The GATE Grind

GATE 2023 CY – Question 28

Physical Chemistry · Hydrogen and Hydrogen-like Atoms: Atomic orbitals; radial and angular distribution functions and their probabilities; Atomic units, Multi-electron atoms: Orbital approximation; electron spin; Pauli exclusion principle; Slater determinants. Variational method and secular determinants; first order non-degenerate perturbation techniques. · 1 mark · Multiple choice

Which statement is INCORRECT?

  1. The zero-point energy of a quantum harmonic oscillator of frequency $\nu$ is $h\nu/2$
  2. A quantum rigid rotor’s energy level is inversely proportional to its moment of inertia
  3. The time-independent Schrödinger equation for $Li^{2+}$ cannot be solved exactly
  4. Total angular momentum equals the sum of orbital and spin angular momenta

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Correct answer: (C) The time-independent Schrödinger equation for $Li^{2+}$ cannot be solved exactly

Explanation

Li2+ contains only one electron, so its Coulomb Schrödinger equation is hydrogenic and can be solved exactly. Statement **C**, claiming otherwise, is false. The oscillator zero-point energyhν/2 and rotor energy inversely proportional to I are correct, and total angular momentum combines orbital and spin contributions.