Quantum

Quanta

Light can be modelled as photons - discrete quanta of light that carry fixed quantities of energy. The energy of a photon with frequency $f$ is given by: $E=hf=\frac{hc}{\lambda }$ Where $h$ is the Planck constant ($6.63\times 10^{-34}\text{J s}$).

One electronvolt ($\mathrm{eV}$) is equal to the energy transferred when one election is moved through a potential difference of one volt. $1\mathrm{eV}=1.6\times 10^{-19}\mathrm{J}$.

Evidence for photons exchanging energy in quanta includes the photoelectric effect, light-emitting diodes, and line spectra.

Photoelectric effect

The photoelectric effect occurs when light is incident on the surface of a metal. It is a one-to-one interaction between photons and electrons on the metal, and electrons can be emitted as photoelectrons.

Intensity

The intensity of light does not cause higher energy electrons (energy doesn’t accumulate), instead intensity is proportional to the number of electrons emitted.

Frequency

The energy of the emitted electrons depends on the frequency of the light. The frequency must be above the threshold frequency for electrons to be emitted. The corresponding energy, the work function, is the amount of energy needed to liberate electrons from the surface of the metal.

Energy of photoelectrons

$E=hf-\varphi$ Where $h$ is Planck’s constant, $f$ is frequency, and $\varphi$ is the work function.

Light-emitting diodes

In a light-emitting diode (LED), there is a one-to-one interaction between electrons and photons. A single photon is emitted when a electron loses energy (‘falling across a band gap’). As shown by the I-V graph of a diode, it only begins conducting above a certain voltage. This voltage corresponds to the energy per electron being sufficient to produce a photon with that amount of energy. Similar to the photoelectric effect, it is the voltage, and not current, which matters, because light is formed of quanta.

For example, red light has a wavelength of $\sim 700$ nm. By $E=hf=\frac{hc}{\lambda }$, this corresponds to an energy of around $2.84\times 10^{-19}\text{J}=1.77\text{eV}$. Red LEDs do indeed require around 1.8V to turn on. This principle can be used in a practical to find Planck’s constant:

• For a range of LEDs emitting known wavelengths, find the voltage required for the LED to just turn on.
• Apply $E=\frac{hc}{\lambda }$, plot the points, and find $h$.

Line spectra

As described in Probing and Ionising, atoms emit or absorb electromagnetic radiation of specific wavelengths, corresponding to electrons falling or gaining energy levels.

Quantum behaviour

Quanta have a certain probability of arrival at a detector. This probability is found by considering all possible paths, where every path from an emitter to detector can be represented by a phasor. For most paths, the phasors curl up and cancel out, producing a small or zero resultant. For places with a high probability of arrival, the phasors line up, producing a large resultant.

The probability of arrival of quanta (or for light, intensity) is proportional to the square of the length of the resultant phasor.

Electron diffraction

Electrons have been observed to diffract, where a thin layer of atoms is used as a diffraction grating. The diffraction pattern shows concentric circles, suggesting wave-like behaviour. The de Broglie equation relates the momentum to the wavelength:

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

This equation shows that matter can exhibit wave-like properties.