Derive the Schrodinger’s Time Dependent differential equation for matter waves

A free particle moving along x is described by the wave function

\[ \psi(x,t)=A\,e^{-i(Et-px)/\hbar} \]

using \(E=\hbar\omega\) and \(p=\hbar k\).

Step 1 (time derivative): \(\dfrac{\partial\psi}{\partial t}=-\dfrac{iE}{\hbar}\psi\Rightarrow E\psi=i\hbar\dfrac{\partial\psi}{\partial t}\) ... (1)

Step 2 (space derivative): \(\dfrac{\partial\psi}{\partial x}=\dfrac{ip}{\hbar}\psi\), so \(\dfrac{\partial^2\psi}{\partial x^2}=-\dfrac{p^2}{\hbar^2}\psi\Rightarrow p^2\psi=-\hbar^2\dfrac{\partial^2\psi}{\partial x^2}\) ... (2)

Step 3 (energy): Total energy \(E=\dfrac{p^2}{2m}+V\). Multiplying by ψ: \(E\psi=\dfrac{p^2}{2m}\psi+V\psi\) ... (3)

Substituting (1) and (2) into (3):

\[ \boxed{i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V\psi} \]

This is the one-dimensional time-dependent Schrödinger equation. In three dimensions: \(i\hbar\dfrac{\partial\psi}{\partial t}=\Big(-\dfrac{\hbar^2}{2m}\nabla^2+V\Big)\psi=\hat H\psi\), where \(\hat H\) is the Hamiltonian operator. The operator correspondences are \(E\to i\hbar\,\partial/\partial t\) and \(p\to-i\hbar\,\partial/\partial x\).