Which one of the following about an electron occupying the 1s orbital in a hydrogen atom is incorrect? (The Bohr radius is represented by a$$_0$$).

We need to identify which statement about an electron in the 1s orbital of a hydrogen atom is incorrect.

Option A: "The total energy of the electron is maximum when it is at a distance $$a_0$$ from the nucleus."

For the hydrogen atom, the most probable distance of the electron from the nucleus in the 1s orbital is the Bohr radius $$a_0$$. At this distance, the electron is in its ground state (n = 1), which is the state of minimum total energy (most negative energy), not maximum. The total energy at this distance is $$E_1 = -13.6$$ eV, which is the lowest possible energy. Therefore, this statement is incorrect.

Option B: "The electron can be found at a distance $$2a_0$$ from the nucleus."

The probability density function $$|\psi_{1s}|^2$$ for the 1s orbital is non-zero for all distances $$r > 0$$. Although the probability decreases exponentially with distance, there is still a non-zero probability of finding the electron at $$r = 2a_0$$. This statement is correct.

Option C: "The probability density of finding the electron is maximum at the nucleus."

For the 1s orbital, $$\psi_{1s} = \frac{1}{\sqrt{\pi}}\left(\frac{1}{a_0}\right)^{3/2} e^{-r/a_0}$$. The probability density $$|\psi_{1s}|^2$$ is maximum at $$r = 0$$ (at the nucleus), since $$e^{-r/a_0}$$ is maximum when $$r = 0$$. Note: the radial probability distribution $$4\pi r^2|\psi|^2$$ peaks at $$r = a_0$$, but the probability density itself is maximum at the nucleus. This statement is correct.

Option D: "The magnitude of the potential energy is double that of its kinetic energy on an average."

By the virial theorem for a Coulombic potential, $$\langle V \rangle = 2\langle E \rangle$$ and $$\langle T \rangle = -\langle E \rangle$$. Therefore $$|\langle V \rangle| = 2|\langle T \rangle|$$. The magnitude of the average potential energy is indeed double the average kinetic energy. This statement is correct.

The incorrect statement is Option A.