Problem Class Week 13 Problems
Updated: 27 Jan 2026
These are the problems that we will solve during the problem class in week 13. You are not expected to attempt these questions in advance of the problem class. We will work through them in sequence, going through the solutions after everybody has time to attempt the questions.
I have included additional questions for you to attempt / discuss if you complete the questions before we go over the solution.
Problems ¶ Consider a particle confined to the region x ≥ 0 x \geq 0 x ≥ 0
∣ ψ ⟩ = 2 L ∫ 0 ∞ e − x / L ∣ x ⟩ d x , \ket{\psi} = \sqrt{\frac{2}{L}}\int_0^\infty e^{-x/L} \ket{x} dx, ∣ ψ ⟩ = L 2 ∫ 0 ∞ e − x / L ∣ x ⟩ d x , where L > 0 L > 0 L > 0
Write down (read off) the wavefunction ψ ( x ) \psi(x) ψ ( x )
Write down the normalisation condition (i) in terms of the quantum state (ii) in terms of the wavefunction.
Using part 2. confirm that the quantum state Equation
Sketch the wavefunction ψ ( x ) \psi(x) ψ ( x ) ℘ ( x ) \pd(x) ℘ ( x )
Calculate the expected value of the position of the particle ⟨ x ⟩ \langle x \rangle ⟨ x ⟩
What are the units of L L L L L L
Use Desmos L L L
Calculate the probability to find the particle in the region 0 ≤ x ≤ L 0\leq x\leq L 0 ≤ x ≤ L
Calculate the uncertainty (standard deviation) Δ x \Delta x Δ x
Consider a particle confined to the region x ≥ 0 x \geq 0 x ≥ 0
∣ ψ ⟩ = 2 L ∫ 0 ∞ e − x / L ∣ x ⟩ d x , \ket{\psi} = \sqrt{\frac{2}{L}}\int_0^\infty e^{-x/L} \ket{x} dx, ∣ ψ ⟩ = L 2 ∫ 0 ∞ e − x / L ∣ x ⟩ d x , where L > 0 L > 0 L > 0
Write down the relationship between the spatial and momentum wavefunctions.
Using part 1. to calculate the momentum wavefunction ψ ~ ( p ) \tilde{\psi}(p) ψ ~ ( p )
Use Desmos ψ ~ ( p ) \tilde{\psi}(p) ψ ~ ( p ) ℘ ( p ) \pd(p) ℘ ( p )
Use Desmos to explore how the momentum wavefunction of the particle changes as L L L
Use Desmos to calculate the average momentum ⟨ p ⟩ \langle p \rangle ⟨ p ⟩ Δ p \Delta p Δ p
Confirm that the Heisenberg Uncertainty Principle is satisfied.
Solutions ¶
Desmos ¶