In ?7, we showed that the energy of a particle of mass M in an L ? L ? L box in the state labeled |n, m, l? is E _{n, m, l} = ?²?²(n² + m² + l²)/2ML². The probability of finding a particle in the state |n, m, l? at temperature T is given by the Boltzmann distribution P(n, m, l) ? exp(?E _{n, m, l}/k_BT). In a hot gas the energy levels are very close together, so E can be regarded as a continuous variable. Show that the probability of finding the particle with energy between E and E + dE at temperature T is given by the Maxwell?Boltzmann distribution, dP/dE = 2??Ee^?E/kBT /(?k_BT)^3/2. Employ the methods used to compute the spectrum of blackbody radiation, including eq. (22.20)?

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dn1 dn2 dn3 8. n?dn 0. 0.