Questions: Quantum Numbers and Spherical Harmonics

Question 1

Why are there exactly five d orbitals at any given energy level?

• A. Because d is the fourth type of orbital and 4 + 1 = 5 by a conventional counting rule
• B. Because d orbitals have a pentagonal symmetry that requires five distinct spatial orientations
• C. Because d orbitals have ℓ = 2, giving magnetic quantum number values m_ℓ = −2, −1, 0, +1, +2 — five distinct quantum states
• D. Because five d orbitals are needed to accommodate the 10 electrons that fill the d subshell

Answer: (C)

The number of orbitals for a given ℓ is always 2ℓ + 1, one for each allowed value of m_ℓ. For d orbitals (ℓ = 2), m_ℓ ranges from −2 to +2: five values, five orbitals. Option D reverses the causation — the 10-electron capacity follows from having 5 orbitals × 2 spins, not the other way around.

Question 2

Which physical constraint gives rise to the magnetic quantum number m_ℓ?

• A. The requirement that the radial wavefunction approach zero at large distances from the nucleus
• B. The requirement that the wavefunction be single-valued — returning to the same value after a full 2π rotation in the azimuthal (φ) direction
• C. The requirement that the total energy of the electron be negative (bound state)
• D. The requirement that the angular momentum magnitude be an integer multiple of ℏ

Answer: (B)

The φ-dependence of the wavefunction has the form e^{im_ℓφ}. For this to be single-valued — ψ(φ) = ψ(φ + 2π) — we need e^{im_ℓ·2π} = 1, which requires m_ℓ to be an integer. Option D describes a consequence of quantization rather than the mathematical constraint that generates it. Option A constrains the radial quantum number n, not m_ℓ.

Question 3

All hydrogen atom states with the same principal quantum number n have the same energy, regardless of their ℓ and m_ℓ values.

• T. True
• F. False

Answer: True

In the ideal hydrogen atom, energy depends only on n: Eₙ = −13.6 eV/n². States with the same n but different ℓ and m_ℓ are degenerate. This n-degeneracy is a special feature of the 1/r Coulomb potential and is broken in multi-electron atoms, where electron-electron repulsion makes energy depend on ℓ as well (which is why the 2s and 2p orbitals have different energies in helium and heavier elements).

Question 4

The electron's spin quantum number m_s = ±1/2 arises naturally from solving the Schrödinger equation in spherical coordinates, just as n, ℓ, and m_ℓ do.

• T. True
• F. False

Answer: False

Spin does not emerge from the non-relativistic Schrödinger equation. It must be added as an independent degree of freedom — a two-component spinor describing intrinsic angular momentum with no classical analog. It arises naturally from the Dirac equation (relativistic quantum mechanics) but must be grafted onto the Schrödinger framework by hand. This is why it is described separately from the three spatial quantum numbers.

Question 5

How do the four quantum numbers (n, ℓ, m_ℓ, m_s) together with the Pauli exclusion principle produce the shell structure of the periodic table?

Model answer: For each n, ℓ can range from 0 to n−1 (n values). For each ℓ, m_ℓ ranges from −ℓ to +ℓ (2ℓ+1 values). Summing over all ℓ gives n² spatial states per shell. With two spin states (m_s = ±1/2), each shell holds 2n² electrons. Pauli forbids any two electrons from sharing all four quantum numbers, so electrons fill these distinct slots sequentially. This yields 2 for n=1, 8 for n=2, 18 for n=3 — directly matching the periods of the periodic table.

The counting is not arbitrary: it follows necessarily from the mathematical constraints that generate each quantum number. Understanding the derivation makes the periodic table's structure predictable from first principles rather than a memorized pattern.