So, when we calculate density of states of the electrons/ phonons, or while during a similar calculation of no. of modes per frequency in the Blackbody radiation, we assume Standing wave solutions:
$c_1Sin(k_x x)Sin(k_yy)Sin(k_zz)e^{-iwt}$, which on applying the boundary conditions give us $k_x=n_x\pi/L$ and similarily for $k_y,k_z $ also. I have seen many books taking these $n_i$ to be $\textbf{POSITIVE}$ integers only, but I don't understand why. This actually forces to take only the first octant in the $k$-space as the total available volume and a 1/8 factor comes.
In short, why do the $n_i$ in the standing wave solutions need to be positive integers and not negative?
Best Answer
You need to count linearly independent eigenfunctions. Switching the sign of $n_i$ can be absorbed in a change of $c_1$ since $\sin$ is odd. You are therefore only overcounting $8$ times the same eigenfunction. Thankfully, this means that the book was correct!
Note that you have a similar situation if you had $\cos$ instead, were the function itself does not even change since it is even. In general it is often easier to consider instead periodic boundary conditions and you can check that the density of states do coincide asymptotically (at high energy) with your hard boundary problem (without your $8$ factor), justifying the infinite volume limit.
Hope this helps.