• Tags: de-broglie, bohr, standing-wave
• Difficulty: Moderate

The Bohr quantization condition L = nℏ is equivalent to the de Broglie standing wave condition 2πr = nλ. Proof: 2πr = nλ = $\frac{nh}{mv}$, so mvr = $\frac{nh}{2π}$ = nℏ. This provides physical insight: stable orbits are those where the electron wave forms a standing wave (constructive interference around the orbit). Non-integer number of wavelengths would lead to destructive interference, eliminating the wave and hence the orbit. The de Broglie wavelength of the electron in the nth orbit: λ_n = 2π$r_n$/n = 2π$na_{0}$/Z (for hydrogen-like atoms). For n=1 in hydrogen: λ = 2π × 0.529 Å = 3.33 Å.