Question

The INCORRECT statement is

• A. Zero-point energy of a quantum mechanical harmonic oscillator of frequency \(\nu\) is \(\tfrac{h\nu}{2}\)
• B. Energy level of a quantum mechanical rigid rotor is inversely proportional to its moment of inertia
• C. The time independent Schrödinger equation for Li\(^ {2+}\) \textbf{cannot} be solved exactly
• D. Total angular momentum of an atomic system is equal to the sum of orbital angular momentum and spin angular momentum

Hint:

Hydrogen-like ions (\(H, He^+, Li^{2+}, Be^{3+}\), etc.) are exactly solvable in quantum mechanics.
Multi-electron atoms cannot be solved exactly due to electron–electron repulsion; approximations are used.
Always recall: Zero-point energy for a harmonic oscillator = \(\tfrac{1}{2} h\nu\).

Answer 1

The Correct Option is C

Step 1: Statement (A) is correct. The zero-point energy of a quantum harmonic oscillator is \(\tfrac{1}{2}h\nu\).
Step 2: Statement (B) is correct. The energy of a rigid rotor is given by \[ E_J = \frac{h^2}{8\pi^2 I} J(J+1) \] where \(I\) is the moment of inertia. Clearly, \(E \propto \tfrac{1}{I}\).
Step 3: Statement (C) is incorrect. The Li\(^ {2+}\) ion is a hydrogen-like species (one electron system with \(Z=3\)). The Schrödinger equation for hydrogen-like atoms can be solved exactly. Hence, this statement is false.
Step 4: Statement (D) is correct. The total angular momentum of an atom is indeed the sum of orbital (\(L\)) and spin (\(S\)) angular momenta: \(\vec{J} = \vec{L} + \vec{S}\).
Thus, the incorrect statement is (C).