Why are (2,1) and (1,2) degenerate only in a square?

E ∝ n_x²/L_x² + n_y²/L_y². If L_x = L_y, swapping n_x ↔ n_y leaves E unchanged. If L_x ≠ L_y, the weights differ and the levels split. The degeneracy is ‘accidental’ relative to a generic rectangle; it reflects the square’s higher symmetry.

Is the ground state ever degenerate here?

No. The ground state is uniquely (n_x, n_y) = (1,1). Degeneracy in this model appears only among excited pairs that can be swapped when L_x = L_y. Confusing ground-state uniqueness with excited-state doublets is a common mistake.

Do the wavefunctions mix when levels split?

For a pure rectangular infinite well the Hamiltonian still separates in x and y, so eigenstates remain product sines even after the aspect ratio changes; only the energies move. Mixing would require a perturbation that couples different (n_x, n_y) labels.

2D Box: Eigenstates & Degeneracy

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.

About this model

A particle in a two-dimensional rectangular infinite well has product eigenstates ψ_{n_x,n_y} ∝ sin(n_x π x / L_x) sin(n_y π y / L_y) with energies E ∝ (n_x/L_x)² + (n_y/L_y)². When the box is square (L_x = L_y), states (n_x, n_y) and (n_y, n_x) share the same energy for n_x ≠ n_y — accidental degeneracy from the extra symmetry. Deforming the rectangle splits those doublets. The model is the ideal infinite-wall Hamiltonian: no finite barriers, no interactions, and no spin. You toggle square versus rectangular aspect ratio and select quantum numbers to see nodal patterns and the lifting of degeneracy as the box is stretched.

Who it's for: Undergraduate quantum mechanics courses on multidimensional wells, separation of variables, and degeneracy.

Key terms

Particle in a 2D box
Energy degeneracy
Separation of variables
Quantum numbers
Infinite well
Nodal pattern

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.

Frequently asked questions

Why are (2,1) and (1,2) degenerate only in a square?: E ∝ n_x²/L_x² + n_y²/L_y². If L_x = L_y, swapping n_x ↔ n_y leaves E unchanged. If L_x ≠ L_y, the weights differ and the levels split. The degeneracy is ‘accidental’ relative to a generic rectangle; it reflects the square’s higher symmetry.
Is the ground state ever degenerate here?: No. The ground state is uniquely (n_x, n_y) = (1,1). Degeneracy in this model appears only among excited pairs that can be swapped when L_x = L_y. Confusing ground-state uniqueness with excited-state doublets is a common mistake.
Do the wavefunctions mix when levels split?: For a pure rectangular infinite well the Hamiltonian still separates in x and y, so eigenstates remain product sines even after the aspect ratio changes; only the energies move. Mixing would require a perturbation that couples different (n_x, n_y) labels.

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