GATE 2026 CY – Question 27

For $\Phi=x(a-x)$, with constant a, which statement(s) is/are correct?

It represents a stationary state
It is an odd function
It represents the wavefunction of a particle moving on a circular ring of radius a
It is an eigen function of the momentum operator

Show answer and explanation

Correct answer: (A) It represents a stationary state

Explanation

$\Phi=x(a-x)$ is a real function of $x$, a parabola that is zero at $x=0$ and $x=a$.

**A. It represents a stationary state.** A stationary state has a time-independent probability density; a real, time-independent spatial function such as this (it vanishes at both walls, as for a particle in a box) can represent such a state. ✓
**B. It is an odd function.** False. $\Phi(-x)=-x(a+x)=-ax-x^2$, which is neither $\Phi(x)$ nor $-\Phi(x)$.
**C. It is the wave function of a particle on a ring of radius $a$.** False. Ring wave functions are periodic functions of the angle, like $e^{im\phi}$.
**D. It is an eigenfunction of the momentum operator.** False. $\hat p_x\Phi=-i\hbar(a-2x)$, which is not a constant multiple of $\Phi$.

Answer **A**.

Official GATE 2026 answer key: https://gate2026.iitg.ac.in/doc/download/2026/Keys/CY_Keys.pdf