59. (a) È210 is real. Squaring it, we obtain the probability density:
r2
|È210|2 = e-r/a cos2 ¸ .
32Ä„a5
Each of the other functions is multiplied by its complex conjugate, obtained by replacing i with
-i in the function. Since eiĆe-iĆ = e0 = 1, the result is the square of the function without the
exponential factor:
r2
|È21+1|2 = e-r/a sin2 ¸
64Ä„a5
and
r2
|È21-1|2 = e-r/a sin2 ¸ .
64Ä„a5
The last two functions lead to the same probability density.
(b) The total probability density for the three states is the sum:

r2 1 1
|È210|2 + |È21+1|2 + |È21-1|2 = e-r/a cos2 ¸ + sin2 ¸ + sin2 ¸
32Ä„a5 2 2
r2
= e-r/a .
32Ä„a5
The trigonometric identity cos2 ¸ + sin2 ¸ = 1 is used. We note that the total probability density
does not depend on ¸ or Ć; it is spherically symmetric.