GATE 2020 PH – Question 50
The radial wave function in a central potential is $R(r)=A(r/a)e^{-r/(2a)}$, with normalization constant A and positive constant a. If the most probable distance from the force centre is $\gamma a$, find γ.
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Correct answer: 4 to 4
Explanation
The radial probability includes the spherical volume factor: $P(r)\propto r^2|R(r)|^2\propto r^4e^{-r/a}$. For its positive maximum, set $d\ln P/dr=4/r-1/a=0$, giving r = 4a. Therefore γ = 4. Maximizing the radial wave-function magnitude alone would omit the volume element and give the wrong distance.