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Schrödinger

Schrödinger

−ℏ22m∇2ψ+Vψ=Eψ-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi

Quantum counterpart of Newton’s second law in classical mechanics

F=maF=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

H^=T^+V^\hat{H}=\hat{T}+\hat{V}
  • V^\hat{V}
    • Potential Energy
  • T^=p^⋅p^2m=−ℏ22m∇2\hat{T}=\frac{\hat{p}\cdot\hat{p}}{2m}=-\frac{\hbar^2}{2m}\nabla^2
    • Kinetic Energy
  • p^=−iℏ∇\hat{p}=-i\hbar\nabla
    • Momentum operator

Wavefunction Normalisation

  • Adds up to 1 under the curve
Orbitals