Schrodinger equation

The Schrödinger equation is the time-evolution law for a quantum state. In abstract form,

$$i\hslash \frac{d}{dt}|\psi (t)⟩=\hat{H}|\psi (t)⟩,$$

where $\hat{H}$ is the Hamiltonian operator. In position space for one nonrelativistic particle,

$$i\hslash \frac{\partial \psi }{\partial t}=\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V(\mathbf{x},t)\right]\psi .$$

The left side is time change; the right side encodes kinetic energy and Potential energy.

Stationary states

If the Hamiltonian is time-independent, energy eigenstates satisfy

$$\hat{H}{\psi }_n=E_n{\psi }_n.$$

Their time dependence is a phase factor, while measurement probabilities can remain steady.

Classical bridge

In Hamiltonian mechanics, $H$ generates motion through Phase space. In quantum mechanics, $\hat{H}$ generates time evolution through Hilbert space. The analogy is close, but the state is not a phase-space point.

Common pitfalls

• The Wavefunction is complex; do not interpret $\psi$ itself as a probability density.
• $|\psi {|}^2$ gives probabilities only after normalisation and with the correct measure.
• The equation is nonrelativistic unless replaced by a relativistic quantum equation.

Source trail: Susskind The Theoretical Minimum index; reference book: Quantum Mechanics: The Theoretical Minimum by Leonard Susskind and Art Friedman.