aura said:
oops! a big printing mistake...thats eigen vector...nx,ny,nz
now can u solve this at least the explanation...
E_{n_x,n_y,n_z} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[n_x^2+n_y^2+n_z^2\right]
E_{1,1,1} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[3\right]
E_{2,1,1} = E_{1,2,1} = E_{1,1,2} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[6\right]
E_{2,2,1} = E_{2,1,2} = E_{1,2,2} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[9\right]
E_{3,1,1} = E_{1,3,1} = E_{1,1,3} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[11\right]
E_{2,2,2} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[12\right]
E_{1,2,3} = E_{1,3,2} = E_{2,1,3} = E_{2,3,1} = E_{3,1,2} = E_{3,2,1} = \left[\frac{2\ \pi^2}{mL^2}\right] \left[14\right]
etc, etc.
I believe you are supposed to be figuring out all possible energies and how many degenerate states there are for each energy, and then divide the number of states by some energy interval to find the density. Unless I have missed some, the density is a bit erratic for these low numbered states. For larger n, perhaps you can come up with a general expression for how many states there are between some energy E and and a slightly higher level to come up with a number of states per unit energy interval. The sum of squares is suggestive that thinking in terms of the number of states contained within a spherical energy surface might prove helpful.
