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 Eigenspace s of each Eigenvalue λ (see Spectral Theorem For Hermitian Operator s for the mathematical background):
	:<math>\lang\psi {\alpha}\psi {\beta} Ang	\delta_{\alpha\beta}</math>
	:<math>\left \hat{H}\psi \lambda Ight Angle	\left \lambda\psi_\lambda ight angle </math>
	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	"http://wwwinformationdelightinfo/encyclopedia/entry/greatest_lower_bound" class="copylinks">Greatest Lower Bound is ''E<sub>0</sub>'' Suppose also that we know the corresponding state &psi> The Expectation Value of '''''H''''' is then
	:<math>\left\langle\psiH\psi Ight Angle	\sum_{\lambda_1,\lambda_2 \in \mathrm{Spec}(H)} \left\langle\psi\psi_{\lambda_1} ight angle \left\langle\psi_{\lambda_1}H\psi_{\lambda_2} ight angle \left\langle\psi_{\lambda_2}\psi ight angle</math>
	::::<math>	\sum_{\lambda\in \mathrm{Spec}(H)}\lambda \left\langle\psi_\lambda\psi ight angle^2\ge\sum_{\lambda \in \mathrm{Spec}(H)}E_0 \left\langle\psi_\lambda\psi ight angle^2=E_0</math>
	:<math> \left\langle \psi(\alpha I) \psi(\alpha I) Ight Angle	1</math>

Information About