The wave functions for the three states of hydrogen atom, which have , , , +1, and -1, are

in which the subscripts on give the values of the quantum numbers , , . Note that the first wave function is real but the others, which involve the imaginary number , are complex.

We have to find the probability density for each function and by adding the three probability densities show that their sum is spherically symmetric, depending on the radial coordinate .

Solution:             Click For PDF Version

(a)

The probability density for a quantum state is given by the modulus square of the normalized wave function. The probability densities of the states corresponding to the quantum numbers, , , , +1, and -1, will therefore be

and

.

Therefore, we get