Consider a system similar to the one in the preceding problem but consisting of \(3 N\) particles moving in one dimension. Show that the partition function in this case is given by

\[
Q_{3 N}(L, T)=\frac{1}{(3 N) !}\left[2 L\left(\frac{k T}{h c}ight)ight]^{3 N}
\]

\(L\) being the "length" of the space available. Compare the thermodynamics and the density of states of this system with the corresponding quantities obtained in the preceding problem.