Introduction to Quantum Mechanics

Many problems that we come across in quantum systems cannot be solved exactly. In fact, exactly solvable problems form only a small fraction of quantum problems that one encounters in realistic physical systems. In the absence of such exact solutions of the Schrödinger equation, one needs to resort to several methods providing approximate solutions. In this chapter we shall list some of these methods for time-independent Hamiltonians.

Variational method

The variational method is often used for a quick estimation of the ground state energy of a Hamiltonian whose exact eigenvalue and eigenstates are unknown. We first describe this method in general terms and then follow the discussion up with examples.

The method

To elucidate the technique, we first note that a trial wavefunction ψ trial for any quantum Hamiltonian H always satisfies the inequality

where E0 is the exact ground state energy of H. To see why this holds, first let us expand ψ triali in the eigenbasis of H:

where |n〉 is nth eigenstate of H with energy En. Since the eigenstates of H form a complete basis, this can always be done. The eigenstates |n〉, as well as the coeficients cn that appear in Eq. (11.2), can be determined only if we can solve the Hamiltonian problem exactly.