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Home/Chemistry/2D Box: Eigenstates & Degeneracy

2D Box: Eigenstates & Degeneracy

This interactive simulator explores 2D Box: Eigenstates & Degeneracy in Chemistry. Particle in a 2-D rectangular infinite well: ψ_{n_x,n_y} ∝ sin(n_xπx/L_x)sin(n_yπy/L_y), E ∝ (n_x/L_x)² + (n_y/L_y)². Toggle a square box (L_x = L_y) to expose the (n_x, n_y) ↔ (n_y, n_x) accidental degeneracy and watch the doublets split as the box deforms. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: For learners comfortable with heavier math or second-level detail. Typical context: Chemistry.

Key terms

  • box
  • eigenstates
  • degeneracy
  • box 2d degeneracy
  • chemistry

2D rectangular box

1
1
2
1

Particle in a 2-D infinite rectangular well: |ψ_{n_x,n_y}|² = (4/L_xL_y) sin²(n_xπx/L_x) sin²(n_yπy/L_y), E = (π²/2)((n_x/L_x)² + (n_y/L_y)²) (ℏ = m = 1). For a square well (L_x = L_y) the swap (n_x, n_y) ↔ (n_y, n_x) gives an exact two-fold accidental degeneracy whenever n_x ≠ n_y; deforming the box to a rectangle splits the doublets — a clean introduction to symmetry-induced degeneracy.

Measured values

E_{2,1}24.6740
Degeneracy at this E2×
Aspect L_x/L_y1.000
Ground E_{1,1}9.8696

How it works

2-D infinite square well: explore the eigenstates ψ_{n_x,n_y} with the energy ladder E ∝ (n_x/L_x)² + (n_y/L_y)². Toggle a square box (L_x = L_y) to see the (n_x, n_y) ↔ (n_y, n_x) accidental degeneracy and the colormap of |ψ|² for each level.