In this problem you will show that the charge density of the filled n = 2, l = 1 subshel...

In this problem you will show that the charge density of the filled n = 2, l = 1 subshell is spherically symmetrical and that therefore L = 0. The angular distribution of the electron charge is simply the sum of the squares of the magnitude of the angular part of the wave functions for l = 1 and m_l = -1, 0, and 1.

a. Given that the angular part of these wave functions is

write an expression for

.

b. Show that does not depend on θ and Φ.

c. Why does this result show that the charge density for the filled n = 2, l = 1 , subshell is spherically symmetrical?