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de Broglie

de Broglie wavelength

An electron moving at 5.4 × 10⁶ m/s has momentum 4.92 × 10⁻²⁴ kg m/s, so its de Broglie wavelength is λ = h / p = 1.35 × 10⁻¹⁰ m = 0.135 nm, about the size of an atom.

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 λ0.135 nm1.35 × 10⁻¹⁰ m
Momentum4.92 × 10⁻²⁴kg m/s
Kinetic energy83 eV1.33 × 10⁻¹⁷ J
Speed5.4 × 10⁶ m/sElectron

Electron: de Broglie wavelength 0.135 nm.

Show the working
  1. Given: electron, m = 9.11 × 10⁻³¹ kg, v = 5.4 × 10⁶ m/s.
  2. p = mv = 9.11 × 10⁻³¹ × 5.4 × 10⁶ = 4.92 × 10⁻²⁴ kg m/s
  3. λ = h / p = 6.63 × 10⁻³⁴ / 4.92 × 10⁻²⁴ = 1.35 × 10⁻¹⁰ m = 0.135 nm
  4. Its kinetic energy is ½mv² = 1.33 × 10⁻¹⁷ J = 83 eV.
Working

Step by step

  1. Given: electron, m = 9.11 × 10⁻³¹ kg, v = 5.4 × 10⁶ m/s.
  2. p = mv = 9.11 × 10⁻³¹ × 5.4 × 10⁶ = 4.92 × 10⁻²⁴ kg m/s
  3. λ = h / p = 6.63 × 10⁻³⁴ / 4.92 × 10⁻²⁴ = 1.35 × 10⁻¹⁰ m = 0.135 nm
  4. Its kinetic energy is ½mv² = 1.33 × 10⁻¹⁷ J = 83 eV.
NCERT example

An electron and a cricket ball

ParticleMassSpeedλ
Electron9.11 × 10⁻³¹ kg5.4 × 10⁶ m/s0.135 nm
Ball0.15 kg30 m/s1.47 × 10⁻³⁴ m
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: quick answers

What is the de Broglie wavelength?

The wavelength of the wave that goes with a moving particle: λ = h / p, where p is its momentum. de Broglie suggested it in 1924 and electron diffraction confirmed it.

Why do we not see the waves of everyday objects?

Their momentum is huge, so λ is tiny. A 150 g ball at 30 m/s has λ = 1.47 × 10⁻³⁴ m, far smaller than anything that could diffract it.

How does λ change with speed?

λ = h / mv, so doubling the speed halves the wavelength. A heavier particle at the same speed has a shorter wavelength.

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