Questions: Orbital Angular Momentum in Quantum Mechanics

Question 1

A quantum particle is in the state l = 2, m_l = 2 — the maximum m_l for this l value. A student concludes the angular momentum vector points exactly along the z-axis since m_l is at its maximum. What is wrong?

• A. Nothing is wrong — when m_l = l, the angular momentum is fully aligned with the z-axis
• B. The student forgot that L_z = ℏm_l is negative for positive m_l values
• C. Even at maximum m_l, L_z = ℏl is always strictly less than |L| = ℏ√(l(l+1)), so the vector cannot be fully z-aligned
• D. The angular momentum vector doesn't exist as a geometric object in quantum mechanics; only eigenvalues exist

Answer: (C)

For l = 2, m_l = 2: L_z = ℏ·2 = 2ℏ, but |L| = ℏ√(2·3) = ℏ√6 ≈ 2.45ℏ. Since L_z < |L|, the vector cannot be fully along the z-axis — there must be components in x and y directions, which are genuinely indeterminate. This is a purely quantum effect with no classical counterpart. Classically, you could align a spinning object precisely with any axis; quantum mechanically, the non-commutativity of components prevents this.

Question 2

Why can only ONE component of the angular momentum vector be known precisely at a time in quantum mechanics?

• A. Only L_z has a well-defined mathematical operator; L_x and L_y are undefined
• B. The Heisenberg uncertainty principle prohibits simultaneously knowing both position and momentum
• C. The angular momentum components do not commute: [L̂_x, L̂_y] = iℏL̂_z, so measuring one component disturbs the others
• D. Electron spin interferes with orbital angular momentum measurement for all but the z-component

Answer: (C)

The non-commutativity of L̂_x, L̂_y, L̂_z is the direct cause: [L̂_x, L̂_y] = iℏL̂_z (and cyclically). Two operators that don't commute cannot share a complete set of simultaneous eigenstates — so you cannot have a state where both L_x and L_y have definite values simultaneously. By contrast, L̂² commutes with each component ([L̂², L̂_z] = 0), so you CAN simultaneously know the total magnitude squared and one component.

Question 3

When l = 0, all three components L_x, L_y, and L_z are simultaneously zero and thus simultaneously well-defined, which is the only case where all components are simultaneously measurable.

• T. True
• F. False

Answer: True

When l = 0, the only allowed value is m_l = 0, giving L_z = 0 and |L|² = ℏ²·0·1 = 0, so all components are zero. A vector that is identically zero has no directional ambiguity — L_x = L_y = L_z = 0 is simultaneously definite. This is consistent with the commutation relations: if [L̂_x, L̂_y] = iℏL̂_z and L_z = 0, the uncertainty relation allows both L_x and L_y to be zero simultaneously. The l = 0 (s-orbital) state is spherically symmetric for exactly this reason.

Question 4

The orbital quantum number l can take any non-negative real value, including fractions, as long as |m_l| ≤ l.

• T. True
• F. False

Answer: False

Both l and m_l must be non-negative integers (l = 0, 1, 2, ...) and integers (m_l = -l, ..., 0, ..., +l) respectively. The quantization to integers arises from the requirement that the wavefunction be single-valued: the azimuthal dependence e^{im_lφ} must return to the same value after a full rotation φ → φ + 2π, which forces m_l to be an integer. The normalizability of the polar part then forces l to be a non-negative integer with |m_l| ≤ l.

Question 5

Explain the key difference between knowing L_z and knowing the full angular momentum vector L⃗ for a quantum particle, and why this difference has no classical counterpart.

Model answer: Knowing L_z gives you the projection of angular momentum along the z-axis (ℏm_l) and the total magnitude (ℏ√(l(l+1))), but the x and y components are genuinely indeterminate — not merely unknown, but without definite values. The angular momentum vector cannot be fully specified in any direction simultaneously because the components don't commute. Classically, L⃗ has definite components in all three directions at once; quantum mechanically, this is forbidden by the algebra of the operators.

This is a genuine departure from classical intuition, not just a measurement limitation. It's not that L_x and L_y are hidden from us — it's that the particle doesn't have simultaneous definite values for them. The angular momentum vector 'points' in an indeterminate direction within a cone around the z-axis, which is why the vector model of atomic orbitals shows L⃗ precessing around the z-axis rather than sitting still.