Matching Atomic Orbital Properties with Quantum Numbers

To solve this matching problem, we analyze each entry of Column I in relation to the quantum numbers given in Column II:

1. (A) Orbital angular momentum of the electron in a hydrogen-like atomic orbital:

   • The magnitude of the orbital angular momentum is given by:
     $L = \sqrt{l(l+1)}\hbar$
     which is determined by the azimuthal quantum number $l$ (entry q).
   • The component of the orbital angular momentum along a chosen quantization axis (usually the $z$-axis) is given by:
     $L_z = m\hbar$
     which is determined by the magnetic quantum number $m$ (entry r).
   • Therefore, the orbital angular momentum is directly related to both the azimuthal and magnetic quantum numbers.
   • (A) $\to$ (q), (r)

2. (B) A hydrogen-like one-electron wave function obeying Pauli principle:

   • According to the Pauli exclusion principle, the complete state of an electron in an atom is uniquely specified by a set of four quantum numbers:
     • Principal quantum number, $n$ (entry p)
     • Azimuthal quantum number, $l$ (entry q)
     • Magnetic quantum number, $m$ (entry r)
     • Spin quantum number, $s$ (or $m_s$) (entry s)
   • A complete one-electron wave function (spin-orbital) is a product of its spatial wave function $\psi_{n, l, m}(r, \theta, \phi)$ and its spin wave function $\chi(s)$. Thus, it involves all four quantum numbers.
   • (B) $\to$ (p), (q), (r), (s)

3. (C) Shape, size, and orientation of hydrogen-like atomic orbitals:

   • Size: Determined primarily by the principal quantum number $n$ (entry p), as the average distance of the electron from the nucleus increases with $n$ ($\langle r \rangle \propto n^2$).
   • Shape: Determined by the azimuthal quantum number $l$ (entry q) (e.g., $l=0$ is spherical, $l=1$ is dumbbell-shaped, $l=2$ is double-dumbbell-shaped).
   • Orientation: Determined by the magnetic quantum number $m$ (entry r), which specifies the spatial orientation of the orbital along the coordinate axes ($p_x, p_y, p_z$, etc.).
   • (C) $\to$ (p), (q), (r)

4. (D) Probability density of electron at the nucleus in hydrogen-like atom:

   • The radial wave function for a hydrogen-like atom near $r = 0$ behaves as $R_{nl}(r) \propto r^l$.
   • Consequently, for orbitals with $l > 0$ ($p, d, f$ orbitals), the wave function vanishes at the nucleus:
     $\psi_{nlm}(0) = 0 \implies |\psi_{nlm}(0)|^2 = 0$
   • For $s$-orbitals ($l = 0$), the wave function at the nucleus is non-zero, and the probability density is given by:
     $|\psi_{n00}(0)|^2 = \frac{Z^3}{\pi n^3 a_0^3}$
   • Thus, the probability density at the nucleus depends on:
     • The azimuthal quantum number $l$ (entry q), which dictates whether the density is non-zero ($l=0$) or zero ($l > 0$).
     • The principal quantum number $n$ (entry p), which determines the non-zero value of the probability density for $s$-orbitals.
   • (D) $\to$ (p), (q)

---

Final Matching:

• (A) $\to$ (q), (r)
• (B) $\to$ (p), (q), (r), (s)
• (C) $\to$ (p), (q), (r)
• (D) $\to$ (p), (q)