_z)})$ as ```U_direct```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.89894119-0.25291945j, -0.25291945-0.25291945j],\n",
       "       [ 0.25291945-0.25291945j,  0.89894119+0.25291945j]])"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_direct = expm(-1j*((np.pi/12)*sx + (np.pi/12)*sy + (np.pi/12)*sz))\n",
    "U_direct"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Calculate $e^{-i (\\theta_x \\sigma_x)} e^{-i (\\theta_y \\sigma_y)} e^{-i(\\theta_z \\sigma_z)}$ as product of three individual rotation as ```U_prod```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[0.92387953-0.38268343j, 0.        +0.j        ],\n",
       "        [0.        +0.j        , 0.92387953+0.38268343j]]),\n",
       " array([[0.96592583+0.j        , 0.        -0.25881905j],\n",
       "        [0.        -0.25881905j, 0.96592583+0.j        ]]),\n",
       " array([[ 0.95105652+0.j, -0.30901699+0.j],\n",
       "        [ 0.30901699+0.j,  0.95105652+0.j]]))"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ux = expm(-1j*np.pi/12*sx)\n",
    "Uy = expm(-1j*np.pi/10*sy)\n",
    "Uz = expm(-1j*np.pi/8*sz)\n",
    "Uz,Ux,Uy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.81811516-0.42544356j, -0.18156837-0.34164059j],\n",
       "       [ 0.18156837-0.34164059j,  0.81811516+0.42544356j]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_prod = np.dot(Ux,np.dot(Uy,Uz))\n",
    "U_prod"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Are ```U_direct``` and ```U_prod``` same Operators?\n",
    "Why? \n",
    "\n",
    "We can see these two operators are not same by implement them in same initial state ```psi0``` vector and observe the final states are not same."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.70710678, 0.70710678])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi0 = 1/np.sqrt(2)*np.array([1,1])\n",
    "psi0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0.45680635-3.57682117e-01j, 0.81448847+8.32667268e-17j]),\n",
       " array([0.45010655-0.5424104j , 0.706883  +0.05925765j]))"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(U_direct,psi0), np.dot(U_prod,psi0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- In fact, both of them are Unitary operators with determinant 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((0.9999999999999999+1.1102230246251564e-16j), (1.0000000000000002+0j))"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(U_direct), LA.det(U_prod)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####  Angular Momentum $l =1, m = -1,0,1$\n",
    "Matrix size  = 3x3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0.         0.70710678 0.        ]\n",
       "  [0.70710678 0.         0.70710678]\n",
       "  [0.         0.70710678 0.        ]],\n",
       " Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0.+0.j         0.-0.70710678j 0.+0.j        ]\n",
       "  [0.+0.70710678j 0.+0.j         0.-0.70710678j]\n",
       "  [0.+0.j         0.+0.70710678j 0.+0.j        ]],\n",
       " Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[ 1.  0.  0.]\n",
       "  [ 0.  0.  0.]\n",
       "  [ 0.  0. -1.]])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jmat(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "LX = np.array(jmat(1,'x'))\n",
    "LY = np.array(jmat(1,'y'))\n",
    "LZ = np.array(jmat(1,'z'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Do $L_x, L_y$ commute?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[False,  True,  True],\n",
       "       [ True, False,  True],\n",
       "       [ True,  True, False]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(LX,LY) == np.dot(LY,LX)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- What is matrix element of $L^{2}$ ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2.+0.j, 0.+0.j, 0.+0.j],\n",
       "       [0.+0.j, 2.+0.j, 0.+0.j],\n",
       "       [0.+0.j, 0.+0.j, 2.+0.j]])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L_square = (np.dot(LX,LX) + np.dot(LY,LY) +np.dot(LZ,LZ))\n",
    "L_square"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Experiment 2\n",
    "\n",
    "- Rotaion $ R(\\theta) = e^{-i \\theta n.L} = e^{-i (\\theta_x L_x + \\theta_y L_y + \\theta_z L_z)} $\n",
    "\n",
    "- Generic quantum state in ($l =0$) subspace: $|\\psi \\rangle = \\alpha |\\psi_{10}\\rangle + \\beta |\\psi_{11} \\rangle + \\gamma |\\psi_{1-1} \\rangle$\n",
    "\n",
    "Let us find rotation matrix for subspace (l=1) with different values of $\\theta_x, \\theta_y, \\theta_z$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "Rx = expm(-(1.0j)*0.1*LX)\n",
    "Ry = expm(-(1.0j)*0.2*LY)\n",
    "Rz = expm(-(1.0j)*0.3*LZ)\n",
    "R_prod = np.dot(Rx,np.dot(Ry,Rz))\n",
    "R_direct = expm(-(1.0j)*(0.1*LX + 0.2*LY + 0.3*LZ))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[ 0.94049792-3.01310652e-01j, -0.14048043-6.91857281e-02j,\n",
