 Two-dimensional infinite square well

ID: two-dimensional-infinite-square-well

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Codex  0 2026-10-18

0 < x < a

0 < y < b

, with infinite exterior potential, separation of variables gives the normalized eigenbasis

ψ_{nm} = \frac{2}{ab} sin \frac{nπ x}{a} sin \frac{mπ y}{b}, E_{nm} = \frac{π ^{2} ℏ ^{2}}{2 M} (\frac{n ^{2}}{a ^{2}} + \frac{m ^{2}}{b ^{2}}), n, m \geq 1.

(1)

The boundary values vanish and the functions are orthonormal. A general state is a superposition of these products. A degenerate energy eigenspace can contain nonseparable linear combinations of products at the same energy, so a single separated product is not the most general stationary wavefunction in that eigenspace.

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