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# Quantum physics summary

## Quantum physics summary

Quantum: smallest possible amount of a physical quantity.

Photon: quantum of electromagnetic radiation $E=hf=\frac{hc}{\lambda}$

Photoelectric effect: 1 photon hitting the metal=1 photoelectron emitted

Threshold frequency $(f_{0})$ minimal frequency of a photon to trigger the photoelectric effect (depends on the metal).

Work function : minimal energy of the photon $\phi=hf_{0}$

Einstein's photoelectric equation $$K_\text{max}=\frac{1}{2}mv_{\text{max}}^{2}=hf-\phi$$

Photoelectric current between an emitter and a collector: $I=\frac{ne}{t}$

Intensity $=\frac{P}{A}=\frac{N(hf)/t}{A}$

$n\propto N \quad \Rightarrow \quad I\propto \text{Intensity}$

Stopping potential $(V_{\text{s}})$: potential difference required to stop a photoelectron with the maximum possible kinetic energy exactly at the collector's surface. $$V_{\text{s}}=\left (\frac{h}{e}\right )f-\frac{\phi}{e}$$

Wave-particle duality (observable at quantum scales) all matter is both a wave and a particle.

de Broglie relation:

$$\lambda=\frac{h}{p}$$

Electrons are in atomic levels around the atom.

Photons emitted by the de-excitation of an atom (electronic change to a lower level): emission spectrum (coloured parallel spectral lines on a dark background)

Photons absorbed excite an atom (electronic change to a higher level): absorption spectrum (dark parallel lines on a coloured background)

Difference between the energy levels : $hf=E_{2}-E_{1}$

X-ray spectrum:

• intensity spikes : excited metal atoms return to their ground state (discrete energies).
• continuous spectrum : electrons decelerating on a metal surface (bremsstrahlung).

Minimum wavelength: $$h\left(\frac{c}{\lambda_{\text{min}}}\right)=\frac{1}{2}mv^{2}$$

Heisenberg uncertainty principle :

$$\Delta x\Delta p\geq\frac{\hbar}{2} \qquad \Delta E\Delta t\geq\frac{\hbar}{2}$$

Wave function $\psi$ (Schrödinger model): carries all the wave-particle information.

$\vert\psi\vert^{2}=$ probability density: probability of finding a particle at particular a point.

Potential barrier : region in space where the potential energy of a particle is higher than its surroundings.

Barrier height : minimum energy to pass through the barrier.

Quantum tunnelling : quantum phenomenon of passing through a potential barrier without sufficient energy.

Transmission coefficient $(T)$ probability of tunnelling $$T=e^{-2kd} \qquad k=\sqrt{\frac{8\pi^{2}(U-E)}{h^{2}}}$$

Reflection coefficient $(R)$ : $R+T=1$

Scanning tunnelling microscope (STM) : device that gives atomic-scale images of surface by using quantum tunnelling.

Symbol Meaning
$h$ Planck constant
$c$ Speed of light
$f$ Frequency
$v$ Velocity
$\lambda$ Wavelength
$m$ Mass of an electron
$K_{\text{max}}$ Maximum kinetic energy of a photoelectron
$N$ Number of incident photons
$n$ Number of emitted photoelectrons
$p$ Momentum
$e$ Charge of an electron
$\hbar=h/2\pi$ Reduced Planck constant
$d$ Barrier width
$h$ Barrier height
$E$ Energy