Question

The correct option with regard to the following statements is 
(a) Time-independent Schrödinger equation can be exactly solved for Be\(^{2+}\). 
(b) For a particle confined in a one-dimensional box of length \( l \) with infinite potential barriers, the trial variation function \( \phi = \left[ \left( \frac{3}{l^3} \right)^{1/2} x \right] \) 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.

• (a) True (b) False (c) False (d) True
• (a) True (b) True (c) False (d) False
• (a) False (b) True (c) True (d) False
• (a) False (b) True (c) True (d) True

Hint:

- Exact Schrödinger equation solutions are for one-electron systems. - Variational trial wavefunctions must meet boundary conditions. - Fermion wavefunctions are antisymmetric under particle exchange. - Born-Oppenheimer approximation separates electronic and nuclear motion.

Answer 1

The Correct Option is C

(a) Time-independent Schrödinger equation can be exactly solved for Be²⁺.
Be²⁺ has an electronic configuration of 1s², meaning it has two electrons.
The time-independent Schrödinger equation can be solved exactly only for one-electron systems.
For systems with two or more electrons, the electron-electron interaction term makes it impossible to obtain an exact analytical solution.
We rely on approximation methods like Hartree-Fock.
Therefore, statement (a) is False.

(b) For a particle confined in a one-dimensional box of length l with infinite potential barriers, the trial variation function φ(x) = [(3/l³)^1/2] × x is not an acceptable trial wavefunction for 0 ≤ x ≤ l.
For a trial wavefunction to be acceptable in the variational method for a 1D box with infinite barriers at x = 0 and x = l, it must satisfy the boundary conditions: φ(0) = 0 and φ(l) = 0.
Given: φ(x) = (3/l³)^1/2 × x
At x = 0: φ(0) = 0 — boundary condition satisfied.
At x = l: φ(l) = (3/l)^1/2 ≠ 0 — boundary condition not satisfied.
Therefore, statement (b) is True.

(c) Wavefunctions for a system of Fermions must be anti-symmetric with respect to exchange of any two Fermions in the system.
Fermions are particles with half-integer spin.
The total wavefunction of a system of identical fermions must be anti-symmetric under exchange — this is the Pauli Exclusion Principle.
Therefore, statement (c) is True.

(d) Born-Oppenheimer approximation can be used to separate the vibrational and rotational motion of a molecule.
The Born-Oppenheimer approximation separates electronic and nuclear motion (not directly vibration and rotation).
Separation of vibrational and rotational motion is a further approximation based on treating the nuclei as a semi-rigid rotor.
Therefore, statement (d) is False.

Summary:
(a) False
(b) True
(c) True
(d) False

\[ \boxed{\text{Correct option is (C)}} \]