Abstract

This interactive visualization explores quantum mechanical Hilbert spaces and special relativity through massively parallel eigenvalue decompositions displayed as rotating kaleidoscopic eigenvector projections. Real-time Fourier spectral decomposition, Minkowski spacetime diagrams with light cone structures, Schrödinger equation numerical integration, particle collision simulations, double-slit interference patterns, uncertainty principle Gaussian wave packet spreading, Bloch sphere qubit trajectories, Fock space ladder operators, and Dirac comb diffraction gratings are all rendered simultaneously at 60fps.

Mathematical Foundation

The Schrödinger equation iℏ∂ψ/∂t = Ĥψ governs wavefunction evolution. Eigenvalue decomposition Ĥ|n⟩ = Eₙ|n⟩ yields energy eigenvalues. The Minkowski metric ds² = c²dt² − dx² − dy² − dz² defines spacetime intervals. Lorentz transformations with rapidity φ = arctanh(v/c) boost between inertial frames. Von Neumann entropy S = −Tr(ρ log ρ) measures quantum information. Bloch sphere coordinates θ,φ parameterize qubit states |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩. Fock space ladder operators a†|n⟩ = √(n+1)|n+1⟩ create particle number states. Heisenberg uncertainty ΔxΔp ≥ ℏ/2 bounds simultaneous observables.