de Broglie standing wave: $2\pi r_n = n\lambda$
$$\lambda = \frac{2\pi r_n}{n}$$$n=2$: $r_2 = (2)^2 a_0 = 4a_0$
$\approx 6.64$ Å $= 0.664$ nm
Louis de Broglie (1924): every moving particle has associated wavelength $\lambda = h/p = h/(mv)$. Inversely proportional to momentum. For macroscopic objects (large $m$): $\lambda$ negligibly small → no observable wave effects. For electrons, protons: $\lambda$ comparable to atomic distances → wave effects observable. Confirmed by Davisson and Germer (1927): electron beam diffracted by nickel crystal → electrons have wave nature.
Bohr quantization $mvr = n\hbar$ is equivalent to standing wave condition $2\pi r = n\lambda$. Substituting $\lambda = h/(mv)$: $2\pi r = n \cdot h/(mv) \Rightarrow mvr = nh/(2\pi) = n\hbar$. The allowed orbits are those where an integer number of electron wavelengths fit exactly around the orbit (standing wave condition). This gives the wave-mechanical interpretation of Bohr's quantum condition.
Einstein (1905): light comes in packets (photons) of energy $E = h\nu$. Photoelectric equation: $KE_{max} = h\nu - \phi$ where $\phi = h\nu_0$ = work function. Threshold frequency $\nu_0$: minimum needed for emission. Stopping potential $V_0$: $eV_0 = KE_{max}$. Key: $KE_{max}$ depends on frequency (not intensity). More intense light → more electrons (not faster ones). Classical wave theory failed; quantum theory succeeded.
Davisson-Germer experiment (1927): electron beam (54 eV) fired at nickel crystal. Sharp diffraction peaks at expected angles (consistent with $\lambda = h/p$). First direct proof of electron wave nature. G.P. Thomson (separately, 1927): electrons diffracted through thin gold foil → diffraction rings. Both Davisson and Thomson won Nobel Prize (1937). Modern electron microscopes exploit this: $\lambda \sim 0.004$ nm for 100 keV electrons → 1000× better resolution than light.
$$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$ Fundamental limit on simultaneous measurement of position and momentum. $\Delta E \cdot \Delta t \geq \hbar/2$ (energy-time uncertainty). Origin: wave nature of particles. To measure position precisely: need short wavelength photon → high energy → disturbs momentum. Not due to instrument limitations — fundamental to nature. Consequences: electrons cannot exist inside nucleus (confinement gives $\Delta p$ → $KE$ too large). Explains ground state energy of hydrogen (zero-point energy).
Light shows: wave nature (interference, diffraction, polarization) and particle nature (photoelectric effect, Compton scattering). Electrons show: particle nature (definite mass, charge, deflection) and wave nature (diffraction). The nature that appears depends on the experiment. Complementarity principle (Bohr): wave and particle aspects cannot be observed simultaneously. Which-way information destroys interference pattern. Quantum eraser experiments: restoring interference by erasing which-way information.
When X-rays scatter off electrons: wavelength increases ($\Delta\lambda = \lambda_C(1-\cos\phi)$ where $\lambda_C = h/(m_e c) = 0.00243$ nm = Compton wavelength). Treated as photon-electron collision (billiard ball collision with relativistic energy-momentum conservation). Proved photons have momentum $p = h/\lambda = E/c$. Classic wave theory predicted no wavelength change → Compton scattering was decisive proof for photon particle nature of light.
Energy levels $E_n = -13.6/n^2$ eV. Spectral series: Lyman ($n_f=1$, UV): $n_i = 2,3,4...$. Balmer ($n_f=2$, visible): $H_\alpha = 656$ nm (red), $H_\beta = 486$ nm (blue-green), $H_\gamma = 434$ nm (violet). Paschen ($n_f=3$, IR). Brackett ($n_f=4$). Pfund ($n_f=5$). Rydberg formula: $1/\lambda = R_H(1/n_f^2 - 1/n_i^2)$. For $n=\infty \to n=1$: ionization energy = 13.6 eV = 2.18×10⁻¹⁸ J.