Variational method quantum mechanics: Encyclopedia II - Variational method quantum mechanics - Introduction

Variational method quantum mechanics - Introduction

Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous spectra, we look at the discrete spectrum of H and the corresponding eigenspaces of each eigenvalue λ (see spectral theorem for Hermitian operators for the mathematical background):

with

and

Physical states are normalized, meaning that their norm is equal to 1. Once again ignoring complications involved with a continuous spectrum of H, suppose it is bounded from below and that its greatest lower bound is E₀. Suppose also that we know the corresponding state |ψ>. The expectation value of H is then

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