# Photons: Light Waves behaving as Particles

Photons:  Electromagnetic radiation behaves as both waves and particles.  The energy in an electromagnetic wave is carried in units called photons.  The energy $E$ of one photon is proportional to the wave frequency $f$ and inversely proportional to the wavelength $\lambda$, and is proportional to a universal quantity $h$ called Planck’s constant: $E=hf=\frac{hc}{\lambda}$.  The momentum of a photon has magnitude $E/c$: $p=\frac Ec=\frac{hf}c=\frac h{\lambda}$.

The photo-electric effect:  In the photo-electric effect, a surface can eject an electron by absorbing a photon whose energy $hf$ is greater than or equal to the work function $\phi$ of the material.  The stopping potential $V_0$ is the voltage required to stop a current of ejected electrons from reaching an anode: $eV_0=hf-\phi$.

Photon production, photon scattering, and pair production:  X rays can be produced when electrons accelerated to high kinetic energy across a potential increase $V_{AC}$ strike a target.  The photon model explains why the maximum frequency and minimum wavelength produced are given by the equation: $eV_{AC}=hf_{\max}=\frac{hc}{\lambda_{\min}}$ (bremsstrahlung).  In Compton scattering a photon transfers some of its energy and momentum to an electron with which it collides.  For free electrons (mass $m$), the wavelengths of incident and scattered photons are related to the photon scattering angle $\phi$: $\lambda'-\lambda=\frac{h}{mc}(1-\cos\phi)$ (Compton scattering).  In pair production a photon of sufficient energy can disappear and be replaced by electron-positron pair.  In the inverse process, an electron and positron can annihilate and be replaced by a pair of photons.

The Heisenberg uncertainty principle:  It is impossible to determine both a photon’s position and its momentum at the same time to arbitrarily high precision.  The precision of such measurements for the $x$-components is limited by the Heisenberg uncertainty principle, $\Delta x\Delta p_x\geq\hbar/2$; there are corresponding relationships for the $y$– and $z$-components.  The uncertainty $\Delta E$ in the energy of a state that is occupied for a time $\Delta t$ is given by equation $\Delta t\Delta E\geq\hbar/2$.  In these expressions, $\hbar=h/2\pi$.