Q2(a). Uncertainty principle; orthogonality of ψ₁ and ψ₂ in a box
State Heisenberg's uncertainty principle. Show that for a particle in a one-dimensional box having length from 0 to L, the two normalised eigenfunctions corresponding to the eigenvalues E₁ and E₂ (characterised by quantum numbers 1 and 2 respectively) are orthogonal to each other.
Answer in brief
Δx Δpₓ ≥ h/4π; and ∫₀ᴸ ψ₁ψ₂ dx = 0, so ψ₁ and ψ₂ are orthogonal.
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Topic notes: particle in a one-dimensional box, with derivations, figures and every PYQ →
CoversHeisenberg uncertainty principleparticle in a boxorthogonalityeigenfunctionszero-point energy
All parts of Question 2
- 2(a) Uncertainty principle; orthogonality of ψ₁ and ψ₂ in a box10 marks
- 2(b)(i) Bond order of O₂, O₂⁻, O₂²⁻ and O₂⁺10 marks
- 2(b)(ii) Molecular orbital diagram of CO10 marks
- 2(c)(i) Limiting radius ratio for coordination numbers 3 and 610 marks
- 2(c)(ii) Stoichiometric defects10 marks