A student argues: 'The z-component of angular momentum is mℏ, so the total angular momentum magnitude must also be mℏ (just taking m = l for the maximum case).' What is wrong with this reasoning?
ANothing — the total magnitude is indeed lℏ when m = l
BThe eigenvalue of L̂² is ℏ²l(l+1), not ℏ²l², so the total magnitude is ℏ√(l(l+1)), which is always greater than lℏ
CThe z-component and total magnitude cannot be simultaneously definite, so the premise is invalid
DAngular momentum in quantum mechanics has no definite total magnitude — only the z-component has eigenvalues
The eigenvalue equation is L̂²|l,m⟩ = ℏ²l(l+1)|l,m⟩, so the total magnitude is ℏ√(l(l+1)), not lℏ. For l = 1: magnitude = ℏ√2 ≈ 1.41ℏ, but the maximum z-component is only 1·ℏ. The discrepancy ℏ²l(l+1) vs. ℏ²l² is not a technicality — it reflects the fact that you cannot align the angular momentum vector exactly with the z-axis. The x and y components have zero expectation value but nonzero uncertainty, contributing to the total. Option C is wrong because L̂² and L̂_z commute, so they can be simultaneously definite. Option D is wrong because L̂² does have eigenvalues.
Question 2 Multiple Choice
Why is it impossible to simultaneously know the x and y components of angular momentum for a quantum system with l > 0?
ABecause angular momentum is conserved and so its components cannot change — but measuring one fixes it forever
BBecause [L̂ₓ, L̂ᵧ] = iℏL̂_z ≠ 0, so measuring one component necessarily disturbs the other
CBecause quantum mechanics allows knowing at most two quantum numbers simultaneously
DBecause L̂ₓ and L̂ᵧ are not Hermitian and therefore have no real eigenvalues
The Heisenberg uncertainty principle applies whenever two operators do not commute: [L̂ₓ, L̂ᵧ] = iℏL̂_z. For a state with l > 0, L̂_z has nonzero eigenvalues, so the right-hand side is not zero, and the commutator is non-trivial. This means measuring L̂ₓ collapses the state in a way that introduces uncertainty in L̂ᵧ, and vice versa. Options A and C are confused. Option D is wrong — L̂ₓ and L̂ᵧ are both Hermitian, which is required for physical observables.
Question 3 True / False
The quantization of the angular momentum quantum number l — that it should be a non-negative integer or half-integer — is imposed as a physical postulate rather than derived from the algebra of the operators.
TTrue
FFalse
Answer: False
The quantization of l emerges from the algebra itself, via the ladder operators L̂₊ and L̂₋. These raise and lower the m quantum number by 1. Since m is bounded above and below (the z-component cannot exceed the total magnitude), the ladder must terminate: L̂₊|l,l⟩ = 0 and L̂₋|l,−l⟩ = 0. Working through the algebra of these termination conditions forces l to be a non-negative integer or half-integer and restricts m to the 2l+1 values from −l to +l. No additional postulate is needed.
Question 4 True / False
The eigenvalue of L̂² for the state |l, m⟩ is ℏ²l², which reduces to ℏ² for l = 1.
TTrue
FFalse
Answer: False
The correct eigenvalue is ℏ²l(l+1), not ℏ²l². For l = 1 this gives ℏ²·1·2 = 2ℏ², not ℏ². The l(l+1) form is a direct output of the ladder operator algebra and reflects the fact that you cannot simultaneously know all three components — the total magnitude is always larger than any single component's maximum value. The difference between l² and l(l+1) is not cosmetic: it implies the angular momentum vector can never be perfectly aligned with any axis.
Question 5 Short Answer
Why can you simultaneously measure L̂² and L̂_z but not L̂_z and L̂ₓ? Explain using commutation relations.
Think about your answer, then reveal below.
Model answer: Two observables can be simultaneously measured (have simultaneous eigenstates) if and only if they commute. L̂² and L̂_z commute: [L̂², L̂_z] = 0. This can be verified using the commutation relations [L̂ᵢ, L̂ⱼ] = iℏεᵢⱼₖL̂ₖ and the fact that L̂² = L̂ₓ² + L̂ᵧ² + L̂_z² is rotationally invariant. In contrast, [L̂_z, L̂ₓ] = iℏL̂ᵧ ≠ 0, so measuring L̂_z and L̂ₓ simultaneously is forbidden by the uncertainty principle. The physical picture is that L̂² measures the total length of the angular momentum vector (which is rotation-invariant), while L̂_z measures its projection onto one axis — these are compatible. But knowing two different components would completely specify the vector's direction, which would violate the uncertainty principle for the remaining component.
The strategy of identifying which operators commute is the general method for finding complete sets of commuting observables (CSCOs) in quantum mechanics. {L̂², L̂_z} forms a CSCO for angular momentum; their shared eigenstates |l, m⟩ are the basis for atomic orbital labels and for the hydrogen atom energy levels.