Questions: Principal, Angular, and Magnetic Quantum Numbers in Atoms

Question 1

Two electrons both have n=2 and ℓ=1, but one has m_ℓ = +1 and the other has m_ℓ = 0. In a hydrogen atom with no applied magnetic field, which electron has higher energy?

• A. m_ℓ = +1 has higher energy because its z-component of angular momentum is larger
• B. m_ℓ = 0 has higher energy because it corresponds to the 'central' orbital orientation
• C. They have the same energy — without an applied field, all m_ℓ states with the same n and ℓ are degenerate
• D. Energy depends on m_ℓ but the calculation requires knowing the spin quantum number too

Answer: (C)

In a free hydrogen atom with no external field, the energy depends only on n (E_n = −13.6 eV/n²). All states with the same n are degenerate — the 2p subshell's three m_ℓ = −1, 0, +1 states all have the same energy. The z-axis has no physical preferred status in the absence of a field; only when a magnetic field defines a preferred direction do the m_ℓ states acquire different energies via the Zeeman effect. The common misconception is treating m_ℓ as directly connected to energy.

Question 2

What is the magnitude of the orbital angular momentum for an electron with ℓ = 2?

• A. 2ℏ
• B. 4ℏ
• C. ℏ√6
• D. ℏ√4 = 2ℏ (same as answer A)

Answer: (C)

|L| = ℏ√(ℓ(ℓ+1)) = ℏ√(2·3) = ℏ√6 ≈ 2.45ℏ. This is a key formula to internalize: the angular momentum magnitude is NOT ℓℏ. The extra term in √(ℓ(ℓ+1)) versus √(ℓ²) = ℓ comes from quantum mechanics — the uncertainty principle prevents all three angular momentum components from being sharp simultaneously, so the magnitude must exceed the maximum z-component (m_ℓ·ℏ = 2ℏ at most for ℓ=2). The formula |L| = ℏ√(ℓ(ℓ+1)) encodes this.

Question 3

For an electron with ℓ = 1 and m_ℓ = 1, the total orbital angular momentum magnitude is ℏ, since L_z = m_ℓ·ℏ = ℏ implies the full angular momentum is ℏ.

• T. True
• F. False

Answer: False

m_ℓ gives only the z-component: L_z = m_ℓ·ℏ = ℏ. The total magnitude is |L| = ℏ√(ℓ(ℓ+1)) = ℏ√(1·2) = ℏ√2 ≈ 1.41ℏ, which is larger than L_z. The total angular momentum always exceeds its z-component (unless ℓ = 0) because the uncertainty principle requires L_x and L_y to have nonzero expectation of their squares. Setting |L| = L_z = m_ℓ·ℏ confuses the component with the magnitude — the most common error with magnetic quantum numbers.

Question 4

In a hydrogen atom with no applied magnetic field, an electron in the 2p subshell (n=2, ℓ=1) exists in one of three distinct energy levels corresponding to m_ℓ = −1, 0, +1.

• T. True
• F. False

Answer: False

Without an applied field, the three m_ℓ states of the 2p subshell are energetically degenerate — they all have the same energy E_2 = −3.4 eV. The z-axis is only meaningful when a magnetic field defines a preferred spatial direction; in a free atom, all orientations are equivalent. It is the Zeeman effect — the splitting of degenerate m_ℓ levels in an applied field — that makes the magnetic quantum number experimentally accessible.

Question 5

Why can we simultaneously know |L|² (the squared magnitude of orbital angular momentum) and L_z (one component), but not all three components L_x, L_y, L_z simultaneously? What physical principle prevents this?

Model answer: The angular momentum components satisfy the commutation relations [L_x, L_y] = iℏL_z (and cyclic permutations). By the Heisenberg uncertainty principle, two observables can be simultaneously sharp only if they commute. L_x and L_y do not commute, so they cannot both have definite values at once. However, L² = L_x² + L_y² + L_z² commutes with each component individually, so L² and one component (conventionally L_z) can be simultaneously sharp. This is why quantum numbers ℓ (giving |L|²) and m_ℓ (giving L_z) can both be well-defined, while L_x and L_y remain inherently indefinite.

This is not a measurement limitation but a fundamental feature of the quantum state. An eigenstate of L² and L_z genuinely has indefinite L_x and L_y — not merely unknown. The consequence is that the angular momentum vector cannot 'point' in a definite direction; it precesses around the z-axis, with fixed |L| and L_z but uncertain transverse components.