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‎How_to_Postulate_an_Equation_Schrodinger_Example.ipynb

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"name": "How_to_Postulate_an_Equation_Schrodinger_Example.ipynb",
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"<a href=\"https://colab.research.google.com/github/OJB-Quantum/Math-and-Physics-How-To/blob/main/How_to_Postulate_an_Equation_Schrodinger_Example.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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"Below is a postulation of the Schrodinger equation."
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],
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"To postulate the Schrödinger equation step by step, we can use the insights from classical physics, wave mechanics, and the heat equation analogy. Below is a structured approach:\n",
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"\n",
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"---\n",
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"\n",
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"### **Step 1: Start with Classical Energy Concepts**\n",
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"In classical mechanics, the total energy $E$ of a free particle (no external forces) is given by the Hamiltonian:\n",
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"\n",
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"$\n",
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"E = \\frac{p^2}{2m}\n",
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"$\n",
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"\n",
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"where:\n",
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"- $ p $ is the momentum,\n",
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"- $ m $ is the mass of the particle.\n",
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"\n",
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"This equation will serve as the foundation for our quantum mechanical formulation.\n",
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"\n",
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"---\n",
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"\n",
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"### **Step 2: Introduce De Broglie’s Hypothesis**\n",
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"Louis de Broglie proposed that matter behaves like waves, with:\n",
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"\n",
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"$\n",
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"p = \\hbar k\n",
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"$\n",
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"\n",
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"$\n",
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"E = \\hbar \\omega\n",
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"$\n",
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"\n",
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"where:\n",
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"- $ k $ is the wave number ($ k = 2\\pi/\\lambda $),\n",
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"- $ \\omega $ is the angular frequency ($ \\omega = 2\\pi f $),\n",
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"- $ \\hbar $ is Planck’s reduced constant.\n",
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"\n",
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"These relations suggest that energy and momentum can be represented using wave functions.\n",
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"\n",
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"---\n",
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"\n",
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"### **Step 3: Define Quantum Operators**\n",
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"In quantum mechanics, we replace classical quantities with operators:\n",
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"\n",
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"- **Momentum operator**: $ p = -i\\hbar \\frac{\\partial}{\\partial x} $\n",
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"- **Energy operator**: $ E = i\\hbar \\frac{\\partial}{\\partial t} $\n",
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"\n",
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"These operators act on the wave function $ \\psi(x,t) $.\n",
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"\n",
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"---\n",
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"\n",
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"### **Step 4: Apply Operators to the Classical Energy Equation**\n",
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"Using the classical energy equation:\n",
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"\n",
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"$\n",
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"E = \\frac{p^2}{2m}\n",
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"$\n",
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"\n",
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"and replacing $ E $ and $ p $ with their quantum operators:\n",
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"\n",
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"$\n",
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"i\\hbar \\frac{\\partial}{\\partial t} \\psi = \\frac{(-i\\hbar \\frac{\\partial}{\\partial x})^2}{2m} \\psi\n",
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"$\n",
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"\n",
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"Simplifying:\n",
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"\n",
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"$\n",
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"i\\hbar \\frac{\\partial}{\\partial t} \\psi = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} \\psi\n",
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"$\n",
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"\n",
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"This is the **time-dependent Schrödinger equation** for a free particle.\n",
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"\n",
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"---\n",
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"\n",
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"### **Step 5: Generalize to Include Potentials**\n",
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"If a particle moves in a potential $ V(x,t) $, we modify the Hamiltonian:\n",
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"\n",
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"$\n",
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"E = \\frac{p^2}{2m} + V\n",
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"$\n",
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"\n",
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"Applying the same substitution:\n",
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"\n",
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"$\n",
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"i\\hbar \\frac{\\partial}{\\partial t} \\psi = \\left(-\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} + V(x,t)\\right) \\psi\n",
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"$\n",
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"\n",
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"This is the general time-dependent **Schrödinger equation**:\n",
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"\n",
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"$\n",
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"i\\hbar \\frac{\\partial}{\\partial t} \\psi = \\hat{H} \\psi\n",
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"$\n",
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"\n",
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"where $ \\hat{H} $ (the Hamiltonian operator) is:\n",
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"\n",
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"$\n",
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"\\hat{H} = -\\frac{\\hbar^2}{2m} \\nabla^2 + V(x,t).\n",
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"$\n",
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"\n",
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"---\n",
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"\n",
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"### **Step 6: Connection to the Heat Equation**\n",
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"- The Schrödinger equation is structurally similar to the heat equation:\n",
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"\n",
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" $\n",
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" \\frac{\\partial u}{\\partial t} = D \\frac{\\partial^2 u}{\\partial x^2}\n",
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" $\n",
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"\n",
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"- However, the Schrödinger equation has an **imaginary** time derivative ($ i\\hbar \\partial/\\partial t $), leading to **wave-like behavior** instead of pure diffusion.\n",
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"- If we perform a Wick rotation ($ t \\to i\\tau $), the Schrödinger equation transforms into a heat-like equation. This analogy is useful in statistical mechanics and path integrals.\n",
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"\n",
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"---\n",
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"\n",
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"### **Final Thoughts**\n",
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"- The Schrödinger equation is postulated based on wave-particle duality and operator substitutions.\n",
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"- The similarity to the heat equation helps us understand its structure, but the imaginary unit $ i $ is what makes quantum mechanics fundamentally different.\n",
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"- The equation is not strictly \"derived\" but is **motivated** by classical mechanics and wave physics.\n",
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"\n",
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"By following these steps, you can systematically arrive at the Schrödinger equation and appreciate its physical significance."
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