{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "ad74460d",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import simplify, expand, dsolve, Function , Derivative, Eq, sqrt, sin, I, exp, pi, \\\n",
    "    expand_power_exp, symbols, integrate, oo, diff, powsimp, solveset\n",
    "from sympy.abc import k,A,B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "87551240",
   "metadata": {},
   "outputs": [],
   "source": [
    "x,t,m,E = symbols(\"x t m E\", positive=True, real=True)\n",
    "C3 = symbols(\"C_3\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c0560e4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\hbar$"
      ],
      "text/plain": [
       "hbar"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hbar = symbols(\"hbar\")\n",
    "hbar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "019d8c59",
   "metadata": {},
   "outputs": [],
   "source": [
    "Psi = Function(\"Psi\")\n",
    "psi = Function(\"psi\")\n",
    "phi= Function(\"phi\")\n",
    "v = Function(\"V\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "ce39b2a5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i \\hbar \\frac{\\partial}{\\partial t} \\Psi{\\left(x,t \\right)} = - \\frac{\\hbar^{2} \\frac{\\partial^{2}}{\\partial x^{2}} \\Psi{\\left(x,t \\right)}}{2 m} + \\Psi{\\left(x,t \\right)} V{\\left(x \\right)}$"
      ],
      "text/plain": [
       "Eq(I*hbar*Derivative(Psi(x, t), t), -hbar**2*Derivative(Psi(x, t), (x, 2))/(2*m) + Psi(x, t)*V(x))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "seqn = Eq(I*hbar*Derivative(Psi(x,t),t), -hbar**2/(2*m)*Derivative(Psi(x,t),x,x) + Psi(x,t)*v(x))\n",
    "seqn"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0cd0d925",
   "metadata": {},
   "source": [
    "Subsitute $ \\Psi(x,t)  $ with $ \\psi(x) \\phi(t) $ and then perform the differentiation and divide through by the same functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "c8eb2d2a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i \\hbar \\frac{\\partial}{\\partial t} \\phi{\\left(t \\right)} \\psi{\\left(x \\right)} = - \\frac{\\hbar^{2} \\frac{\\partial^{2}}{\\partial x^{2}} \\phi{\\left(t \\right)} \\psi{\\left(x \\right)}}{2 m} + V{\\left(x \\right)} \\phi{\\left(t \\right)} \\psi{\\left(x \\right)}$"
      ],
      "text/plain": [
       "Eq(I*hbar*Derivative(phi(t)*psi(x), t), -hbar**2*Derivative(phi(t)*psi(x), (x, 2))/(2*m) + V(x)*phi(t)*psi(x))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sseqn = seqn.subs(Psi(x,t), psi(x)*phi(t))\n",
    "sseqn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "f04910ba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i \\hbar \\psi{\\left(x \\right)} \\frac{d}{d t} \\phi{\\left(t \\right)} = - \\frac{\\hbar^{2} \\phi{\\left(t \\right)} \\frac{d^{2}}{d x^{2}} \\psi{\\left(x \\right)}}{2 m} + V{\\left(x \\right)} \\phi{\\left(t \\right)} \\psi{\\left(x \\right)}$"
      ],
      "text/plain": [
       "Eq(I*hbar*psi(x)*Derivative(phi(t), t), -hbar**2*phi(t)*Derivative(psi(x), (x, 2))/(2*m) + V(x)*phi(t)*psi(x))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "seqn2 = sseqn.doit()\n",
    "seqn2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6e8cad71",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{i \\hbar \\frac{d}{d t} \\phi{\\left(t \\right)}}{\\phi{\\left(t \\right)}}$"
      ],
      "text/plain": [
       "I*hbar*Derivative(phi(t), t)/phi(t)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "slhs = seqn2.args[0]/(phi(t)*psi(x))\n",
    "slhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "bc71359e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{\\hbar^{2} \\frac{d^{2}}{d x^{2}} \\psi{\\left(x \\right)}}{2 m \\psi{\\left(x \\right)}}$"
      ],
      "text/plain": [
       "-hbar**2*Derivative(psi(x), (x, 2))/(2*m*psi(x))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "srhs = seqn2.args[1]/(phi(t)*psi(x))\n",
    "srhs = srhs.simplify()\n",
    "# assume v(x) == 0 for a free particle\n",
    "srhs = srhs.subs(v(x),0)\n",
    "srhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "d9b8ed96",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\phi{\\left(t \\right)} = C_{1} e^{- \\frac{i k^{2} t}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(phi(t), C1*exp(-I*k**2*t/hbar))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# dsolve seems to need phi(t) not just phi otherwise an obscure error occurs\n",
