Show that the average of the Hamiltonian is an upper bound to the ground state energy.
Let |psi_n> be the normalized eigenstates with eigenvalue E_n of the Hamiltonian H:
H|psi_n> = E_n |psi_n> (1)
If |psi> is any arbitrary state we can always expand it in terms of the eigenstates:
|psi> = sum from n = 0 to infinity of c_n |psi_n> (2)
The expansion coefficients c_n are given by:
c_n = <psi_n|psi>
If |psi> is normalized then:
Inserting (2) in here and using that |psi_n> are orthonormal ...
We give a proof of the variational principle for the ground state energy.