Zero-Point Energy: Oscillator Ladder & Mode Sum

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 simulation renders that ladder in 3D — pick a frequency ω and a quantum number n to see the rung light up and, on request, animate a particle oscillating between the classical turning points at that energy. Alongside it, a bank of field modes ω_k = k·Δω each contribute their own ½ħω_k of zero-point energy; 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.