Use particle in a box wavefunctions to construct the \(N\)-body quantum canonical partition function \(Q_{N}(V, T)\) for indistinguishable noninteracting particles with mass \(m\) in a rectangular box with hard walls and volume \(V=L_{x} L_{y} L_{z}\). Use the delabeling factor \(1 / N\) ! to account for indistinguishability. Show that the partition function factorizes: \(Q_{N}(V, T)=Q_{1}^{N}(V, T) / N\) !. Show that if the thermal de Broglie wavelength \(\lambda=h / \sqrt{2 \pi m k T}\) is small compared with all three dimensions of the box, then the partition function approaches the classical ideal gas partition function \(Q_{N}(V, T) \approx\left(V / \lambda^{3}\right)^{N} / N\) !. What happens for the cases in which one or two of the dimensions of the box are reduced until they are much smaller than \(\lambda\) ?