    "d1 = dsolve(Eq(slhs, k**2), phi(t))\n",
    "d1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c4f41dcc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{i C_{1} k^{2} e^{- \\frac{i k^{2} t}{\\hbar}}}{\\hbar}$"
      ],
      "text/plain": [
       "-I*C1*k**2*exp(-I*k**2*t/hbar)/hbar"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(d1.args[1],t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "20436c84",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\phi{\\left(t \\right)} = C_{1} e^{- \\frac{i E t}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(phi(t), C1*exp(-I*E*t/hbar))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d1 = d1.subs(k**2, E)\n",
    "d1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "29ec466d",
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_symbol(name, symset):\n",
    "    syms = list(symset)\n",
    "    for sym in syms:\n",
    "        if sym.name == name:\n",
    "            return sym\n",
    "    return None\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "0d9cea11",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{1}$"
      ],
      "text/plain": [
       "C1"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C1 = get_symbol(\"C1\", d1.free_symbols)\n",
    "C1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "eba588f5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\phi{\\left(t \\right)} = C_{3} e^{- \\frac{i E t}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(phi(t), C_3*exp(-I*E*t/hbar))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d1 = d1.subs(C1, C3)\n",
    "d1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "77e3ea67",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\psi{\\left(x \\right)} = C_{1} e^{- \\frac{\\sqrt{2} i k \\sqrt{m} x}{\\hbar}} + C_{2} e^{\\frac{\\sqrt{2} i k \\sqrt{m} x}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(psi(x), C1*exp(-sqrt(2)*I*k*sqrt(m)*x/hbar) + C2*exp(sqrt(2)*I*k*sqrt(m)*x/hbar))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d2 = dsolve(Eq(srhs, k**2), psi(x))\n",
    "d2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "7745f005",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\psi{\\left(x \\right)} = C_{1} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(psi(x), C1*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d2 = d2.subs(k, sqrt(E))\n",
    "d2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "9097959c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\psi{\\left(x \\right)} = C_{1} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(psi(x), C1*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d2i = d2.subs(I, -I)\n",
    "d2i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "cc8300ca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{2 E m \\left(C_{1} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}}\\right)}{\\hbar^{2}}$"
      ],
      "text/plain": [
       "-2*E*m*(C1*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar))/hbar**2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(d2i.args[1], x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "9af2e36b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\phi{\\left(t \\right)} = C_{3} e^{\\frac{i E t}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(phi(t), C_3*exp(I*E*t/hbar))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d1i = d1.subs(I, -I)\n",
    "d1i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "a32cc1c6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{3}^{2}$"
      ],
      "text/plain": [
       "C_3**2"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d1conj = d1i.args[1]*d1.args[1]\n",
    "d1conj"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "dcb7496b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{1}^{2} + C_{1} C_{2} e^{\\frac{2 \\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{1} C_{2} e^{- \\frac{2 \\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2}^{2}$"
      ],
      "text/plain": [
