stem/Quantum/Schrödinger.md

-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi

• Time Independent
• \psi is the wave function

Quantum counterpart of Newton's second law in classical mechanics

F=ma

From

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, the quantum-mechanical characterization of an isolated physical system.

From

Time–Independent Schrödinger Equation 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