GerritJan Linker

Variational theorem The variational theorem is: <ΦHΦ> ≥ E_{0} where: H is the time independent Hamiltonian E_{0} is the exact Energy of the ground state (the lowest eigenvalue of H) Φ is a normalised trial wave function satisfying the same boundary conditions as the axact wave function. In words the theorem says that the mean value of H from that wavefunction Φ is alway above or equal to the exact energy of the ground state of the system. When the exact wave function ψ_{0} is used <ψ_{0}Hψ_{0}> = <ψ_{0}E_{0}ψ_{0}> = E_{0} Derivation: Let us use the trial wave function Φ again. Let F = <ΦH  E_{0}Φ> We can expand Φ in the basis set of eigen functions {ψ_{i}} of H: Φ = ∑_{i}c_{i}ψ_{i} When we use that in the above formula we get: F = <ΦH  E_{0}Φ> = <∑_{i}c_{i}ψ_{i}H  E_{0}∑_{j}c_{j}ψ_{j}> = ∑_{i}∑_{j}c^{*}_{i}c_{j}<ψ_{i}H  E_{0}ψ_{j}> = ∑_{i}∑_{j}c^{*}_{i}c_{j}(<ψ_{i}Hψ_{j}>  <ψ_{i}E_{0}ψ_{j}>) = ∑_{i}∑_{j}c^{*}_{i}c_{j}(<ψ_{i}E_{j}ψ_{j}>  <ψ_{i}E_{0}ψ_{j}>) = ∑_{i}∑_{j}c^{*}_{i}c_{j}(E_{j}<ψ_{i}ψ_{j}>  E_{0}<ψ_{i}ψ_{j}>) = ∑_{i}∑_{j}c^{*}_{i}c_{j}(E_{j}δ_{ij}  E_{0}δ_{ij}) = ∑_{i}∑_{j}c^{*}_{i}c_{j}δ_{ij}(E_{j}  E_{0}) = ∑_{i}c^{*}_{i}c_{i}(E_{i}  E_{0}) = ∑_{i}c_{i}^{2}(E_{i}  E_{0}) We arrive at F = ∑_{i}c_{i}^{2}(E_{i}  E_{0}) Since E_{i} ≥ E_{0} it follows that F > 0
