GATE 2022 PH – Question 40

Correct answer: (C) $-i(\partial_s+1/(2s))$

Explanation

An operator is Hermitian if $\langle f|\hat Ag\rangle=\langle\hat Af|g\rangle$ for all well-behaved functions. In cylindrical coordinates the volume element contains the factor $s$: $dV=s\,ds\,d\phi\,dz$, so the inner product of radial functions is $\int f^*g\,s\,ds$.

**Test $\hat p=-i(\partial_s+c/s)$.**
$$\int f^*\left[-i\left(g^\prime+\frac cs g\right)\right]s\,ds=-i\int f^*sg^\prime\,ds-ic\int f^*g\,ds .$$

Integrate the first term by parts (the boundary terms vanish):
$$-i\int f^*sg^\prime\,ds=i\int(f^*s)^\prime g\,ds=i\int\left(f^{*\prime}s+f^*\right)g\,ds .$$

Collect:
$$\langle f|\hat pg\rangle=\int\left[i f^{*\prime}s+if^*-icf^*\right]g\,ds=\int\left[i\,f^{*\prime}+i(1-c)\frac{f^*}{s}\right]g\,s\,ds .$$

For Hermiticity this must equal $\int\left[-i\left(f^\prime+\dfrac csf\right)\right]^*g\,s\,ds=\int\left[i\,f^{*\prime}+i\dfrac cs f^*\right]g\,s\,ds$. Matching the coefficients of $f^*/s$:
$$1-c=c\;\Rightarrow\;c=\tfrac12 .$$

The Hermitian radial momentum operator is $-i\left(\partial_s+\dfrac1{2s}\right)$ (option C).