       "          0.00420456+1.16811741e-02j],\n",
       "        [ 0.11267399-1.08747365e-01j,  0.97517033+6.69983947e-18j,\n",
       "         -0.11267399-1.08747365e-01j],\n",
       "        [ 0.00420456-1.16811741e-02j,  0.14048043-6.91857281e-02j,\n",
       "          0.94049792+3.01310652e-01j]]),\n",
       " array([[ 0.94316771-2.93048837e-01j, -0.14862798-4.81054052e-02j,\n",
       "          0.00741291+9.88387642e-03j],\n",
       "        [ 0.12766111-9.00391414e-02j,  0.97529031-1.50304582e-18j,\n",
       "         -0.12766111-9.00391414e-02j],\n",
       "        [ 0.00741291-9.88387642e-03j,  0.14862798-4.81054052e-02j,\n",
       "          0.94316771+2.93048837e-01j]]))"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_prod, R_direct"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Mini Assignment:**\n",
    "- Roate a random vector  $|\\psi \\rangle $,i.e. ($|\\psi \\rangle = \\alpha |\\psi_{10}\\rangle + \\beta |\\psi_{11} \\rangle + \\gamma |\\psi_{1-1} \\rangle)$ by implementing ```R_prod``` and ``` R_direct``` calculated above and compere the final state vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Angular Momentum plus Spin: $l = 3/2, m = -3/2,-1/2,1/2,3/2$\n",
    "Matrix size = 4x4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.866 & 0.0 & 0.0\\\\0.866 & 0.0 & 1.0 & 0.0\\\\0.0 & 1.0 & 0.0 & 0.866\\\\0.0 & 0.0 & 0.866 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0.        0.8660254 0.        0.       ]\n",
       " [0.8660254 0.        1.        0.       ]\n",
       " [0.        1.        0.        0.8660254]\n",
       " [0.        0.        0.8660254 0.       ]]"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jmat(3/2,'x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & -0.866j & 0.0 & 0.0\\\\0.866j & 0.0 & -1.0j & 0.0\\\\0.0 & 1.0j & 0.0 & -0.866j\\\\0.0 & 0.0 & 0.866j & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0.+0.j        0.-0.8660254j 0.+0.j        0.+0.j       ]\n",
       " [0.+0.8660254j 0.+0.j        0.-1.j        0.+0.j       ]\n",
       " [0.+0.j        0.+1.j        0.+0.j        0.-0.8660254j]\n",
       " [0.+0.j        0.+0.j        0.+0.8660254j 0.+0.j       ]]"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jmat(3/2,'y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Summary: Rotation in Hilbert Space \n",
    "\n",
    "Structure of a general [rotation matrix (R) in Hilbert space](https://www2.ph.ed.ac.uk/~ldeldebb/docs/QM2/chap4.pdf)\n",
    "\n",
    "j = 1/2,1,3/2,2,5/2,3,...\n",
    "\n",
    "- A general rotation matrix ($R$) in hilbert space of basis $|l,m\\rangle$ appears as block diagonal matrix, wher every block represents the rotation with specific subspaces called irriducible subspace.\n",
    "- In the same basis $|l,m\\rangle$  Hamiltonian Matrix $H$, $J^2$ and $J_z$ are simultaneously diagonalized."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![img](https://physicscourses.colorado.edu/phys5250/phys5250_fa19/images/block-diag.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Quantum Harmonic Oscillator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[5], [5]], shape = (5, 5), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & -0.707j & 0.0 & 0.0 & 0.0\\\\0.707j & 0.0 & -1.0j & 0.0 & 0.0\\\\0.0 & 1.0j & 0.0 & -1.225j & 0.0\\\\0.0 & 0.0 & 1.225j & 0.0 & -1.414j\\\\0.0 & 0.0 & 0.0 & 1.414j & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[5], [5]], shape = (5, 5), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0.+0.j         0.-0.70710678j 0.+0.j         0.+0.j\n",
       "  0.+0.j        ]\n",
       " [0.+0.70710678j 0.+0.j         0.-1.j         0.+0.j\n",
       "  0.+0.j        ]\n",
       " [0.+0.j         0.+1.j         0.+0.j         0.-1.22474487j\n",
       "  0.+0.j        ]\n",
       " [0.+0.j         0.+0.j         0.+1.22474487j 0.+0.j\n",
       "  0.-1.41421356j]\n",
       " [0.+0.j         0.+0.j         0.+0.j         0.+1.41421356j\n",
       "  0.+0.j        ]]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "momentum(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[5], [5]], shape = (5, 5), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.707 & 0.0 & 0.0 & 0.0\\\\0.707 & 0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0 & 1.225 & 0.0\\\\0.0 & 0.0 & 1.225 & 0.0 & 1.414\\\\0.0 & 0.0 & 0.0 & 1.414 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[5], [5]], shape = (5, 5), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0.         0.70710678 0.         0.         0.        ]\n",
       " [0.70710678 0.         1.         0.         0.        ]\n",
       " [0.         1.         0.         1.22474487 0.        ]\n",
       " [0.         0.         1.22474487 0.         1.41421356]\n",
