stem/Quantum/Schrödinger.md

---
tags:
  - quantum
---
$$-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi$$
- Time Independent
- $\psi$ is the [Wave Function](Wave%20Function.md)

Quantum counterpart of Newton's second law in classical mechanics

$$F=ma$$
[From](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)

Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a [wave function](https://en.wikipedia.org/wiki/Wave_function), the quantum-mechanical characterization of an isolated physical system.

[From](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)

[Time–Independent Schrödinger Equation](https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_4.pdf)
[RadialEquation.pdf](https://physics.weber.edu/schroeder/quantum/RadialEquation.pdf)

## Hamiltonian
-   Operator
-   Total energy of a system

-   Kinetic + Potential energy

$$\hat{H}=\hat{T}+\hat{V}$$
- $\hat{V}$
	- Potential Energy
- $\hat{T}=\frac{\hat{p}\cdot\hat{p}}{2m}=-\frac{\hbar^2}{2m}\nabla^2$
	- Kinetic Energy
- $\hat{p}=-i\hbar\nabla$
	- Momentum operator

## Wavefunction Normalisation
-   Adds up to 1 under the curve