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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