File:Particle in a box wavefunctions 2.svg

English: Initial wavefunctions for the lowest four quantum states of a particle trapped in an infinitely deep quantum well. The spatial position is shown along the horizontal axis. The red regions represent barriers with an infinitely large potential, while the area between the barriers represents a well with zero potential.

The wavefunction for each state is given by

${\displaystyle \psi (x,t)={\begin{cases}{\sqrt {\frac {2}{L}}}\mathrm {e} ^{\mathrm {i} \omega _{n}t}\sin \left({\frac {n\pi x}{L}}\right);&0<x<L\\0;&{\text{otherwise}},\end{cases}}}$

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