Questions: Canonical Commutation Relations and Uncertainty

Question 1

An experimenter argues: 'The Heisenberg uncertainty principle just means our measuring devices disturb particles. A sufficiently gentle measurement could in principle determine both position and momentum precisely.' What is fundamentally wrong with this claim?

• A. Nothing — the uncertainty principle is indeed about measurement disturbance
• B. The principle only applies to subatomic particles, not macroscopic measuring devices
• C. The principle reflects that no quantum state can simultaneously have definite position and definite momentum — it is not about measurement clumsiness
• D. The principle only applies when the particle is not in an energy eigenstate

Answer: (C)

The Heisenberg uncertainty principle ΔxΔp ≥ ℏ/2 is a statement about the spread of outcomes in repeated measurements on identically prepared systems — it is a property of quantum *states*, not of measurement technology. A state that is a sharp position eigenstate (Dirac delta in position space) must be a superposition of all momentum eigenstates with equal amplitude — it literally has no definite momentum to be 'disturbed.' No matter how gentle the measurement, you cannot extract precise momentum information that isn't there. The 'disturbance' picture is a popular but incorrect gloss; the principle was rigorously established by Kennard (1927) as a statement about state preparation.

Question 2

Which mathematical property of the relation [x̂, p̂] = iℏ directly implies that position and momentum cannot be simultaneously measured with arbitrary precision?

• A. That iℏ is imaginary, which means their eigenvalues cannot both be real
• B. That the commutator is nonzero, meaning x̂ and p̂ cannot be simultaneously diagonalized in the same basis
• C. That ℏ is very small, so the uncertainty is negligible for macroscopic objects
• D. That p̂ contains a derivative operator, making it unbounded and unphysical

Answer: (B)

Two Hermitian operators share a complete set of simultaneous eigenstates — and can therefore be simultaneously measured with precision — if and only if their commutator is zero. Since [x̂,p̂] = iℏ ≠ 0, they cannot be simultaneously diagonalized. Any position eigenstate spreads across infinitely many momentum eigenstates, and vice versa. The smallness of ℏ (option C) determines the *size* of the minimum uncertainty product but does not affect whether simultaneous precision is possible in principle. Option A is wrong because Hermitian operators always have real eigenvalues regardless of whether i appears in their commutator.

Question 3

The canonical commutation relation [x̂, p̂] = iℏ can be verified in the position representation by showing that applying x̂ then p̂ differs from applying p̂ then x̂ by a term equal to iℏ times the wavefunction — an extra term that emerges from the product rule of differentiation.

• T. True
• F. False

Answer: True

With x̂ψ = xψ and p̂ψ = (ℏ/i)∂ψ/∂x, computing [x̂,p̂]ψ = x̂(p̂ψ) − p̂(x̂ψ) = x·(ℏ/i)∂ψ/∂x − (ℏ/i)∂(xψ)/∂x. Applying the product rule to the second term: ∂(xψ)/∂x = ψ + x∂ψ/∂x. Substituting: x·(ℏ/i)∂ψ/∂x − (ℏ/i)(ψ + x∂ψ/∂x) = −(ℏ/i)ψ = iℏψ. The extra ψ term is precisely the product rule contribution, and it equals iℏψ for any ψ, confirming [x̂,p̂] = iℏ as an operator identity.

Question 4

The Heisenberg uncertainty principle is violated by classical objects like a baseball, which simultaneously have a definite position and a definite momentum.

• T. True
• F. False

Answer: False

Classical objects do not violate the uncertainty principle — they trivially satisfy it. For a 0.1 kg baseball, even a position uncertainty of 10⁻³⁰ m (far below any measurable scale, far smaller than an atomic nucleus) gives a momentum uncertainty of only ~10⁻⁵ kg·m/s via ΔxΔp ≥ ℏ/2 ≈ 5.3×10⁻³⁵ J·s. The principle holds for all physical objects; it just has no detectable consequence at macroscopic scales because ℏ is so tiny. Quantum mechanics does not break down for classical objects — it reduces to classical behavior in the appropriate limit.

Question 5

Explain why the canonical commutation relation [x̂, p̂] = iℏ is considered the defining postulate that distinguishes quantum mechanics from classical mechanics, and what would follow if the commutator were zero instead.

Model answer: In classical mechanics, position and momentum are ordinary numbers (not operators), and their product always commutes: xp = px. The canonical commutation relation [x̂,p̂] = iℏ ≠ 0 is the precise mathematical statement that position and momentum are incompatible observables in quantum theory — their operators do not share eigenstates, so no quantum state can have simultaneously definite values for both. If [x̂,p̂] = 0, the uncertainty principle ΔxΔp ≥ ℏ/2 would collapse to ΔxΔp ≥ 0 — trivially satisfied — and position and momentum could be simultaneously sharp. Wavefunctions would be unnecessary; particles could have definite trajectories; and the theory would reduce to classical mechanics. The nonzero commutator is what forces the probabilistic structure of quantum mechanics and the wave-particle duality it entails.