## 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, $\hbar=h/2\pi$.

## Relativity

Invariance of physical laws, simultaneity:  All of the fundamental laws of physics have the same form in all inertial frames of reference.  The speed of light in vacuum is the same in all inertial frames and is independent of the motion of the source.  Simultaneity is not an absolute concept; events that are simultaneous in one frame are not necessarily simultaneous in a second frame moving relative to the first.

Time dilation:  If two events occur at the same space point in a particular frame of reference, the time interval $\Delta t_0$ between the events as measured in that frame is called a proper time interval.  If this frame moves with constant velocity $u$ relative to a second frame, the time interval $\Delta t$ between the events as observed in the second frame is longer than $\Delta t_0$: $\Delta t=\frac{\Delta t_0}{\sqrt{1-\frac{u^2}{c^2}}}=\gamma\Delta t_0$, $\gamma=\frac1{\sqrt{1-u^2/c^2}}$.

Length contraction:  If two points are at rest in a particular frame of reference, the distance $l_0$ between the points as measured in that frame is called a proper length. If this frame moves with constant velocity $u$ relative to a second frame and the distances are measured parallel to the motion, the distance $l$ between the points as measured in the second frame is shorter than $l_0$.  $l=l_0\sqrt{1-\frac{u^2}{c^2}}=\frac{l_0}{\gamma}$.

The Lorentz transformation:  The Lorentz coordinate transformations relate the coordinates and time of an event in an inertial frame $S$ to the coordinates and time of the same event as observed in a second inertial frame $S'$ moving at velocity $u$ relative to the first.  For one-dimensional motion, a particle’s velocities $v_x$ in $S$ and $v_x'$ in $S'$ are related by the Lorentz velocity transformation.  $x'=\frac{x-ut}{\sqrt{1-u^2/c^2}}=\gamma(x-ut)$, $y'=y$, $z'=z$, $t'=\frac{t-ux/c^2}{\sqrt{1-u^2/c^2}}=\gamma(t-ux/c^2)$, $v_x'=\frac{v_x-u}{1-uv_x/c^2}$, $v_x=\frac{v_x'+u}{1+uv_x'/c^2}$.

The Doppler effect for electromagnetic waves:  The Doppler effect is the frequency shift in light from a source due to the relative motion of source and observer.  For a source moving toward the observer with speed $u$, the received frequency $f$ in terms of the emitted frequency $f_0$ is $f=\sqrt{\frac{c+u}{c-u}}f_0$.

Relativistic momentum and energy:  For a particle of rest mass $m$ moving with velocity $\vec{v}$, the relativistic momentum is $\vec{p}=\frac{m\vec{v}}{\sqrt{1-v^2/c^2}}=\gamma m\vec{v}$, the relativistic kinetic energy is $K=\frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2=(\gamma-1)mc^2$.  The total energy $E$ is the sum of the kinetic energy and the rest energy $mc^2$: $E=K+mc^2=\frac{mc^2}{\sqrt{1-v^2/c^2}}=\gamma mc^2$.  Also, $E^2=(mc^2)^2+(pc)^2$.