Editing Particle in a box (section)

=== Energy levels ===
[[File:Confined particle dispersion - positive.svg|thumb|upright|The energy of a particle in a box (black circles) and a free particle (grey line) both depend upon wavenumber in the same way. However, the particle in a box may only have certain, discrete energy levels.]]
The energies that correspond with each of the permitted wave numbers may be written as{{sfn|Hall|2013|p=81}}
<math display="block">E_n = \frac{n^2\hbar^2 \pi ^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}.</math>
The energy levels increase with <math>n^2</math>, meaning that high energy levels are separated from each other by a greater amount than low energy levels are.  The lowest possible energy for the particle (its ''[[zero-point energy]]'') is found in state 1, which is given by<ref name="Bransden159">Bransden and Joachain, p. 159</ref>
<math display="block">E_1 = \frac{\hbar^2\pi^2}{2mL^2} = \frac{h^2}{8mL^2}.</math>
The particle, therefore, always has a positive energy.  This contrasts with classical systems, where the particle can have zero energy by resting motionlessly.  This can be explained in terms of the [[uncertainty principle]], which states that the product of the uncertainties in the position and momentum of a particle is limited by
<math display="block">\Delta x\Delta p \geq \frac{\hbar}{2}</math>
It can be shown that the uncertainty in the position of the particle is proportional to the width of the box.<ref name="Davies15">Davies, p. 15</ref>  Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box.<ref name="Bransden159" />  The [[kinetic energy]] of a particle is given by <math>E = p^2/(2m)</math>, and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above.<ref name="Bransden159" />