English: Wavefunctions and energies for particle trapped in an infinitely deep quantum well of width ${\displaystyle L}$. The spatial position is shown along the horizontal axis, and the energy along the vertical axis. The grey regions represent barriers with an infinitely large potential, while the white area represents a well with zero potential.

The particle may only exist in certain discrete states with energy given by

${\displaystyle \varepsilon _{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}}$

where ${\displaystyle n}$ is a positive, whole number. The energy of the particle is shown in the figure as a set of horizontal dotted lines, labeled with the corresponding state number.

The wavefunction for each state is given by

${\displaystyle \psi (x)={\begin{cases}{\sqrt {\frac {2}{L}}}\sin \left({\frac {n\pi x}{L}}\right);&0<x<L\\0;&{\text{otherwise}}\end{cases}}}$

The wavefunctions are shown superimposed on the diagram at the corresponding energies.