       " [0.         0.         0.         1.41421356 0.        ]]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "position(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.0\\\\1.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 1.414 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.732 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[0.         0.         0.         0.        ]\n",
       " [1.         0.         0.         0.        ]\n",
       " [0.         1.41421356 0.         0.        ]\n",
       " [0.         0.         1.73205081 0.        ]]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "create(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[5], [5]], shape = (5, 5), type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 1.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.414 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.732 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 2.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[5], [5]], shape = (5, 5), type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[0.         1.         0.         0.         0.        ]\n",
       " [0.         0.         1.41421356 0.         0.        ]\n",
       " [0.         0.         0.         1.73205081 0.        ]\n",
       " [0.         0.         0.         0.         2.        ]\n",
       " [0.         0.         0.         0.         0.        ]]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "destroy(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 2.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 3.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0. 0. 0.]\n",
       " [0. 1. 0. 0.]\n",
       " [0. 0. 2. 0.]\n",
       " [0. 0. 0. 3.]]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Random Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Random hermitian matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}-0.501 & (-0.172+0.146j) & 0.0 & (-0.060+0.697j)\\\\(-0.172-0.146j) & 0.169 & (-0.305+0.588j) & (0.044+0.641j)\\\\0.0 & (-0.305-0.588j) & 0.0 & (-0.458-0.087j)\\\\(-0.060-0.697j) & (0.044-0.641j) & (-0.458+0.087j) & 0.756\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[-0.50113582+0.j         -0.17222998+0.14571042j  0.        +0.j\n",
       "  -0.06039662+0.6965142j ]\n",
       " [-0.17222998-0.14571042j  0.1689534 +0.j         -0.30474969+0.58835929j\n",
       "   0.04412378+0.64077284j]\n",
       " [ 0.        +0.j         -0.30474969-0.58835929j  0.        +0.j\n",
       "  -0.45812567-0.08748371j]\n",
       " [-0.06039662-0.6965142j   0.04412378-0.64077284j -0.45812567+0.08748371j\n",
       "   0.75578241+0.j        ]]"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rand_herm(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Random Unitary Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}(0.332-0.610j) & (-0.248+0.007j) & (-0.332-0.267j) & (0.468-0.235j)\\\\(-0.062-0.085j) & (0.838-0.353j) & (-0.312-0.253j) & (3.589\\times10^{-05}-0.034j)\\\\(-0.032-0.481j) & (0.145-0.293j) & (0.790+0.106j) & (0.020-0.156j)\\\\(-0.427+0.302j) & (0.069+0.006j) & (0.109+0.097j) & (0.834-0.066j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[4], [4]], shape = (4, 4), type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[ 3.32426091e-01-0.61008781j -2.48216764e-01+0.00742287j\n",
       "  -3.32218003e-01-0.26721645j  4.67803649e-01-0.23453233j]\n",
       " [-6.18056368e-02-0.08520346j  8.37959790e-01-0.35291461j\n",
       "  -3.11514724e-01-0.25291069j  3.58896057e-05-0.03449434j]\n",
       " [-3.17360027e-02-0.48099205j  1.44640852e-01-0.29316018j\n",
       "   7.90378747e-01+0.10591817j  2.00616601e-02-0.15638388j]\n",
       " [-4.27155409e-01+0.30229774j  6.86551818e-02+0.00553252j\n",
       "   1.09189564e-01+0.0968594j   8.34155661e-01-0.06550129j]]"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rand_unitary(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mini Assignment:\n",
    "1. Generate a random Hermitian matrix $H$ of size 10 by 10.\n",
    "3. Diagonalize the Hermitian Operator $H$ and find eigne values and eigen vectors.\n",
    "2. Create a Unitary operator $U$ by exponentiating the Hermitian matrix i.e. $U = e^{-i \\alpha H}$.\n",
    "3. Apply operator $U$ over eigen vectors of operator $H$. Does this operation preserve the norm of eigen vectors?\n",
    "3. Check that $U$ is unitary or not.\n",
    "4. What is the determinant of $U$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reference\n",
    "1. [Rotation](https://www2.ph.ed.ac.uk/~ldeldebb/docs/QM2/chap4.pdf)\n",
    "2. [Spherical Harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics)\n",
    "3. [Angular Momentum](https://theory.physics.manchester.ac.uk/~xian/qm/chapter2.pdf)"
   ]
  }
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