In this problem you will prove that the groundstate energy for a system obtained using t...

In this problem you will prove that the groundstate 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 of the total energy operator in the form . Show that by substituting in the equation

you obtain the result

b. Because the are eigenfunctions of , they are orthogonal and . Show that this information allows us to simplify the expression for E 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 - E₀ ≥ 0