       "C1**2 + C1*C2*exp(2*sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C1*C2*exp(-2*sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2**2"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d2conj = (d2.args[1]*d2i.args[1]).expand()\n",
    "d2conj"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "0549e14d",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{1}^{2} + 2 C_{1} C_{2} \\cos{\\left(\\frac{2 \\sqrt{2} \\sqrt{E} \\sqrt{m} x}{\\hbar} \\right)} + C_{2}^{2}$"
      ],
      "text/plain": [
       "C1**2 + 2*C1*C2*cos(2*sqrt(2)*sqrt(E)*sqrt(m)*x/hbar) + C2**2"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d2expand = d2conj.rewrite(sin).simplify()\n",
    "d2expand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "aaaae1a9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i \\hbar \\psi{\\left(x \\right)} \\frac{d}{d t} \\phi{\\left(t \\right)} = - \\frac{\\hbar^{2} \\phi{\\left(t \\right)} \\frac{d^{2}}{d x^{2}} \\psi{\\left(x \\right)}}{2 m} + V{\\left(x \\right)} \\phi{\\left(t \\right)} \\psi{\\left(x \\right)}$"
      ],
      "text/plain": [
       "Eq(I*hbar*psi(x)*Derivative(phi(t), t), -hbar**2*phi(t)*Derivative(psi(x), (x, 2))/(2*m) + V(x)*phi(t)*psi(x))"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "seqn2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "296ecf28",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{3} E \\left(C_{1} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}}\\right) e^{- \\frac{i E t}{\\hbar}} = C_{3} E \\left(C_{1} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}}\\right) e^{- \\frac{i E t}{\\hbar}} + C_{3} \\left(C_{1} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}}\\right) V{\\left(x \\right)} e^{- \\frac{i E t}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(C_3*E*(C1*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar))*exp(-I*E*t/hbar), C_3*E*(C1*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar))*exp(-I*E*t/hbar) + C_3*(C1*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar))*V(x)*exp(-I*E*t/hbar))"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "seqnsubs = seqn2.subs([(phi(t),d1.args[1]), (psi(x), d2.args[1])]).doit()\n",
    "seqnsubs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "cb97387b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{1} C_{3} E e^{- \\frac{i E t}{\\hbar}} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} C_{3} E e^{- \\frac{i E t}{\\hbar}} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} = C_{1} C_{3} E e^{- \\frac{i E t}{\\hbar}} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{1} C_{3} V{\\left(x \\right)} e^{- \\frac{i E t}{\\hbar}} e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} C_{3} E e^{- \\frac{i E t}{\\hbar}} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}} + C_{2} C_{3} V{\\left(x \\right)} e^{- \\frac{i E t}{\\hbar}} e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}}$"
      ],
      "text/plain": [
       "Eq(C1*C_3*E*exp(-I*E*t/hbar)*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*C_3*E*exp(-I*E*t/hbar)*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar), C1*C_3*E*exp(-I*E*t/hbar)*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C1*C_3*V(x)*exp(-I*E*t/hbar)*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*C_3*E*exp(-I*E*t/hbar)*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar) + C2*C_3*V(x)*exp(-I*E*t/hbar)*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "seqnsubs2 = seqnsubs.simplify().expand()\n",
    "seqnsubs2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "f6402f39",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{1} C_{3} E e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar} - \\frac{i E t}{\\hbar}} + C_{2} C_{3} E e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar} - \\frac{i E t}{\\hbar}}$"
      ],
      "text/plain": [
       "C1*C_3*E*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar - I*E*t/hbar) + C2*C_3*E*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar - I*E*t/hbar)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rhs = powsimp(seqnsubs2.args[1].subs(v(x),0))\n",
    "rhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "6dfc69a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{1} C_{3} E e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar} - \\frac{i E t}{\\hbar}} + C_{2} C_{3} E e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar} - \\frac{i E t}{\\hbar}}$"
