# 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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