• summary_type: concept
• word_count: 160

The Bohr quantization condition L = nhbar is equivalent to the de Broglie standing wave condition 2pir = nlambda. Proof: substituting lambda = h/(mv) gives mvr = nh/(2pi) = nhbar. This physical insight shows stable orbits are those where the electron wave constructively interferes with itself around the orbit. The de Broglie wavelength in the nth orbit: lambda_n = 2pir_n/n = 2pina_0/Z. For reduced mass correction, replace electron mass m with mu = mM/(m+M). For hydrogen: mu approximately equals 0.99946m (tiny correction). For positronium (electron-positron): mu = m/2, so energies are halved and radii doubled. For muonic hydrogen (muon replaces electron, mass = 207m_e): mu approximately equals 186m_e, radii are approximately 200 times smaller, energies approximately 200 times larger. JEE occasionally tests exotic atom calculations using reduced mass.