Photon: massless particle of light. Energy $E = h\nu = hc/\lambda$. Momentum $p = E/c = h/\lambda$. No rest mass. Always travels at $c$. Cannot be at rest. de Broglie wavelength for photon $= h/p = hc/E = \lambda$ (same as wavelength from wave picture — consistent). Photon energy: visible light ($\lambda=500$ nm): $E = 2.5$ eV. X-ray ($\lambda=0.1$ nm): $E = 12.4$ keV. Gamma ray ($E = 1$ MeV): $\lambda = 1.24\times10^{-3}$ nm.

$\lambda = h/p = h/(mv)$ for non-relativistic particles. For particle with kinetic energy $KE$: $p = \sqrt{2m\cdot KE}$, so $\lambda = h/\sqrt{2m\cdot KE}$. For electron accelerated through $V$ volts: $KE = eV$, $\lambda = h/\sqrt{2meV}$. Numerically: $\lambda(\text{Å}) = \sqrt{150/V}$ for electrons. For proton (mass $1836m_e$): $\lambda$ is $\sqrt{1836}$ times smaller than electron at same $KE$. Thermal de Broglie wavelength: $\lambda_{th} = h/\sqrt{2\pi m k_B T}$.

$KE_{max} = h\nu - \phi = h\nu - h\nu_0$. Stopping potential $V_0$: $eV_0 = KE_{max}$. Threshold frequency $\nu_0 = \phi/h$. Key results: $KE_{max}$ depends on frequency, NOT intensity. Intensity determines number of photoelectrons, not their energy. No time delay in emission. Einstein (1905): light consists of photons. Photoelectric effect proved quantum nature of light. Millikan confirmed Einstein equation (1916) — Nobel Prize.

X-ray photon scatters off electron: wavelength increases. $\Delta\lambda = \lambda_C(1-\cos\phi)$ where $\lambda_C = h/(m_e c) = 0.00243$ nm (Compton wavelength), $\phi$ = scattering angle. Proved photons have momentum $p = h/\lambda$. Maximum shift at $\phi = 180°$: $\Delta\lambda = 2\lambda_C = 0.00486$ nm. Compton (1923) — Nobel Prize 1927. Combined with photoelectric effect: definitive proof of wave-particle duality of light.

Light: wave (interference, diffraction, polarization) AND particle (photoelectric, Compton). Electrons: particle (mass, charge) AND wave (diffraction, interference). Principle: nature of observed phenomenon depends on type of experiment. Double-slit with electrons: interference pattern builds up electron by electron (each is both wave and particle). Quantum eraser: restoring wave behavior by erasing which-path information. Bohr complementarity: wave and particle aspects cannot be simultaneously observed.

$\Delta x \cdot \Delta p_x \geq \hbar/2$. $\Delta E \cdot \Delta t \geq \hbar/2$. Fundamental limits — not instrument limitations. Origin: wave nature of particles (localizing a wave requires superposition of many wavelengths → momentum uncertainty). Consequences: electrons cannot exist in nucleus (confinement gives huge $\Delta p$ → kinetic energy too large). Zero-point energy: ground state has $E > 0$ (cannot have $E = 0$ as that would require $\Delta p = 0$ → infinite $\Delta x$).

Davisson-Germer (1927): 54 eV electrons diffracted by Ni crystal. Peak at $\phi = 50°$ consistent with $\lambda = h/p = 1.67$ Å. First experimental proof of electron wave nature. G.P. Thomson (1927): diffraction rings from thin Al foil — electrons behave like X-rays with similar wavelengths. Both won Nobel Prize 1937. Applications: electron microscope ($\lambda \sim 0.004$ nm at 100 kV → 100,000× magnification), low-energy electron diffraction (LEED) for surface structure analysis, transmission electron microscopy (TEM) for atomic-resolution imaging.

de Broglie (1924): particles have associated wavelength $\lambda = h/p$. Schrodinger (1926): wave equation for matter waves. $-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = i\hbar\frac{\partial\psi}{\partial t}$. $|\psi|^2$ = probability density. For stationary states: $-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi$ (time-independent). For hydrogen: solutions give quantized energies $E_n = -13.6/n^2$ eV — same as Bohr, but with correct wavefunctions and selection rules. Solved the problems of Bohr model (multi-electron atoms, intensities, selection rules, spin).