GATE 2024 CY – Question 56

Correct answer: (B) the number of nodes in the radial wave function of 3s orbital of a hydrogen atom is the same as the number of nodes in the angular wave function of a 4d orbital of hydrogen atom.; (C) the energy separation between any two adjacent states is same for a harmonic oscillator, while it is different for a rigid rotor.

Explanation

Check each statement.

• **A. The normalisation factor of a Slater determinant for a 3-electron atom is $\sqrt{1/3}$.** False. For $N$ electrons the factor is $1/\sqrt{N!}$, so $1/\sqrt{3!}=1/\sqrt6$.
• **B. The number of radial nodes in 3s equals the number of angular nodes in 4d.** True. Radial nodes $=n-l-1=3-0-1=2$ for 3s. Angular nodes $=l=2$ for 4d. ✓
• **C. For a harmonic oscillator all adjacent levels are equally spaced, while for a rigid rotor the spacing differs.** True. The oscillator levels are $E_v=(v+\tfrac12)h\nu$ with constant spacing $h\nu$, and the rotor levels $E_J=BJ(J+1)$ have spacing $2B(J+1)$ that grows with $J$. ✓
• **D. The magnitude of the spin angular momentum of an $\alpha$ electron is the negative of a $\beta$ electron's.** False. Both have the same magnitude $\sqrt{s(s+1)}\hbar=\tfrac{\sqrt3}{2}\hbar$; only the $z$-components $\pm\tfrac12\hbar$ differ in sign.

Answer **B and C**.