de Broglie proposed that every particle has a wavelength λ = h/p = h/(mv). For an electron accelerated through potential V: λ = h/√(2meV); in Å with V in volts, λ ≈ 12.27/√V. Phase velocity of the matter wave is v_phase = c²/v (always > c); group velocity v_group = v (particle speed).
Matter wave calculators
λ from p or mv, accelerated electron, relativistic, phase & group velocity
〰️ Matter wave summary
| Concept | Formula | Notes |
|---|---|---|
| de Broglie wavelength | λ = h / p = h / (m v) | p = momentum |
| Accelerated electron | λ = h / √(2 m e V) | V = accelerating voltage |
| λ (Å) for electron (V in volts) | λ (Å) ≈ 12.27 / √V | Useful numerical relation |
| Relativistic particle | λ = h / √(2 m₀ c² (γ − 1)) | γ = 1/√(1 − v²/c²) |
| Phase velocity | v_phase = c² / v | Always > c (de Broglie waves) |
| Group velocity | v_group = v | Equals particle velocity |
About matter waves (de Broglie hypothesis)
de Broglie proposed that every particle with momentum p has an associated wavelength λ = h/p = h/(mv), showing wave-particle duality. For an electron accelerated from rest through potential V, kinetic energy K = eV, so λ = h/√(2meV). With V in volts, λ (Å) ≈ 12.27/√V.
Phase and group velocity
For de Broglie waves, the phase velocity is v_phase = c²/v (always greater than c); the group velocity is v_group = v, the particle speed. Only the group velocity carries energy and information.
Relativistic case
For relativistic particles, λ = h/√(2 m₀ c² (γ − 1)), where γ = 1/√(1 − v²/c²). This reduces to the non-relativistic form when v ≪ c.