In the ground state of the hydrogen atom we have to show that the probability P that the electron lies within a sphere of radius r is given by

in which . (b) We have to evaluate the probability that, in the ground state, the electron lies within a sphere of radius a.

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(a)

The radial probability density in the ground state of the hydrogen atom is given by the expression

,

in which a is the Bohr radius.

Therefore, the probability that the electron lies within a sphere of radius r will be given by the integral

.

We thus have

Therefore,

where .

(b)

The probability that the electron lies within a sphere of Bohr radius a will therefore be given by

The probability that electron will be found within a sphere of one Bohr radius is 32.3%.