In this problem you will prove that the ground-state energy for a system obtained using the variational method is greater than the true energy.

a. The approximate wave function Φ can be expanded in the true (but unknown) eigenfunctions ψ n of the total energy operator in the form Φ = Σ_nc_nψ_n. Show that by substituting Φ = Σ_ncnψ_n in the equation

you obtain the result

b. Because the ψ_n are eigenfunctions of Ĥ, they are orthonormal and Ĥψ_n = E_nψ_n. Show that this information allows us to simplify the expression for D from part (a) to

c. Arrange the terms in the summation such that the first energy is the true ground-state energy E₀ and the energy increases with the summation index m. Why can you conclude that E ˆ’ E₀ ‰¥ 0?

Transcribed Image Text:

арФН, (Ф'НФаr ФФdг ΣΣ Η (,Ψ.) άτι ΣΣΙW(ςm) de п т п т