Included mode / ladder rung
Excluded (above cutoff)
Excited level n / particle
Drag to pan · scroll to zoom (left well view)
Every quantized field mode is a quantum harmonic oscillator with energies E_n = (n + ½)ħω, so even its ground state carries a non-zero zero-point energy E₀ = ½ħω. This 2D simulation draws the oscillator's own parabolic potential well V(x) = ½ω²x² directly, with dashed rungs at each E_n — pick a frequency ω and a quantum number n to light one up, then press Oscillate to watch a particle bounce between its true classical turning points x_max = √(2E_n/ω²), correctly narrower in a stiffer well. Alongside it, a bank of field modes ω_k = k·Δω each contribute their own ½ħω_k of zero-point energy as a bar chart; a tunable UV cutoff k꜀ shows live, as a running sum, why adding up every mode's vacuum energy diverges rather than converges — the same ultraviolet-divergence problem that makes "the energy of the vacuum" a subtle, cutoff-dependent quantity in quantum field theory.