Derivation of Schrodinger Wave Equation | Schrodinger Equation | Schrodinger Wave Equation | Quantum Mechanics | Rizwan Chemistry Classes

Deriving the Schrödinger Equation — Step by Step

A simple, intuitive, and complete derivation for students — by Rizwan Chemistry Classes

Goal

\( i\hbar \frac{\partial \psi(\mathbf{r},t)}{\partial t} = \left( -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t) \right)\psi(\mathbf{r},t) \)

This is the time-dependent Schrödinger equation (TDSE). We’ll build it step-by-step using wave concepts, energy relations, and simple logic.

Step 1: Matter Waves — The Foundation

According to de Broglie and Einstein:

• Momentum–wavenumber relation: \( p = \hbar k \)
• Energy–frequency relation: \( E = \hbar \omega \)

\( \psi(\mathbf r,t) = A\,e^{\,i(\mathbf k\cdot\mathbf r - \omega t)} \)

Step 2: Identify Operator Rules

Time derivative → Energy operator

\( \frac{\partial \psi}{\partial t} = -i\omega \psi \Rightarrow i\hbar \frac{\partial \psi}{\partial t} = E\psi \)

Thus, \( \hat{E} = i\hbar \frac{\partial}{\partial t} \).

Spatial derivatives → Momentum operator

\( \nabla \psi = i\mathbf{k}\psi \Rightarrow \nabla^2\psi = -k^2\psi \)

Using \( p = \hbar k \), we get \( -\hbar^2 \nabla^2 \psi = p^2 \psi \), hence \( \hat{p} = -i\hbar\nabla \).

Step 3: Quantize Classical Energy

\( i\hbar\frac{\partial \psi}{\partial t} = \left( -\frac{\hbar^2}{2m}\nabla^2 + V \right)\psi \)

Step 4: Time-Independent Case

\( \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf r)\right)\phi = E\phi \)

Step 5: Probability Interpretation (Born Rule)

\( \rho = |\psi|^2, \quad \int |\psi|^2\,d\tau = 1 \)

Step 6: Continuity Equation

\( \frac{\partial \rho}{\partial t} + \nabla\cdot \mathbf J = 0 \), where \( \mathbf{J} = \frac{\hbar}{2mi}(\psi^*\nabla\psi - \psi\nabla\psi^*) \)

Step 7: Classical Limit

\( \frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + V + Q = 0 \), with \( Q = -\frac{\hbar^2}{2m}\frac{\nabla^2 R}{R} \)

Step 8: Boundary Conditions

• ψ finite and continuous
• ψ' continuous unless V is infinite
• ψ normalizable: \( \int |\psi|^2 d\tau = 1 \)

Step 9: Spin & Relativity

• Pauli Equation: adds spin and magnetic effects.
• Dirac Equation: relativistic version.

Quick Recap

Matter wave → \( E=\hbar\omega, p=\hbar k \) → replace by operators → apply to energy equation → Schrödinger’s Equation.

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