Quantum rules for light and matter
Quantum physics is required when energy and momentum are exchanged on microscopic scales. Light arrives in quanta called photons. A photon of frequency f and wavelength λ has E = hf = hc/λ and p = E/c = h/λ. Higher frequency means more energy per photon; greater intensity at fixed frequency usually means more photons arriving per unit time, not more energy in each photon.
E = hf, p = h/λ
λ = h/p for a particle
ΔxΔpₓ ≥ ħ/2
The photoelectric effect
An electron bound in a material needs at least the work function φ to escape. Energy conservation gives Kmax = hf − φ. The threshold frequency is f₀ = φ/h. A stopping potential Vs satisfies eVs = Kmax. Above threshold, increasing frequency increases maximum electron kinetic energy. Increasing intensity at fixed frequency increases the emission rate but does not change Kmax. Below threshold, increasing ordinary light intensity does not eject electrons in the basic one-photon model.
Matter waves and uncertainty
de Broglie proposed that a particle with momentum p has wavelength λ = h/p. Electron and atom diffraction confirm this wave behaviour. The wavefunction does not trace a material ripple; its squared magnitude gives a probability density for possible measurement outcomes. A localized particle requires a wave packet containing many wavelengths and momenta. Consequently position and momentum spreads satisfy ΔxΔpₓ ≥ ħ/2. This is an intrinsic property of quantum states, not simply poor instruments or careless observation.
Complementarity: the same quantum object can produce particle-like localized detections and wave-like interference; the experimental arrangement determines which aspect is resolved.