`ℏ(del/(delt))Psi(r,t) = [-ℏ^2/(2m)*grad^2 + V(r,t)] * Psi(r,t)`

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Time Dependent Schrödinger Equation

  The Scrödinger equation describes how the quantum state of a physical system changes with respect to time. This equation is the quantum mechanical analogue of Newton's Second Law.

Notes

This differential equation describes the wave function of the system, also called the quantum state or state vector. Solutions to this equation describe change occurring in macroscopic, molecular, atomic, and subatomic systems.

The Laplacian in this equation is

 

Per Richard Feynman referring to the Schrödinger Equation: "Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.