A photon and an electron have the same kinetic energy $E = 20.2$ eV. The ratio of the momentum of the electron to that of the photon is: (given $m_e c^2 = 0.511$ MeV)

Historical development: Newton (light as particles) → Huygens (wave theory) → Young (interference proved wave nature) → Maxwell (EM waves) → Planck/Einstein (photons, particle nature) → de Broglie (matter waves). Bohr complementarity principle: wave and particle aspects are complementary; cannot observe both simultaneously in same experiment.

Einstein\'s photon model (1905, Nobel 1921): $E = hf = hc/\lambda$. Photoelectric: $hf = \phi + KE_{max}$; stopping potential $V_0 = (hf-\phi)/e$; threshold $f_0 = \phi/h$. Applications: photomultiplier tubes, photodiodes, solar cells, CCD cameras. Photovoltaic effect (solar cells): similar to photoelectric but internal — electron-hole pairs created in semiconductor.

de Broglie wavelength $\lambda = h/p = h/mv$. Large macroscopic objects have extremely small $\lambda$ (unobservable). Electron microscope: uses electron waves ($\lambda \sim 0.001$ nm at 100 keV) for much higher resolution than light microscopes ($\lambda \sim 400-700$ nm). Electron diffraction: electrons diffracting through crystal lattice confirms wave nature. Quantum confinement: when object size approaches $\lambda$, quantum effects dominate (quantum dots, nanoscale devices).

Bohr model works only for one-electron systems (H, He+, Li2+). Fails for: multi-electron atoms, cannot explain intensity of spectral lines, cannot explain fine structure, cannot explain chemical bonding. Schrodinger equation (1926): $H\psi = E\psi$ (wave equation for matter). Wavefunction $\psi$: $|\psi|^2$ gives probability density of finding particle. Quantum numbers: $n$ (principal), $l$ (angular momentum), $m_l$ (magnetic), $m_s$ (spin). Pauli exclusion principle: no two electrons can have same set of all 4 quantum numbers.

Every moving particle has an associated wavelength $\lambda = h/p = h/mv$. For an electron accelerated through a potential V, this becomes the convenient form $\lambda = \frac{12.27}{\sqrt{V}}$ Å. The wavelength is inversely proportional to mass, which is why wave behaviour is observable for electrons but not for a cricket ball — a 150 g ball at 30 m/s has a wavelength around $10^{-34}$ m, far too small to detect.

Electrons accelerated through 54 V were scattered off a nickel crystal and showed a diffraction maximum at 50°, exactly where the Bragg condition predicted for a wavelength of 1.65 Å. The de Broglie formula gives 1.67 Å for that voltage. The agreement was the first direct experimental proof that matter has wave properties, and it is the standard example asked for when a question requests evidence of the dual nature of matter.

A photon has no rest mass but does carry momentum, $p = h/\lambda = E/c$. When light is absorbed by a surface the momentum transferred produces a pressure $I/c$; when it is perfectly reflected the momentum change doubles and so does the pressure, giving $2I/c$. That single factor of 2 is the most frequently dropped mark in this topic, and it is also the principle behind solar sails.