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Hover or tap each term to reveal its role
QUANTUM MECHANICS · WAVEFUNCTION iℏ i = imaginary unit (√−1) ℏ = reduced Planck constant ∂ψ ∂t ∂ψ/∂t = rate of change of wavefunction over time = Ĥ Ĥ = Hamiltonian operator total energy of the system Ĥ = −(ℏ²/2m)∇² + V(x) ψ ψ = wavefunction encodes all measurable info about the quantum state hover each term EXPANDED FORM iℏ ∂ψ ∂t = − ℏ² 2m kinetic energy term m = mass of particle ∇² ∇² = Laplacian operator spatial curvature of ψ ψ (x,t) + V (x,t) V = potential energy external forces on particle ψ (x,t) WAVEFUNCTION ψ(x,t) — REAL PART x ψ 0 λ (wavelength) amplitude |ψ|² probability density SUPERPOSITION ψ = α|0⟩ + β|1⟩ particle exists in multiple states simultaneously until measured PROBABILITY P(x) = |ψ(x)|² |ψ|² gives the probability of finding the particle at position x COLLAPSE measure → |x₀⟩ observation forces the wavefunction to resolve to one definite state iℏ ∂ψ/∂t = −(ℏ²/2m)∇²ψ + V(x,t)ψ ERWIN SCHRÖDINGER · 1926 PRISMARA · PHYSICS ANIMATED

The Schrödinger Equation

what governs the double-slit

The interference pattern in the double-slit experiment is not magic — it is a precise mathematical consequence of the Schrödinger equation. The wavefunction ψ evolves smoothly in time according to this equation, and its squared magnitude |ψ|² at any point gives the probability of finding the particle there. When ψ passes through two slits, the two wavefronts interfere just like water waves, producing the characteristic banded pattern.

Hover each term in the equation above to see what it represents. The Hamiltonian Ĥ encodes all the energy in the system — kinetic (how fast the particle moves) plus potential (what forces act on it). The equation says: the rate at which the quantum state changes in time is completely determined by the total energy of the system. Everything else — the double-slit, the orbital shapes of atoms, the tunnel effect in semiconductors — follows from this single line.