A particle has wavefunction e-(i x+) with real constants ,. In which potential is it moving?

GATE 2024 PH – Question 29

Quantum Mechanics · time independent Schrodinger equation · 1 mark · Multiple choice

A particle has wavefunction $e^{-(i\alpha x+\beta)}$ with real constants $\alpha,\beta$. In which potential is it moving?

$V\propto\alpha^2x^2$
$V\propto e^{-\alpha x}$
$V=0$
$V\propto\sin(\alpha x)$

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Correct answer: (C) $V=0$

Explanation

The Schrödinger equation (time-independent) in one dimension is
$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V(x)\psi=E\psi .$$

**Differentiate the given wave function** $\psi=e^{-(i\alpha x+\beta)}$:
$$\psi^\prime=-i\alpha\,\psi,\qquad\psi^{\prime\prime}=(-i\alpha)^2\psi=-\alpha^2\psi .$$

**Substitute:**
$$\frac{\hbar^2\alpha^2}{2m}\psi+V\psi=E\psi\;\Rightarrow\;V=E-\frac{\hbar^2\alpha^2}{2m}=\text{constant}.$$

A constant potential means the particle is moving in a region of **constant (zero) potential**, i.e. it is a free particle: $V=0$ (taking the constant as the zero of energy) (option C).