Law

Time-independent Schrödinger equation

(Q3.2.1) Q3.2 · The Schrödinger Equation →

- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} + U (x) ψ = E ψ

Statement

Determines the allowed wave functions and energies for a particle of mass m in a potential U(x).

Symbols and units

$ℏ$	reduced Planck constant	1.055×10⁻³⁴ J·s
$m$	particle mass	kg
$U (x)$	potential energy	J
$E$	total energy	J
$ψ$	wave function	—

When it applies

Non-relativistic quantum mechanics, time-independent (stationary) states.

Don't use when

Non-relativistic and specific to time-independent (stationary) states — a time-varying potential or relativistic speeds require the full time-dependent equation or a relativistic wave equation instead.

Derivation

A postulate of quantum mechanics, not derived from anything more basic — it determines how the wave function is shaped by the potential energy landscape a particle sits in.

- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} + U (x) ψ = E ψ

Watch out for

Solutions exist only for certain values of E when a particle is confined — this quantization of allowed energies is a direct mathematical consequence of requiring ψ to be finite and continuous, not a separate assumption layered on top.

Connects to

Particle-in-a-box energies