- #1

- 202

- 2

In the book "Physics of atoms and molecules" by Bransden and Joachain it is said:

We have to calculate the density of final states. To do this let the volume V be a cube of side L. We can impose periodic boundary conditions on the wave function, that is:

[tex]k_x=\frac{2\pi}{L}n_x[/tex]

[tex]k_y=\frac{2\pi}{L}n_y[/tex]

[tex]k_z=\frac{2\pi}{L}n_z[/tex]

where nx, ny and nz are positive or negative integers, or zero. Since L is very large we can treat nx, ny and nz as continuous variables, and the number of states in the range d[tex]\vec{k}=dk_xdk_ydk_z[/tex] is:

[tex]dn_xdn_ydn_z=\left(\frac{L}{2\pi}\right)^3dk_xdk_ydk_z=\left(\frac{L}{2\pi}\right)^3k^2dkd\Omega[/tex]

I can't understand the last equality, [tex]\Omega[/tex] is the solid angle, but how do I relate it to [tex]dk_xdk_ydk_z[/tex]?