Variational Monte Carlo — Quantum Ground State

Trial wavefunction ψ(x;α) for harmonic oscillator. MC sampling of |ψ|². Gradient descent finds E_min.

Current α: 1.00
⟨E⟩(α): —
E_exact: 0.500 ℏω
Optimal α: α* = 1.00
Trial wavefunction:
ψ(x;α) = exp(-α x²/2)
Hamiltonian (HO):
H = -½d²/dx² + ½x²
Local energy:
E_L = α - (α²-1)x²/2 + ½x²
E_L = α/2 + (1-α²)x²/2
⟨E⟩ = α/2 + 1/(4α)
Minimum at α=1: E=½ℏω ✓
Optimization log will appear here