• Tags: de-broglie, matter-waves, wavelength
• Difficulty: Moderate

Louis de Broglie (1924) proposed that all matter has wave-like properties. The de Broglie wavelength of a particle with momentum p: λ = $\frac{h}{p}$ = $\frac{h}{mv}$.

For a particle with kinetic energy KE: p = $\sqrt{2mKE}$, so λ = h/$\sqrt{2mKE}$.

For a charged particle accelerated through potential V: KE = qV, so λ = h/$\sqrt{2mqV}$.

Special cases:

• Electron: λ = 12.27/√V Å
• Proton: λ = 0.286/√V Å
• Alpha particle: λ = 0.101/√V Å

Key comparison: at the same KE, the lighter particle has a longer wavelength (λ ∝ 1/√m). At the same momentum, all particles have the same wavelength (λ = $\frac{h}{p}$). At the same speed, the lighter particle again has a longer wavelength (λ = $\frac{h}{mv}$ ∝ 1/m).