Hamiltonian Truncation & Variational Eigenvalue Bounds
The variational principle guarantees truncated Hamiltonian eigenvalues bound exact eigenvalues from above. Watch convergence as basis size grows.
Variational Principle (Ritz 1908): For any trial state |ψ⟩, ⟨ψ|H|ψ⟩/⟨ψ|ψ⟩ ≥ E₀. By truncating to an N-dimensional subspace, all N eigenvalues of H_N provide upper bounds on the exact eigenvalues: E_n(N) ≥ E_n(∞). The interlacing theorem (Cauchy) guarantees monotone convergence: E_n(N+1) ≤ E_n(N). Basis: harmonic oscillator eigenstates |n⟩ are ideal for smooth potentials. The chart shows convergence of each level as N increases — exact values (red dashes) are computed from large N reference.