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Quantum mechanics

Advanced physics enrichment path through wavefunctions, uncertainty, the Schrödinger equation, square wells, tunnelling, and the quantum oscillator.

  • Advanced Physics
  • 12 lessons
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Before you begin

In quantum mechanics a particle no longer has a definite path. Its wavefunction predicts only the probabilities of measurement results. The lessons start with the uncertainty principle and the Schrödinger equation, then solve it for a particle trapped in a box, a finite well and a harmonic oscillator, and end with the correspondence principle, which links the quantum results back to classical physics. This topic goes beyond the H3 Physics syllabus.

Be comfortable with: quantum physics, the quantum theory of light and matter waves, and with differentiation and basic integration.

Learning goals
  • Use state functions, boundary conditions, stationary states, and measurement to analyse introductory quantum systems.

Lessons

Work through them in order.

Foundations and uncertainty

  1. Introduction to quantum mechanicsExplain why classical physics fails at atomic scales and read |ψ|² as a probability density.
  2. The uncertainty principleInterpret Δ x Δ p ≥ ħ/2 as a property of the state and use it for quick estimates.

Wave functions and probability

  1. The wavefunction and the Schrödinger equationExplain what the wavefunction represents and how the Schrödinger equation restricts allowed states.
  2. Normalisation of the wavefunctionNormalise a wavefunction so the total probability is one and find its normalisation constant.

Stationary states and boundaries

  1. The time-independent Schrödinger equationSolve the time-independent Schrödinger equation in constant-potential regions and apply its boundary conditions.

Infinite well

  1. A particle trapped in a box of length LSet up the infinite square well and use its boundary conditions to find standing-wave states.
  2. Allowed energy states of a trapped particleDerive the energy levels of a particle in a box and explain why its ground-state energy is not zero.
  3. Wavefunctions of a particle trapped in a boxDerive and normalise the infinite-well wavefunctions and locate the nodes of |ψₙ|².
  4. The particle in a box revisitedConnect the particle in a box's wavefunctions, energies, momenta and probabilities, and use them together.

Extensions and classical limit

  1. A particle in a well of finite heightMatch solutions across a finite square well and explain penetration into the walls and tunnelling.
  2. The quantum harmonic oscillatorUse the equally spaced energy levels of the quantum harmonic oscillator and explain its zero-point energy.
  3. The correspondence principleUse large-quantum-number and large-scale limits to check that quantum results approach classical physics.

Practise and check

Recommended nextQuantum mechanics topic check

Topic reference

Quick reference

IdeaRelationNotes
uncertainty principleΔ x Δ p ≥ ħ/2a limit on the spreads, not on measuring skill
time-independent Schrödinger equation-(ħ²/2m)d²ψ/dx² + Vψ = Eψone dimension
probability density|ψ|², with ∫|ψ|² dx = 1normalisation
particle in a boxEₙ = n²h²/8mL², n = 1,2,3,…infinitely high walls
inside a finite wall, E < Vψ ∝ e^(-κ x), κ = (square root of (2m(V-E)))/ħdecays, but is not zero
harmonic oscillatorEₙ = (n + 1/2)ħω, n = 0,1,2,…equally spaced levels

Core knowledge to remember

  • The wavefunction ψ is an amplitude, which can be complex; |ψ|² gives the probability density.
  • A particle confined to a region has quantised energy levels, set by the conditions the wavefunction must meet at the boundaries.
  • The lowest energy of a confined particle is never zero.
  • In a finite well, the wavefunction extends into the walls, where it decays exponentially.
  • At large quantum numbers, quantum predictions approach the classical ones.

Common mistakes

  • Treating ψ itself as a probability.
  • Giving a standing state in a box one momentum. A measurement gives + pₙ or -pₙ.
  • Starting the particle-in-a-box levels at n = 0. The box starts at n = 1; the oscillator starts at n = 0 with E₀ = (1/2)ħω.
  • Calling every exponential tail tunnelling. Tunnelling also needs an allowed region beyond the barrier.