Quantum physics: evidence, states and spectra
Key idea: Keep evidence tied to the claim it supports, add probability amplitudes before probabilities, and derive quantised energies from boundary conditions rather than treating them as arbitrary rules.
Before you start: Waves and Superposition objective chainEnergy and Fields objective chain
By the end, you can
- Connect particle and wave evidence to photons, energy, momentum and matter waves.
- Interpret and normalise wavefunctions, use probability density and apply superposition.
- Apply position–momentum uncertainty and infinite-square-well standing-wave quantisation.
- Explain discrete atomic levels, distinguish line spectra and solve photon-transition problems.
Starting-point self-check
1. Check your starting point
Attempt all five groups without notes and mark the first evidence, amplitude, boundary, uncertainty or transition step you cannot justify. Use the recorded topic diagnostic above when you want scoring and a personalised repair plan.
Wavefunctions, probability density and superposition 19(f)–(g)
Question 1
For ψ = A on 0 < x < L and zero elsewhere, find positive A and interpret |ψ|².
Check the model response
Normalisation gives A²L = 1, so A = 1/√L. |ψ|² is position probability density; its integral over an interval is the probability there.
repair
2. Repair the common breaks
Use only the correction matching an error, then retry the corresponding diagnostic.
Wavefunctions, probability density and superposition 19(f)–(g)
Check this idea
Misconception: ψ itself is position probability.
Repair: |ψ|² is probability density; probability over an interval is its integral.
Check this idea
Misconception: Probabilities from two paths are added before interference.
Repair: Superpose amplitudes first, ψ = ψ₁ + ψ₂, then calculate |ψ|².
worked example
3. Follow five worked models
Follow how each solution ties evidence to a claim, normalises amplitudes, applies boundary conditions or selects an allowed transition.
Wavefunctions, probability density and superposition 19(f)–(g)
Model 1
Normalise ψ = A sin(πx/L) on 0 < x < L.
Check the model response
Set ∫₀ᴸ|ψ|²dx = A²L/2 = 1, giving A = √(2/L). Symmetry then gives probability 1/2 in either half.
guided practice
4. Guided practice
Use each hint only to select the correct proportionality, amplitude rule, boundary condition or level difference.
Wavefunctions, probability density and superposition 19(f)–(g)
Question 1
A normalised ψ is multiplied by −1. State the effect on probability density.
Hint: Probability uses modulus squared.
Check the model response
|−ψ|² = |ψ|², so measured position probabilities are unchanged.
independent practice
5. Independent practice
Solve without repair notes and state the evidence, normalisation, quantum-number and transition assumptions.
Wavefunctions, probability density and superposition 19(f)–(g)
Question 1
Explain ψ, |ψ|², normalisation and superposition for a square or sinusoidal wavefunction.
Check the model response
ψ represents the quantum state. |ψ|² is probability density and ∫|ψ|²dx = 1 fixes the normalisation factor. Amplitudes superpose before taking modulus squared, producing interference terms and standing-wave solutions.
Practice exit check
6. Practice assessment
Use this as extra closed-book practice, then complete the separate recorded assessment in your plan.
Wavefunctions, probability density and superposition 19(f)–(g)
Question 1
For ψ = A sin(2πx/L) on 0 < x < L, find positive A and state the number of internal nodes.
Check the model response
The sine-squared integral is L/2, so A = √(2/L). There is one internal node at x = L/2, in addition to boundary nodes.
Re-test practice
7. Delayed re-test practice
Return after at least three days and solve these fresh contexts without reopening earlier responses. The recorded plan enforces the delay and uses a separate re-test family for selected-response skill-group evidence.
Wavefunctions, probability density and superposition 19(f)–(g)
Question 1
What integral gives probability between a and b?
Check the model response
P(a ≤ x ≤ b) = ∫ₐᵇ|ψ(x)|²dx.