We have to calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are a and 2a, where a is the Bohr radius.

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The radial probability density for a hydrogen atom is defined such that gives the probability for the electron to be found in the volume enclosed by concentric spherical shell of radius r and thickness . For the hydrogen atom in the ground state, we have

,

in which a is the Bohr radius. We recall that

Therefore, the probability of finding electron in the volume between the spherical shells of radii a and 2a will be given by the following integral:

The probability of finding the electron in the hydrogen atom, in its ground state, between spherical shells whose radii are a and 2a will, therefore, be 43.9%.