Revise Wave-particle duality for Cambridge 9702 (22.3)

Light, which we often think of as a wave, also shows distinct particle characteristics. These tiny 'packets' of energy are called photons or quanta. Each photon carries a specific amount of energy that depends on its colour, or more accurately, its frequency. This particle behaviour is brilliantly demonstrated by the photoelectric effect.

$E = hf$ (where $E$ is photon energy, $h$ is Planck's constant (6.63 × 10⁻³⁴ Js), and $f$ is light frequency)

A key piece of evidence from the photoelectric effect is the role of light intensity. According to the wave model, increasing the intensity (brightness) of light should increase the energy of the waves, leading to emitted electrons with higher kinetic energy. However, experiments show that increasing intensity only increases the number of electrons emitted per second (the photoelectric current), while their maximum kinetic energy remains unchanged. This is perfectly explained by the particle model: higher intensity means more photons arriving per second, each with the same energy $E=hf$. More photons can eject more electrons, but the energy of each individual electron depends only on the energy of the single photon that ejected it.

$hf = Φ + KE_max$ (The photoelectric equation, relating photon energy to work function and maximum kinetic energy of emitted electrons)

While light can act as a particle, matter – like electrons, protons, and even atoms – can behave as waves. This concept, known as the de Broglie hypothesis, suggests that every particle has an associated wavelength. This means particles can exhibit wave-like phenomena such as diffraction and interference. The evidence for this is striking: when a beam of electrons is fired at a thin crystal lattice (like graphite), it produces a diffraction pattern of concentric rings, just as X-rays (which are waves) do. This wave behaviour is only significant for microscopic particles, as for macroscopic objects, the momentum $p$ is so large that the de Broglie wavelength $λ = h/p$ becomes immeasurably small.

$λ = h/p$ (where λ is the de Broglie wavelength, $h$ is Planck's constant, and $p$ is the particle's momentum (mv))

The wave nature of electrons confined within an atom has profound implications. An electron's de Broglie wave must 'fit' into its orbit around the nucleus, forming a standing wave. Only specific wavelengths, and therefore specific energies, can form stable standing waves. This is why electrons can only occupy discrete energy levels – a phenomenon called quantisation. Think of it like a guitar string; it can only vibrate at specific harmonic frequencies. An electron can be on one energy 'step' or another, but not in between.

$hf = E_initial - E_final$ (Energy of an emitted photon when an electron de-excites)