Quantum physics
Study H2 Physics photons, matter waves, wavefunctions, uncertainty and the infinite square well, then explain atomic line spectra.
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Before you begin
Light behaves as particles and particles behave as waves. The lessons start with the photoelectric effect, the evidence for photons, and the energy and momentum a photon carries, then turn to electron diffraction and the de Broglie wavelength. The wave description of a particle leads to the wavefunction and probability, the uncertainty principle, and the quantised energies of a particle in a box. They end with the energy levels of atoms and the line spectra they produce.
Be comfortable with: waves and superposition, because diffraction, interference and standing waves are used throughout.
Learning goals
- Use photon energy and momentum and analyse the photoelectric effect.
- Apply de Broglie wavelength and wave-particle evidence.
- Interpret wavefunctions, probability density and superposition.
- Apply uncertainty and infinite-square-well energy quantisation.
- Analyse atomic energy levels and emission or absorption spectra.
Syllabus statements covered
- show an understanding that the existence of a threshold frequency in the photoelectric effect provides evidence that supports the particulate nature of electromagnetic radiation while phenomena such as interference and diffraction provide evidence that supports its wave nature
- state that a photon is a quantum of electromagnetic radiation, and recall and use the equation E = hf for the energy of a photon to solve problems, where h is the Planck constant
- show an understanding that while a photon is massless, it has a momentum given by p = E/c and p = h/λ, where c is the speed of light in free space
- show an understanding that electron diffraction and double-slit interference of single particles provide evidence that supports the wave nature of particles
- recall and use the equation λ = h/p for the de Broglie wavelength to solve problems
- show an understanding that the state of a particle can be represented as a wavefunction ψ, e.g. for an electron cloud in an atom, and that the square of the wavefunction amplitude |ψ|² is the probability density function (including calculation of normalisation factors for square and sinusoidal wavefunctions)
- show an understanding that the principle of superposition applies to the wavefunctions describing a particle’s position, leading to standing wave solutions for a particle in a box and phenomena such as single-particle interference in double-slit experiments
- show an understanding that the Heisenberg position-momentum uncertainty principle ΔxΔp ≳ h relates to the necessity of a spread of momenta for localised particles, and apply this to solve problems
- show an understanding of standing wave solutions ψₙ for the wavefunction of a particle in a one-dimensional infinite square well potential
- solve problems using Eₙ = h²n²/(8mL²) for the allowed energy levels of a particle of mass m in a one-dimensional infinite square well of width L
- show an understanding of the existence of discrete electronic energy levels for the electron’s wavefunction in isolated atoms (e.g. atomic hydrogen) and deduce how this leads to the observation of spectral lines
- distinguish between emission and absorption line spectra
- solve problems involving photon absorption or emission during atomic energy level transitions
Lessons
Work through them in order.
Core lessons
- Photoelectric evidence for photonsExplain how the threshold frequency in the photoelectric effect supports the particle nature of light.
- Photon energy and momentumUse E = hf and p = h/λ for photons, and separate the energy of one photon from the number arriving per second.
- Matter-wave evidence and de Broglie wavelengthApply the de Broglie relation and explain electron diffraction as evidence for matter waves.
- Wavefunctions, probability density and superpositionInterpret |ψ|² as probability density, normalise simple wavefunctions and explain interference by superposition.
- Localisation and momentum uncertaintyExplain why a localised particle needs a spread of momenta, and use Δ x Δ p ≳ h to estimate it.
- Standing states in an infinite square wellUse boundary nodes to find allowed standing wavefunctions, discrete energies and the effects of well width and particle mass.
- Atomic energy levels and line spectraLink atomic energy levels to emission and absorption line spectra using hf = Δ E.
Beyond the syllabus
- Einstein’s equation and stopping potentialApply Kₘₐₓ = hf - Φ and stopping potential, and read work function and h from graphs.Beyond the syllabus
- Bohr’s historical model and atomic transitionsUse the Bohr model to picture discrete energy levels and photon transitions, and state its limits.Beyond the syllabus
- Wavefunctions and the Schrödinger equationInterpret the wavefunction and normalise simple states, with the Schrödinger equation as context.Beyond the syllabus
- Reflection and transmission of matter wavesDescribe how a matter wave is partly reflected and partly transmitted at a potential step or barrier.Beyond the syllabus
- Quantum tunnellingExplain how a particle can pass through a barrier higher than its energy, and what controls the probability.Beyond the syllabus
- Scanning tunnelling microscopeExplain how a scanning tunnelling microscope uses tunnelling current to map a surface at atomic scale.Beyond the syllabus
- How a Coolidge X-ray tube worksFollow thermionic emission, electron acceleration and target interactions in a Coolidge X-ray tube, and distinguish voltage from tube current (optional H2 extension).Beyond the syllabus
- X-ray photon energy limitsUse electron energy gain eV to calculate maximum X-ray photon energy and frequency, minimum wavelength and the voltage needed for a chosen cut-off (optional H2 extension).Beyond the syllabus
- Continuous and characteristic X-ray spectraExplain how bremsstrahlung and characteristic lines form an X-ray spectrum and how the accelerating voltage sets the cut-off wavelength, as optional enrichment.Beyond the syllabus
Practise and check
Or choose
Topic reference
Quick reference
| Idea | Relation | Notes |
|---|---|---|
| photon energy | E = hf = hc/λ | h = 6.63 × 10⁻³⁴ J s |
| photon momentum | p = E/c = h/λ | a photon has no mass |
| photoelectric threshold | hf₀ = Φ | Φ is the work function |
| de Broglie wavelength | λ = h/p | p = mv for a particle |
| probability density | |ψ|² | probability is the area under it |
| uncertainty principle | Δ x Δ p≳ h | an order-of-magnitude estimate |
| infinite square well | Eₙ = n²h²/8mL² | n = 1,2,3,… |
| atomic transition | hf = E_high-E_low | one photon per transition |
1 eV = 1.60 × 10⁻¹⁹ J
Core knowledge to remember
- A photon is a quantum of electromagnetic radiation.
- The threshold frequency in the photoelectric effect is evidence for the particulate nature of electromagnetic radiation; interference and diffraction are evidence for its wave nature.
- Electron diffraction and the double-slit interference of single particles are evidence for the wave nature of particles.
- The state of a particle is described by a wavefunction ψ, and |ψ|² is the probability density.
- A particle confined to a small region must have a spread of momenta.
- Electrons in an isolated atom have discrete energy levels, so atoms emit and absorb photons of particular frequencies only, giving line spectra.
Common mistakes
- Saying brighter light gives photons more energy; it gives more photons.
- Using E = pc for an electron; it applies only to photons.
- Treating ψ itself, or the height of a density graph, as a probability.
- Including n = 0 in the infinite square well, or giving the ground state zero energy.
- Thinking an atom absorbs any photon with enough energy to reach the next level; excitation needs an exact match.