Approximate methods: variation and perturbation (intro)
Quantum Chemistry (UG) · Chemistry
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Estimate hydrogen's ground-state energy using trial function ψ = (α³/π)½·e^(-αr), treating α as adjustable. Step 1: Compute ⟨T⟩ = ℏ²α²/2m (kinetic) and ⟨V⟩ = -e²α/4πε₀ (potential). Step 2: E(α) = ℏ²α²/2m - e²α/4πε₀. Step 3: Minimise: dE/dα = 0 → α = me²/4πε₀ℏ² = 1/a₀. Step 4: E_min = -13.6 eV — exactly the true answer! A good guess recovered the exact energy.