Q7. Degeneracy in a two-dimensional square box
Solution of a two-dimensional box problem gives the following for wave function and energy: ψn₁,n₂(x, y) = (2/L) sin(n₁πx/L) sin(n₂πy/L) for 0 ≤ x ≤ L and 0 ≤ y ≤ L; ψn₁,n₂(x, y) = 0 outside the box, and En₁,n₂ = (n₁² + n₂²)h²/8mL². Show that if the 2D box becomes a square there arises the case of degeneracy.
Answer in brief
In a square box E depends only on n₁² + n₂², so the different functions ψ₁₂ and ψ₂₁ share E = 5h²/8mL² (two-fold degenerate); in a rectangle (a ≠ b) their energies differ.
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Coversparticle in a two-dimensional boxdegeneracysquare boxwave functionaccidental degeneracyquantum numbers