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 , with the following quantum state
∣ 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 is a constant.
Write down (read off) the wavefunction ψ ( x ) \psi(x) ψ ( x ) of the particle.
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 is normalised.
Sketch the wavefunction ψ ( x ) \psi(x) ψ ( x ) and the probability density ℘ ( 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 ? Does L L L have any physical significance/interpretation?
Use Desmos to make an interactive plot of the wavefunction/probability density, to investigate what properties of the particle change as L L L is varied.
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 in the position of the particle.
Consider a particle confined to the region x ≥ 0 x \geq 0 x ≥ 0 , with the following quantum state
∣ 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 is a constant.
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 ) of the particle.
Use Desmos (or otherwise) to sketch the momentum wavefunction ψ ~ ( p ) \tilde{\psi}(p) ψ ~ ( p ) and momentum probability density ℘ ( p ) \pd(p) ℘ ( p ) .
Use Desmos to explore how the momentum wavefunction of the particle changes as L L L varies. How does this compare to how the spatial wavefunction varied?
Use Desmos to calculate the average momentum ⟨ p ⟩ \langle p \rangle ⟨ p ⟩ and uncertainty (standard deviation) Δ p \Delta p Δ p of momentum of the particle.
Confirm that the Heisenberg Uncertainty Principle is satisfied.
Solutions ¶
Desmos ¶