Semiclassical bound-state energies for arbitrary V(x): harmonic, quartic, Morse, double-well, asymmetric and square wells. Bisection on the action ∫√(2m(E−V))dx = (n+½)πℏ gives the WKB ladder; compare with the exact harmonic ladder ℏω(n+½).
About this model
This simulator finds semiclassical bound-state energies for a one-dimensional potential V(x) via the Bohr–Sommerfeld / WKB quantization condition ∫ √(2m(E−V)) dx = (n+½)πħ between classical turning points. Built-in wells include harmonic, quartic, Morse, double-well, asymmetric, and square potentials. Bisection on the action locates each WKB level; for the harmonic case you can compare with the exact ladder ħω(n+½). The method assumes a smooth slowly varying potential, single-well or simple turning-point structure as implemented, and the standard Maslov connection that supplies the +½. You choose the potential shape and quantum number n to see how the WKB ladder tracks — or misses — exact eigenvalues when the potential is highly anharmonic or has tunneling-split double wells.
Who it's for: Advanced undergraduate quantum mechanics and mathematical physics courses covering semiclassical approximation and quantization conditions.
Key terms
WKB approximation
Bohr–Sommerfeld quantization
Action integral
Turning points
Semiclassical energy
Maslov index
How it works
Semiclassical Bohr–Sommerfeld quantization for an arbitrary 1-D potential V(x). Bisection inside the action integral ∫√(2m(E−V))dx gives the WKB ladder E_n; presets include harmonic, quartic, Morse, double well, asymmetric and square wells (ℏ = m = 1).
Frequently asked questions
Why is the right-hand side (n+½)πħ instead of nπħ?
Matching WKB waves to Airy solutions at each soft turning point contributes a π/4 phase (Maslov index). Two turning points therefore add ½ to the integer n, giving ∫ p dx = (n+½)πħ. Hard walls (infinite square well) change the connection and replace ½ by a different offset — a common source of confusion when mixing formulas.
When is WKB exact for the harmonic oscillator?
For V(x) = ½ m ω² x² the WKB condition with the +½ term reproduces E_n = ħω(n+½) exactly. That agreement is special to the quadratic well; for quartic, Morse, or double-well potentials WKB is only asymptotic for large n.
Does WKB capture tunneling splitting in a double well?
Simple single-well Bohr–Sommerfeld quantization does not produce the exponentially small symmetric/antisymmetric splitting. Instanton or coupled-well WKB extensions are needed. This lab’s double-well entry illustrates the ordinary action condition, not a full tunneling multiplet calculation.