Energy levels and wavefunctions of a quantum particle confined to a 2D region
The wave structure of the eigenstate, including sign. Red and blue regions are positive and negative lobes; white lines are nodes (where ψ = 0). The ground state has no interior nodes; higher energy levels develop more.
The physical probability density — where the particle is most likely to be found. Bright regions are high probability, dark are low. For modes carrying angular momentum, the density is rotationally symmetric even when ψ has lobes.
Two parallel ladders of discrete energy levels on a shared scale. The left ladder shows analytical reference values where closed-form or rigorous bounds exist; the right ladder shows our framework's computed values. Paired ticks sit at their own value-proportional y-positions, so perfect agreement shows as horizontal alignment and disagreement shows as a small vertical offset. Hover any tick for both values; click a framework tick to jump to that level.
Method: Link strength is the dipole matrix element |⟨ψᵢ | x | ψⱼ⟩|, the standard quantum-mechanical measure of how strongly an applied field can drive a transition between two states. The framework computes the same quantity computed by every quantum optics simulator and quantum chemistry code, derived from first principles rather than fit to data. Selection rules emerge directly from the integral.