A dust particle (mass ~10⁻¹⁵ kg, speed ~0.01 m/s) and an electron (mass ~9×10⁻³¹ kg, speed ~10⁶ m/s) are both described by de Broglie wavelengths. Which would exhibit observable wave behavior such as diffraction?
AThe dust particle, because its slow speed gives it more time to interact with nearby surfaces
BThe electron, because its de Broglie wavelength (~0.7 nm) is comparable to atomic spacings, enabling diffraction from crystal lattices
CBoth equally, because the formula λ = h/p applies to all matter without distinction
DNeither, because wave behavior is exclusive to photons and massless particles
The de Broglie relation applies to all matter, but observable wave behavior (diffraction, interference) only occurs when λ is comparable to the scale of the physical interaction. The electron's wavelength (~0.7 nm at 10⁶ m/s) is close to atomic spacings (~0.1–0.5 nm), so electrons diffract from crystal lattices — exactly what Davisson and Germer demonstrated. The dust particle's wavelength would be enormous compared to atomic spacings in the other direction... actually, let's compute: λ = h/mv = 6.6×10⁻³⁴/(10⁻¹⁵ × 0.01) = 6.6×10⁻¹⁷ m — far smaller than an atomic nucleus. Its wave character is utterly undetectable. The formula applies universally; detectability depends on relative scale.
Question 2 Multiple Choice
A particle is prepared so that its momentum is known exactly (Δp = 0). The de Broglie relation assigns it a precise wavelength λ = h/p. What does this imply about the particle's position?
AIts position is also precisely defined, since both momentum and position can be determined from the wavefunction
BIts position is completely indefinite — the wavefunction is a plane wave spread uniformly throughout all space
CIts position is uncertain by one wavelength on either side of its classical trajectory
DThe particle has no position because it is purely a wave with no particle-like localization
A particle with definite momentum p has a wavefunction ψ ∝ e^{ipx/ℏ} — a plane wave extending throughout all space with equal amplitude everywhere. The probability density |ψ|² is uniform: the particle is equally likely to be anywhere. This is the Heisenberg uncertainty principle: Δx·Δp ≥ ℏ/2. With Δp = 0, we get Δx → ∞. A precise de Broglie wavelength (definite momentum) is mathematically incompatible with any localization. Option D is a misconception: the particle still has particle-like properties (it is detected at a single point), but its position before measurement is maximally uncertain.
Question 3 True / False
A baseball has a de Broglie wavelength that is, in principle, nonzero, but its wave behavior is completely unobservable in practice.
TTrue
FFalse
Answer: True
The de Broglie relation λ = h/p applies to all matter, including baseballs. For a 0.15 kg baseball at 40 m/s, λ = h/mv ≈ 6.6×10⁻³⁴/(0.15×40) ≈ 10⁻³⁴ m — roughly 20 orders of magnitude smaller than an atomic nucleus (10⁻¹⁵ m). No instrument or physical phenomenon could resolve structure at this scale. The wavelength is technically nonzero but physically meaningless. This is why classical mechanics describes macroscopic objects exactly: quantum effects scale with λ, and at 10⁻³⁴ m those effects are indistinguishable from zero.
Question 4 True / False
A de Broglie matter wave is a classical mechanical wave — a physical disturbance propagating through a medium, similar to sound or water waves.
TTrue
FFalse
Answer: False
A de Broglie wave is the quantum mechanical wavefunction ψ(x, t), not a classical wave in any medium. Its squared amplitude |ψ|² gives the probability density for finding the particle at a given position — there is no physical disturbance propagating through space. Unlike sound (pressure oscillations) or water waves (surface displacement), the wavefunction is not a directly observable field; it is a mathematical object encoding probabilistic information. The confusion between 'matter waves' and classical waves leads to mistaken pictures of particles oscillating in space.
Question 5 Short Answer
A thrown baseball is described by the de Broglie relation λ = h/p, yet it shows no observable wave behavior. Explain why, using the formula to support your reasoning.
Think about your answer, then reveal below.
Model answer: For a 0.15 kg baseball thrown at 40 m/s, λ = h/mv ≈ (6.6×10⁻³⁴)/(0.15×40) ≈ 10⁻³⁴ m. This is roughly 20 orders of magnitude smaller than an atomic nucleus. Wave behavior (diffraction, interference) is only observable when the wavelength is comparable to the scale of the physical system. No material exists with structure at 10⁻³⁴ m, so the baseball cannot diffract or interfere with anything. The larger the momentum, the shorter the wavelength, and the less observable the wave character — which is why quantum effects vanish at macroscopic scales.
This question directly addresses the most natural misconception: if λ = h/p applies to everything, why doesn't everything exhibit wave behavior? The answer is that observability requires λ to be comparable to the relevant physical scale. For electrons, λ ~ atomic spacings, enabling crystal diffraction. For a baseball, λ is fantastically smaller than anything real, so quantum behavior is undetectable. The boundary between quantum and classical behavior is not a sharp line but a practical limit set by the ratio of de Broglie wavelength to the scale of the interaction.