1D harmonic oscillator H = −½∂²/∂x² + ½x² (ħ=m=ω=1), exact E₀ = 0.5. Trial ψ always upper-bounds ground state.
Gaussian: ψ=N·exp(−αx²) → ⟨H⟩=α/2+1/(8α), min at α=½ → ⟨H⟩=0.5=E₀ |
Top-hat: ⟨H⟩=π²/(24a²)+a²/6 |
Parabola: ⟨H⟩=5/(4a²)+a²/14
⟨H⟩ vs parameter (variational curve)
Trial ψ(x) vs exact ψ₀(x) = π^{−1/4}·exp(−x²/2)