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de Broglie

de Broglie wavelength of an electron

An electron accelerated through 100 V has λ = h / √(2meV) = 0.123 nm. The shortcut λ = 1.227 / √V nm gives 0.123 nm.

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 λ0.123 nm1.23 × 10⁻¹⁰ m
Momentum5.4 × 10⁻²⁴kg m/s
Kinetic energy100 eV1.6 × 10⁻¹⁷ J
Speed5.93 × 10⁶ m/sElectron

Electron: de Broglie wavelength 0.123 nm.

Show the working
  1. Given: electron, m = 9.11 × 10⁻³¹ 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 × 9.11 × 10⁻³¹ × 1.6 × 10⁻¹⁷) = 5.4 × 10⁻²⁴ kg m/s
  4. λ = h / p = 6.63 × 10⁻³⁴ / 5.4 × 10⁻²⁴ = 1.23 × 10⁻¹⁰ m = 0.123 nm
  5. Shortcut for an electron: λ = 1.227 / √V nm = 1.227 / √100 = 0.123 nm
Working

Step by step

  1. Given: electron, m = 9.11 × 10⁻³¹ 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 × 9.11 × 10⁻³¹ × 1.6 × 10⁻¹⁷) = 5.4 × 10⁻²⁴ kg m/s
  4. λ = h / p = 6.63 × 10⁻³⁴ / 5.4 × 10⁻²⁴ = 1.23 × 10⁻¹⁰ m = 0.123 nm
  5. Shortcut for an electron: λ = 1.227 / √V nm = 1.227 / √100 = 0.123 nm
Other voltages

λ = 1.227 / √V nm

VoltageKSpeedλ
10 V10 eV1.87 × 10⁶ m/s0.388 nm
50 V50 eV4.19 × 10⁶ m/s0.174 nm
100 V100 eV5.93 × 10⁶ m/s0.123 nm
150 V150 eV7.26 × 10⁶ m/s0.1 nm
500 V500 eV1.33 × 10⁷ m/s0.0549 nm
1,000 V1,000 eV1.87 × 10⁷ m/s0.0388 nm
10,000 V10,000 eV5.93 × 10⁷ m/s0.0123 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 an electron: quick answers

What is the formula for the de Broglie wavelength of an electron?

λ = h / √(2meV) for an electron accelerated through V volts. Putting in h, m and e gives λ = 1.227 / √V nm.

What is λ for an electron through 100 V?

λ = 1.227 / √100 nm = 0.123 nm. Working from h / √(2meV) with NCERT constants gives 0.123 nm.

Why are electron microscopes so sharp?

Electrons accelerated through thousands of volts have wavelengths far shorter than light, so they resolve much finer detail than an optical microscope.

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