Why does the energy level of a particle in a box decrease with length?
In quantum mechanics, particles are waves, and as with any wave the longer the wavelength the less energy is in the wave.  You don't need the box to see this; infrared light has less energy in the photon than ultraviolet (the former makes you warm but the latter can damage your skin) and a slow moving free electron (lower energy) has a shorter associated wavelength than a fast-moving one (higher energy).  In the particle/box problem, the possible wavelengths are set by the dimensions of the box; a bigger box allows longer wavelengths -&gt; lower energies.