de Broglie Wavelength

de Broglie Wavelength

The de Broglie hypothesis associates every moving particle of momentum p with a matter-wave wavelength λ = h/p, where h = 6.62607015 × 10⁻³⁴ J·s is Planck's constant. In the nonrelativistic regime (v/c ≪ 1), a particle of mass m and kinetic energy K has momentum p = √(2mK) and speed v = √(2K/m).

Worked Example: 1 eV Electron

Consider an electron (m = 9.11 × 10⁻³¹ kg) with kinetic energy K = 1 eV:

1. Convert kinetic energy into joules: K = 1 × 1.602176634 × 10⁻¹⁹ J = 1.602176634 × 10⁻¹⁹ J.
2. Calculate momentum: p = √(2mK) = √(2 × 9.11 × 10⁻³¹ × 1.602176634 × 10⁻¹⁹) ≈ 0.540293052625 × 10⁻²⁴ kg·m/s.
3. Calculate speed and relativistic ratio: v = p/m ≈ 0.593076896405 × 10⁶ m/s, giving v/c ≈ 0.00197829158332.
4. Compute the de Broglie wavelength: λ = h / p ≈ 1.226384481127 nm.

Experiment

In the de Broglie Wavelength module, compare how kinetic energy and particle mass determine momentum, speed, and matter-wave wavelength:

1. Select Particle as "Electron" and Kinetic energy eV as 1, then increase energy to 4 and 9. Because p ∝ √K, quadrupling energy doubles momentum and halves the wavelength to 0.613192240564 nm.
2. Switch Particle to "Neutron" (m = 1.675 × 10⁻²⁷ kg) at 1 eV. The heavier mass increases momentum to 23.167416178547 × 10⁻²⁴ kg·m/s and shortens the wavelength to 0.028600816332 nm.
3. Set Kinetic energy eV to 0. At rest, momentum and speed are zero and the wavelength is unbounded (reported as "Unavailable").

Assumptions and Limitations

This module uses the nonrelativistic free-particle relation p = √(2mK) with fixed educational masses (9.11 × 10⁻³¹ kg for the electron and 1.675 × 10⁻²⁷ kg for the neutron). Across the supported energy grid (K ≤ 9 eV), v/c < 0.01 so relativistic corrections are small; at high energies comparable to rest energy mc², the relativistic momentum p = √(K² + 2Kmc²)/c must be used instead.

References

The historical background, wave-particle duality experiments, and nonrelativistic de Broglie wavelength relations are covered in:

• OpenStax University Physics Volume 3, Section 6.5: De Broglie's Matter Waves