Quantum physics

22.1 Energy and momentum of a photon

Electromagnetic radiation has a particulate nature. A photon is a quantum of electromagnetic energy: a discrete packet of it.

$E = hf = \frac{hc}{\lambda}$

where $h = 6.63 \times 10^{-34}$ J s is the Planck constant.

The electronvolt

An electronvolt (eV) is the energy gained by an electron accelerated through a p.d. of 1 V:

$1\text{ eV} = 1.60 \times 10^{-19}\text{ J}$

Photon momentum

A photon has no rest mass, but it does have momentum:

$p = \frac{E}{c}$

Combining this with $E = hc/\lambda$ gives $p = h/\lambda$.

22.2 Photoelectric effect

Electrons may be emitted from a metal surface when it is illuminated with electromagnetic radiation. These are photoelectrons.

Observations and terms

• Threshold frequency $f_0$: the minimum frequency needed to emit electrons from the surface. Below it, none are emitted, however intense the radiation.
• Threshold wavelength $\lambda_0$: the maximum wavelength that will emit electrons.
• Work function $\Phi$: the minimum energy needed to remove an electron from the surface. $\Phi = hf_0$.
• Emission is instantaneous, even in faint light.

Einstein's equation

One photon gives all of its energy to one electron. If it has enough, the electron escapes. Any remaining energy is its kinetic energy:

$hf = \Phi + \tfrac{1}{2}mv_{\text{max}}^2$

Electrons deeper in the metal lose energy on the way out, so this is the maximum kinetic energy.

A graph of $E_{K,\text{max}}$ against $f$ has gradient $h$, x-intercept $f_0$, and an extrapolated y-intercept of $-\Phi$. All metals have the same gradient.

Why intensity does not change the maximum kinetic energy

22.3 Wave-particle duality

• The photoelectric effect shows the particulate nature of electromagnetic radiation.
• Interference and diffraction show its wave nature.

Electron diffraction

Electrons fired at a thin slice of graphite produce rings on a fluorescent screen, which is a diffraction pattern. Diffraction is a wave property, so particles can behave as waves. Faster electrons have a smaller wavelength, so the rings get closer together.

The de Broglie wavelength

Any moving particle has an associated wavelength:

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

For an electron accelerated through a p.d. $V$: $\tfrac{1}{2}mv^2 = eV$, so $p = \sqrt{2meV}$.

22.4 Energy levels in atoms and line spectra

Electrons in isolated atoms can only have certain discrete energies. These levels are negative, with zero meaning the electron has just escaped.

When an electron drops between levels, it emits one photon:

$hf = E_1 - E_2$

• Emission line spectrum: bright lines on a dark background, from a hot, low-pressure gas. Excited electrons fall to lower levels and emit only particular photon energies.
• Absorption line spectrum: dark lines on a continuous spectrum, when white light passes through a cool gas. Atoms absorb only the photons that exactly match a gap between levels, then re-emit them in all directions.

Since levels are discrete, the photon energies are discrete, which gives lines rather than a continuous spectrum.