Quantum Physics: Complete A Level H2 Physics Cheatsheet
A Level Physics (H2), Chapter 19 · 9478 Topic 19 · Read time: ~13 minutes
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Key Formulas
Light as particles: the photoelectric effect
A photon is a quantum of electromagnetic radiation with energy E = hf. Though massless, it carries momentum p = E/c = h/λ.
Light above a threshold frequency ejects electrons from a metal; below it, no electrons are emitted however intense the light. This threshold is evidence for the particulate nature of light. Interference and diffraction are evidence for its wave nature.
One photon gives all its energy to one electron: hf = Φ + KE(max).
Trap: Brighter light does not give electrons more energy. Intensity changes how many electrons per second; only frequency changes the maximum kinetic energy.
Particles as waves
Electron diffraction and double-slit interference built up one particle at a time show that particles have wave properties.
The de Broglie wavelength is λ = h/p. Heavier or faster particles have shorter wavelengths.
Wavefunctions
A particle's state is described by a wavefunction ψ, for example the electron cloud in an atom. |ψ|² is the probability density.
Normalisation: the particle must be somewhere, so the area under |ψ|² equals 1. For a square wavefunction of height A over width L, A = 1/√L. For ψ = A sin(nπx/L), A = √(2/L).
Wavefunctions obey superposition, which gives standing wave solutions in a box and single-particle interference in a double slit.
Heisenberg uncertainty principle: Δx Δp ≳ h. A localised particle must have a spread of momenta.
Trap: ψ itself is not the probability and can be negative. Only |ψ|² is a probability density.
Particle in a box and atomic energy levels
In a 1-D infinite square well of width L, ψ is zero at the walls, so only standing waves with λ = 2L/n fit. This gives Eₙ = n²h²/8mL²: quantised energies with a non-zero ground state.
Electrons in isolated atoms likewise have discrete energy levels. A transition between two levels emits or absorbs a photon with hf = ΔE, giving spectral lines.
Emission spectrum: bright lines on a dark background, from excited atoms. Absorption spectrum: dark lines on a continuous spectrum, at the same frequencies.
Trick: Eₙ ∝ n², so E₂ = 4E₁ and E₃ = 9E₁. Find E₁ once and scale.
Worked Example
A metal has work function 3.20 × 10⁻¹⁹ J. Separately, an electron is confined in a 1-D infinite well of width 0.50 nm. h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, mₑ = 9.11 × 10⁻³¹ kg.
(a) Light of wavelength 450 nm falls on the metal. Calculate the maximum kinetic energy of the emitted electrons. [3]
E = hc/λ = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) ÷ 450 × 10⁻⁹ = 4.42 × 10⁻¹⁹ J
KE(max) = 4.42 × 10⁻¹⁹ − 3.20 × 10⁻¹⁹ = 1.22 × 10⁻¹⁹ J
(b) The light intensity is doubled. State the effect on KE(max). [1]
No change. Each photon still has the same energy hf; only the emission rate increases.
(c) Calculate the ground state energy of the electron in the well. [2]
E₁ = h² / 8mL² = (6.63 × 10⁻³⁴)² ÷ (8 × 9.11 × 10⁻³¹ × (0.50 × 10⁻⁹)²)
= 2.4 × 10⁻¹⁹ J
(d) Calculate the wavelength of the photon emitted when it drops from n = 2 to n = 1. [2]
ΔE = E₂ − E₁ = 3E₁ = 7.24 × 10⁻¹⁹ J
λ = hc/ΔE = 2.7 × 10⁻⁷ m (ultraviolet)
Why this question is a good test of the topic: it covers both halves of 9478 Quantum Physics, light as particles and electrons as waves, and the n² scaling saves a full second calculation.
Frequently Asked Questions
Does increasing light intensity increase photoelectron kinetic energy?
No. Intensity only changes the rate of emission. The maximum kinetic energy depends on frequency.
Does a photon have momentum?
Yes. Although massless, a photon has momentum p = E/c = h/λ.
What does |ψ|² represent?
The probability density: the probability per unit length of finding the particle at that position.
Why is the ground state energy in a box not zero?
Only standing waves fitting the box are allowed, and the lowest has n = 1. This agrees with the uncertainty principle: a confined particle can't be completely at rest.
What causes spectral lines?
Electrons moving between discrete energy levels emit or absorb photons with hf equal to the energy difference, so only specific frequencies appear.
Finding the new quantum topics abstract?
Small group A Level H2 Physics classes at TGC Academy's Bishan, Bukit Timah and Potong Pasir centres, helmed by Andrew Seah, MOE Award-Winning Teacher and Marshall Cavendish textbook author, with a team of tutors.