The wave function of the hydrogen atom in the (n, l, m_l) = (2,1,0) state, y(r, q, f), is

.

Note that the wave function has no f dependence.
The figure below is a grayscale contour plot of this function. Dark means high density.

a) Along which axis is the probability density greatest? Answer x = 1, y = 2, or z = 3.

The probability density is greatest along the axis.

b) On the z axis, at z = a_o, what is the probability density, |y|²? You must give a symbolic answer, expressed in terms of the appropriate combination of h, c, ao, and numbers (pi, etc). Note that we have chosen convenient units: per a_o³.

Probability density = per a_o³.

c) Consider all points a distance r = a_o from the origin as (the point in part b is one such point). At what polar angle, q, between 0 and 90 degrees is the probability density one-half of the value in part b?

q = degrees.