GATE PH 2022 ā¢ Q40
MCQ ā¢ 2 Marks
Section 1: Mathematical Physics
Linear Algebra & Vector Calculus
Subtopic: Vector Differential Operators (Grad, Div, Curl)
Question Statement:
In cylindrical coordinates (s, š, z), which of the following is a Hermitian
operator?
Options:
-
(A) 1
i
š
šs
-
(B) 1
i \left( š
šs + 1
s\right)
-
(C) 1
i \left( š
šs + 1
2s\right)
-
(D) \left( š
šs + 1
s\right)
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.40)**
**1. Quantum Operator Hermiticity in Curvilinear Coordinates**:
- In cylindrical coordinates $(s, \phi, z)$, the volume element is:
$d^3r = s\, ds\, d\phi\, dz$
- An operator $\hat{A}$ acting on radial coordinate $s$ is Hermitian if:
$\int_0^\infty s\, ds\, \psi^*(s) [\hat{A} \phi(s)] = \int_0^\infty s\, ds\, [\hat{A} \psi(s)]^* \phi(s)$
**2. Step-by-Step Integration by Parts**:
- Let $\hat{p}_s = \frac{1}{i}\left(\frac{\partial}{\partial s} + f(s)\right)$.
- Compute the inner product:
$\int_0^\infty s \psi^* \frac{1}{i}\frac{\partial \phi}{\partial s}\, ds = \left[ \frac{s\psi^*\phi}{i} \right]_0^\infty - \frac{1}{i} \int_0^\infty \frac{\partial(s\psi^*)}{\partial s} \phi\, ds$
- Using product rule on $\frac{\partial(s\psi^*)}{\partial s} = \psi^* + s\frac{\partial\psi^*}{\partial s}$:
$= -\frac{1}{i}\int_0^\infty \psi^* \phi\, ds - \frac{1}{i}\int_0^\infty s \frac{\partial\psi^*}{\partial s}\phi\, ds$
$= \int_0^\infty s\, ds \left[\left(\frac{1}{i}\frac{\partial\psi}{\partial s}\right)^* - \frac{1}{is} \psi^*\right] \phi$
- To balance the extra term $-\frac{1}{is}$, the symmetric Hermitian radial momentum operator requires adding half of the metric divergence $\frac{1}{2s}$:
$\hat{p}_s = \frac{1}{i}\left(\frac{\partial}{\partial s} + \frac{1}{2s}\right)$
**3. Official Verification**:
The verified official IIT Master Answer is **Option (C)** (`1/i (\partial/\partial s + 1/(2s))`).
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