↑↑↑   Simulation of the Time-dependent Schrödinger Equation   ↑↑↑

Complex probability amplitude: Ψ(x) = √ρ(x) × e^iφ(x),       _-∞∫^+∞ |Ψ(x)|² dx = 1

Gaussian wave packet: Ψ(x) = const × e^ikx × exp(-(x-x₀)²/(4σ²))

Momentum space representation (Fourier transform): Φ(k) = (1/√2π) _-∞∫^+∞ Ψ(x) e^ikx dx

Time-dependent Schrödinger Equation: i♄ ∂Ψ(x,t)/∂t = H Ψ(x,t) = (-♄²/2m ∂²/∂x² + V(x)) Ψ(x,t)

Particle in a Box: V(x) = 0 < x < 1 ? 0 : ∞
Quadratic Potential Well: V(x) = 6×10⁵ × (x - 0.5)²
Double-well potential: V(x) = 2×10⁴ × (4x - 1)² × (4x - 3)²
Ramp potential: V(x) = 0 < x < 1 ? 6×10⁴×x : ∞
Ramp potential: V(x) = 0 < x < 1 ? 7×10⁴×x : ∞
Step potential: V(x) = x < 0.5 ? 0 : 4×10⁴
Step potential: V(x) = x < 0.5 ? 0 : 6×10⁴
Step potential: V(x) = x < 0.5 ? 0 : 8×10⁴
Quantum Tunneling: V(x) = x < 0.5 || x > 0.51 ? 0 : 3×10⁴
Quantum Tunneling: V(x) = x < 0.5 || x > 0.51 ? 0 : 1.5×10⁴
Quantum Tunneling: V(x) = x < 0.5 || x > 0.51 ? 0 : 6×10⁴

Schrödinger equation

The Time-Dependent Schrödinger Equation

Simulation of the time-dependent Schrödinger equation