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 $a \times b \times c$ ?

The answer is 60.

1 solution

David Mattingly Staff
May 13, 2014

The particle is trapped in the box, hence there are a half-integer number of wavelengths $\lambda$ in L. Therefore, as I slowly increase L the wavelength increases linearly. The wavenumber $k \propto 1/\lambda$ . The momentum is proportional to wavenumber (via quantum mechanics) and so we have the energy of the particle $E \propto p^2 \propto k^2 \propto \lambda^{-2} \propto L^{-2}$ . Hence the energy density scales as $E/V \propto L^{-5}$ . Therefore c=5. Similar logic for a,b yield a=3 and b=4. Hence $a \times b \times c=60$ .

Packing a particle in

The answer is 60.

1 solution

0 pending reports