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de Broglie

de Broglie wavelength of a proton

A proton accelerated through 100 V has λ = h / √(2meV) = 2.87 pm, about 43 times shorter than an electron's 0.123 nm at the same voltage.

de Broglie wavelength

Every moving particle has a wave with λ = h / p. Give the speed, the kinetic energy, or the voltage a charged particle was accelerated through. Speeds are taken well below the speed of light.

Particle
Given
Wavelength λ2.87 pm2.87 × 10⁻¹² m
Momentum2.31 × 10⁻²²kg m/s
Kinetic energy100 eV1.6 × 10⁻¹⁷ J
Speed1.38 × 10⁵ m/sProton

Proton: de Broglie wavelength 2.87 pm.

Show the working
  1. Given: proton, m = 1.67 × 10⁻²⁷ kg, accelerated through V = 100 V.
  2. K = eV = 100 eV = 100 × 1.6 × 10⁻¹⁹ = 1.6 × 10⁻¹⁷ J
  3. p = √(2mK) = √(2 × 1.67 × 10⁻²⁷ × 1.6 × 10⁻¹⁷) = 2.31 × 10⁻²² kg m/s
  4. λ = h / p = 6.63 × 10⁻³⁴ / 2.31 × 10⁻²² = 2.87 × 10⁻¹² m = 2.87 pm
Working

Step by step

  1. Given: proton, m = 1.67 × 10⁻²⁷ kg, accelerated through V = 100 V.
  2. K = eV = 100 eV = 100 × 1.6 × 10⁻¹⁹ = 1.6 × 10⁻¹⁷ J
  3. p = √(2mK) = √(2 × 1.67 × 10⁻²⁷ × 1.6 × 10⁻¹⁷) = 2.31 × 10⁻²² kg m/s
  4. λ = h / p = 6.63 × 10⁻³⁴ / 2.31 × 10⁻²² = 2.87 × 10⁻¹² m = 2.87 pm
Same voltage

A proton and an electron through 100 V

ParticleMassSpeedλ
Proton1.67 × 10⁻²⁷ kg1.38 × 10⁵ m/s2.87 pm
Electron9.11 × 10⁻³¹ kg5.93 × 10⁶ m/s0.123 nm
Derivation

Why it works

  1. de Broglie proposed that a moving particle has a wave, with λ = h / p, the same relation that holds for a photon.
  2. For a slow particle p = mv, so λ = h / mv.
  3. Kinetic energy K = p² / 2m gives p = √(2mK), so λ = h / √(2mK).
  4. A charge q accelerated through V gains K = qV, so λ = h / √(2mqV).
  5. For an electron, putting in h, m and e gives λ = 1.227 / √V nm.
Questions

de Broglie wavelength of a proton: quick answers

What is the de Broglie wavelength of a proton?

λ = h / √(2m qV). Through 100 V a proton gets 100 eV, so λ = 2.87 pm.

Why is a proton's wavelength shorter than an electron's?

At the same energy λ goes as 1 / √m, and a proton is about 1836 times heavier, so its wavelength is about √1836 ≈ 43 times shorter.

Can the 1.227 / √V shortcut be used for a proton?

No. The 1.227 comes from the electron's mass. For a proton use λ = h / √(2meV) with the proton mass, or 0.0286 / √V nm.

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