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Schrödinger Equation

Time-dependent form

Time-independent form

The Schrödinger equation is the equation of motion for a non-relativistic quantum state. The time-dependent form says how the wavefunction evolves; the time-independent form is the energy eigenvalue problem for a static Hamiltonian.

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PhysicsQuantum mechanicsWave mechanics
H^ψ=Eψ\htmlData{term=hamiltonian}{\hat{H}} \htmlData{term=stationary-state}{\psi} = \htmlData{term=energy-eigenvalue}{E} \htmlData{term=stationary-state}{\psi}

Notes

Erwin Schrödinger published the wave equation in 1926. The most general statement uses the Hamiltonian operator H^\hat{H} and a state Ψ\Psi. When H^\hat{H} does not depend explicitly on time, stationary states satisfy H^ψ=Eψ\hat{H}\psi = E\psi.

This entry stays with the abstract operator forms. A later expression can expand H^\hat{H} in position space as −ℏ22m∇2+V-\frac{\hbar^2}{2m}\nabla^2 + V.

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