Questions: de Broglie Wavelength

Question 1

An electron is moving at 1×10⁶ m/s. A proton is also moving at 1×10⁶ m/s. Which particle has the longer de Broglie wavelength?

• A. The proton — it is more massive and therefore has more wave-like character
• B. The electron — it has less mass and therefore less momentum, giving a longer wavelength
• C. They are equal — same speed means same wavelength
• D. The proton — faster particles in quantum mechanics have longer wavelengths

Answer: (B)

λ = h/p = h/(mv). At the same speed, the more massive proton has far greater momentum (m_proton ≈ 1836 × m_electron), so its wavelength is about 1836 times shorter. The electron's tiny mass means tiny momentum, which gives a large wavelength. This is why electron diffraction is experimentally observable but proton diffraction requires much more effort.

Question 2

A baseball and an electron are traveling at the same speed. Which has a shorter de Broglie wavelength and why does this explain why we don't see the baseball diffract?

• A. The electron — because macroscopic objects don't have de Broglie wavelengths
• B. The baseball — because it is more massive, giving it far greater momentum and therefore a far shorter wavelength
• C. The baseball — because classical objects travel faster, reducing their wavelength
• D. They are the same — de Broglie wavelength depends only on speed, not mass

Answer: (B)

λ = h/(mv). The baseball's mass (~0.15 kg) is ~10²⁸ times greater than an electron's, so its momentum is ~10²⁸ times larger and its wavelength is ~10²⁸ times shorter — on the order of 10⁻³⁴ m, far smaller than any atomic nucleus. No physical obstacle or slit can produce diffraction at that scale. The de Broglie relation applies to ALL matter; quantum effects simply vanish at macroscopic scales because h is so small.

Question 3

A slower particle usually has a shorter de Broglie wavelength than a faster particle of the same mass.

• T. True
• F. False

Answer: False

This is the most common misconception about the de Broglie relation. λ = h/p = h/(mv): wavelength is *inversely* proportional to momentum. A slower particle has less momentum, so it has a *longer* wavelength, not shorter. The intuition fails because in everyday wave experience (e.g., sound), speed is tied to wavelength differently. For matter waves, slower means more wavelike (longer λ), which is why ultra-cold atoms — slowed to near absolute zero — exhibit dramatic quantum wave effects like Bose-Einstein condensation.

Question 4

The de Broglie matter wave is a physical oscillation of a medium, similar to a water wave or sound wave.

• T. True
• F. False

Answer: False

The matter wave is a probability wave — there is no medium oscillating. The squared magnitude of the wave function at any location gives the probability of finding the particle there. This is a fundamentally non-classical concept: there is nothing physically waving, yet the wave predicts interference patterns (observed in double-slit and Davisson-Germer experiments). Confusing the matter wave with a physical oscillation leads to paradoxes; the correct interpretation is purely probabilistic.

Question 5

Explain how de Broglie's hypothesis gives a physical reason for Bohr's quantization rule, rather than treating it as an arbitrary postulate.

Model answer: Bohr postulated that angular momentum is quantized in units of ħ without explaining why. de Broglie's hypothesis says an electron with momentum p has wavelength λ = h/p. For a stable circular orbit of radius r, the electron's matter wave must form a standing wave — an integer number of wavelengths must fit the circumference: 2πr = nλ = nh/p. This gives mvr = nħ, exactly Bohr's condition. Quantization is no longer arbitrary — it is a resonance condition. Only orbits where the matter wave closes on itself constructively are stable.

This derivation reveals a deep principle: quantization throughout quantum mechanics arises from wave boundary conditions. Just as a guitar string can only sustain standing waves at specific frequencies (harmonics), an electron orbit can only sustain a standing matter wave for specific radii. The 'arbitrary' Bohr postulate turns out to be the statement that non-resonant orbits self-destructively interfere and disappear — exactly analogous to destructive interference in any wave system.