# Packing a particle in

**Classical Mechanics**Level 5

If a very low energy proton is trapped in a box with sides of length L and I slowly increase L then the energy density in \(J/m^3\) decreases as \(L^{-a}\), where a is some number. Similarly if a very low energy photon is trapped in the box the energy density scales as \(L^{-b}\), where b is a different number. Consider now a particle for which the energy E and momentum p are related by \(E^2=p^4 c^2/p_0^2\), where \(p_0\) is a constant with dimension of momentum and \(c\) is the speed of light. If I trap such a low energy particle in our box, the energy density scales as \(L^{-c}\) where c is still a different number. What is \(a \times b \times c\)?

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