WKB Quantization Condition Calculator (Quantum Bouncer)

The WKB (Wentzel-Kramers-Brillouin) quantization condition is a semiclassical recipe, ∮p dx = (n + phase)h, for finding approximate energy levels of a quantum system directly from its classical motion, without solving the Schrödinger equation exactly. It works well whenever the potential varies slowly compared to the particle's local de Broglie wavelength.

The general WKB condition needs the classical turning points and the exact shape of the potential V(x), which for an arbitrary potential requires numerical integration with no simple algebraic formula. This calculator focuses on the one case with a clean, well-known closed-form result: a uniform force (a linear potential, V(x)=Fx) acting against a hard floor, often called the 'quantum bouncer'.

The quantum bouncer describes a particle falling under a constant force (like gravity) that reflects off a hard floor at x=0, forming a series of bound, quantized 'bounce' states instead of falling continuously as a classical object would. Its WKB energy levels are E_n = [3πħF(n−1/4) / (2√(2m))]^(2/3), where F is the force and m is the particle's mass.

Yes. The GRANIT experiment (Nesvizhevsky et al., published in Nature in 2002) measured the quantized gravitational states of ultracold neutrons bouncing on a horizontal mirror, finding a ground-state energy of roughly 1.4 picoelectronvolts, in close agreement with this exact WKB formula. It remains one of the cleanest demonstrations of quantum mechanics acting under gravity.

This is the Langer correction for a hard, rigid wall: the WKB phase accumulated at a hard wall is π/2, different from the π/4 phase accumulated at a smooth classical turning point (like the top of the bounce). Combining these two different boundary phases in the full quantization condition produces the (n − 1/4) factor rather than the (n − 1/2) or (n + 1/2) seen in other WKB problems.

l0 = (ħ² / (2m²g))^(1/3) sets the natural height scale of the bounce, roughly how high the particle's wavefunction extends above the floor. For an ultracold neutron under Earth's gravity, l0 works out to about 5.9 micrometres, remarkably large for a quantum length scale, which is exactly why the effect is measurable in a tabletop experiment.

A classical ball under gravity can have any energy and bounces to any height. The quantum bouncer is restricted to a discrete ladder of allowed energies, E_n ∝ (n−1/4)^(2/3), each corresponding to a specific quantized bounce height. Unlike the evenly-spaced harmonic oscillator or the n²-scaled particle in a box, this ladder grows relatively slowly with n.

Yes, in principle it applies to any particle under any uniform force, gravity, a uniform electric field, or any constant applied force, with the mass and force adjusted accordingly. In practice the gravitational quantum bouncer is most famous with neutrons because they are electrically neutral, so gravity (not electromagnetic forces) dominates their motion near a surface.

It is extremely accurate but not perfectly exact, the true eigenvalues come from Airy functions (the exact solution of the Schrödinger equation for a linear potential), and WKB reproduces them to within a fraction of a percent even for the ground state, improving further for higher n. This makes it one of the best-known examples of the WKB approximation's surprising accuracy even outside its formal 'large n' regime.

Because E_n scales as (n−1/4)^(2/3), a power less than 1, each additional quantum number adds progressively less energy than the last, in contrast to the particle in a box (which scales as n², growing faster than linearly). This reflects the shape of the linear potential: climbing higher against a constant force takes proportionally less extra 'phase space' per level than climbing the steep walls of a box.