Q1(a). Expectation value of position in the ground state of a box

Answer in brief

⟨x⟩ = ∫₀ˡ ψ₁ x ψ₁ dx = l/2: the average position in the ground state is the centre of the box.

Key points

• Inside the box the normalised wave functions are ψₙ = √(2/l) sin(nπx/l); the ground state is n = 1.
• The expectation value is ⟨A⟩ = ∫ ψ* Â ψ dτ, and the position operator is multiplication by x.
• Using sin²θ = ½(1 − cos 2θ), the cosine integral vanishes over one complete wave, leaving ⟨x⟩ = (1/l)(l²/2) = l/2.
• Symmetry: ψ² is the same at x and at l − x, so ⟨x⟩ = l/2 in every state, whatever the value of n.
• ⟨x⟩ is the mean of many measurements, not where the particle sits; the spread is Δx = 0.181 l.