GATE 2025 CY – Question 45
The correct option with regard to the following statements is (a) Time-independent Schrödinger equation can be exactly solved for $\mathrm{Be}^{2+}$. (b) For a particle confined in a one-dimensional box of length $l$ with infinite potential barriers, the trial variation function $\phi=\sqrt{3/l^3}\,x$ is not an acceptable trial wavefunction for $0\le x\le l$. (c) Wavefunctions for system of Fermions must be anti-symmetric with respect to exchange of any two Fermions in the system. (d) Born-Oppenheimer approximation can be used to separate the vibrational and rotational motion of a molecule.
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Correct answer: (C) (a) False (b) True (c) True (d) False
Explanation
Check each statement.
- **(a) The time-independent Schrödinger equation can be solved exactly for $\text{Be}^{2+}$.** $\text{Be}^{2+}$ has $4-2=2$ electrons, so it is a two-electron ion with electron-electron repulsion. Only one-electron (hydrogen-like) ions are exactly solvable. **False.**
- **(b) The trial function $\phi=\sqrt{3/l^3}\,x$ is not acceptable for the particle in a box.** An acceptable trial function must vanish at the walls, $\phi(0)=\phi(l)=0$. This one is $0$ at $x=0$ but equals $\sqrt{3/l}\neq0$ at $x=l$. **True.**
- **(c) Wavefunctions of fermions must be antisymmetric under exchange of two fermions.** This is the Pauli principle. **True.**
- **(d) The Born-Oppenheimer approximation separates vibrational and rotational motion.** It separates **electronic** from **nuclear** motion. **False.**
So (a) False, (b) True, (c) True, (d) False: option **C**.