Derive (i) an asymptotic expression for the number of ways in which a given energy \(E\) can be distributed among a set of \(N\) one-dimensional harmonic oscillators, the energy eigenvalues of the oscillators being \(\left(n+\frac{1}{2}ight) \hbar \omega ; n=0,1,2, \ldots\), and (ii) the corresponding expression for the "volume" of the relevant region of the phase space of this system. Establish the correspondence between the two results, showing that the conversion factor \(\omega_{0}\) is precisely \(h^{N}\).