Questions: Quantum Numbers

Question 1

For an electron in the n = 3 shell of hydrogen, how many distinct values of ℓ are allowed?

• A. 1 (only ℓ = 0)
• B. 2 (ℓ = 0 and 1)
• C. 3 (ℓ = 0, 1, and 2)
• D. 4 (ℓ = 0, 1, 2, and 3)

Answer: (C)

ℓ ranges from 0 to n−1 inclusive. For n = 3, the allowed values are ℓ = 0, 1, 2 — three values. The common error is allowing ℓ = n = 3, but the constraint is ℓ ≤ n−1. Each value of ℓ corresponds to a subshell: s (ℓ=0), p (ℓ=1), d (ℓ=2).

Question 2

In hydrogen, knowing the principal quantum number n is sufficient to mostly specify the electron's quantum state.

• T. True
• F. False

Answer: False

n determines the energy level of hydrogen, but specifying the quantum state requires all four quantum numbers: n, ℓ, m_ℓ, and m_s. For n = 2 alone, there are 8 distinct states: the ℓ=0 subshell has 1 orbital × 2 spin states = 2; the ℓ=1 subshell has 3 orientations (m_ℓ = −1, 0, +1) × 2 spin states = 6. Total: 8 states all sharing the same energy in hydrogen.

Question 3

Why did the Bohr model need to be replaced by the full quantum mechanical treatment, even though it gives the correct energy levels for hydrogen?

Model answer: The Bohr model gives correct energies but cannot explain orbital shapes, the degeneracy structure of subshells, fine structure of spectral lines, or multi-electron atoms. The full treatment with ℓ, m_ℓ, and m_s accounts for these and provides a consistent framework for all atoms.

Bohr quantized angular momentum by fiat (L = nℏ) without a physical justification, predicted only circular orbits, and failed for any atom beyond hydrogen. The wave-mechanical treatment solves the Schrödinger equation in 3D, from which ℓ and m_ℓ emerge naturally and explain orbital shapes and the Zeeman effect. Spin (m_s) is needed for the Pauli exclusion principle and the periodic table.