      ],
      "text/plain": [
       "C1*C_3*E*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar - I*E*t/hbar) + C2*C_3*E*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar - I*E*t/hbar)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lhs = powsimp(seqnsubs.args[0].expand())\n",
    "lhs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1224c0e2",
   "metadata": {},
   "source": [
    "Both the left hand side and right hand sides of the equation are in agreement showing that our result is consistent with getting a valid expression for the solution of the Schrodinger equation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "1e197055",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{C1, C2, C_3, E, hbar, m, t, x}"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lhs.free_symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "92c01e07",
   "metadata": {},
   "outputs": [],
   "source": [
    "c1 = get_symbol(\"C1\", lhs.free_symbols)\n",
    "c2 = get_symbol(\"C2\", lhs.free_symbols)\n",
    "c3 = get_symbol(\"C_3\", lhs.free_symbols)\n",
    "A,B,L = symbols(\"A B L\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "582f2760",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle A E e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar} - \\frac{i E t}{\\hbar}} + B E e^{\\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar} - \\frac{i E t}{\\hbar}}$"
      ],
      "text/plain": [
       "A*E*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar - I*E*t/hbar) + B*E*exp(sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar - I*E*t/hbar)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psieqn = lhs.subs([(c1*c3, A ), (c2*c3, B)])\n",
    "psieqn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "a42e36ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{2} i \\sqrt{E} \\hbar \\left(A e^{\\frac{4 \\sqrt{2} i \\sqrt{E} L \\sqrt{m}}{\\hbar}} - B\\right) e^{- \\frac{i \\left(2 \\sqrt{2} \\sqrt{E} L \\sqrt{m} + E t\\right)}{\\hbar}}}{2 \\sqrt{m}} = 1$"
      ],
      "text/plain": [
       "Eq(sqrt(2)*I*sqrt(E)*hbar*(A*exp(4*sqrt(2)*I*sqrt(E)*L*sqrt(m)/hbar) - B)*exp(-I*(2*sqrt(2)*sqrt(E)*L*sqrt(m) + E*t)/hbar)/(2*sqrt(m)), 1)"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psiint = Eq( integrate(psieqn, (x, -L- L)).simplify(), 1)\n",
    "psiint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "f2193e9c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle A e^{\\frac{4 \\sqrt{2} i \\sqrt{E} L \\sqrt{m}}{\\hbar}} + \\frac{\\sqrt{2} i \\sqrt{m} e^{\\frac{i \\left(2 \\sqrt{2} \\sqrt{E} L \\sqrt{m} + E t\\right)}{\\hbar}}}{\\sqrt{E} \\hbar}$"
      ],
      "text/plain": [
       "A*exp(4*sqrt(2)*I*sqrt(E)*L*sqrt(m)/hbar) + sqrt(2)*I*sqrt(m)*exp(I*(2*sqrt(2)*sqrt(E)*L*sqrt(m) + E*t)/hbar)/(sqrt(E)*hbar)"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bval = solveset(psiint,B).args[0]\n",
    "bval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "1051089c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\Psi{\\left(x,t \\right)} = A E e^{- \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar} - \\frac{i E t}{\\hbar}} + A E e^{\\frac{4 \\sqrt{2} i \\sqrt{E} L \\sqrt{m}}{\\hbar} + \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar} - \\frac{i E t}{\\hbar}} + \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} e^{\\frac{2 \\sqrt{2} i \\sqrt{E} L \\sqrt{m}}{\\hbar} + \\frac{\\sqrt{2} i \\sqrt{E} \\sqrt{m} x}{\\hbar}}}{\\hbar}$"
      ],
      "text/plain": [
       "Eq(Psi(x, t), A*E*exp(-sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar - I*E*t/hbar) + A*E*exp(4*sqrt(2)*I*sqrt(E)*L*sqrt(m)/hbar + sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar - I*E*t/hbar) + sqrt(2)*I*sqrt(E)*sqrt(m)*exp(2*sqrt(2)*I*sqrt(E)*L*sqrt(m)/hbar + sqrt(2)*I*sqrt(E)*sqrt(m)*x/hbar)/hbar)"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(Psi(x,t), powsimp(psieqn.subs(B, bval).expand()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "115b642e",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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