GATE 2021 PH – Question 61
The source labels the following normalized hydrogen radial wave function as a “second excited state”: $$R(r)=\frac1{\sqrt{24}}a^{-3/2}\frac r a e^{-r/(2a)},$$ where $a$ is the Bohr radius. The most probable distance of the electron from the centre is $y\,a$. Find integer $y$.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 4 to 4
Explanation
The radial probability density includes the spherical volume factor: $P(r)\,dr\propto r^2|R(r)|^2dr\propto r^4e^{-r/a}dr$. Setting its logarithmic derivative to zero gives $4/r-1/a=0$, so $r_{\max}=4a$ and $y=4$. Maximizing $|R|^2$ alone would omit the volume factor. The supplied function is the 2p radial form, irrespective of the source’s excited-state label.