13.1: Variational Principle

Suppose that we wish to solve the time-independent Schrödinger equation where is a known (presumably complicated) time-independent Hamiltonian. Let be a properly normalized trial solution to the previous equation. The variational principle states, quite simply, that the ground-state energy, , is always less than or equal to the expectation value of calculated with the trial wavefunction: that is,

Thus, by varying until the expectation value of is minimized, we can obtain approximations to the wavefunction and the energy of the ground-state.

Let us prove the variational principle. Suppose that the and the are the true eigenstates and eigenvalues of : that is,

Furthermore, let

so that is the ground-state, the first excited state, et cetera. The are assumed to be orthonormal: that is,

If our trial wavefunction is properly normalized then we can write where

Now, the expectation value of , calculated with , takes the form

where use has been made of Equations and . So, we can write

However, Equation can be rearranged to give

Combining the previous two equations, we obtain

The second term on the right-hand side of the previous expression is positive definite, because for all (Equation ). Hence, we obtain the desired result

Excited States

Suppose that we have found a good approximation, , to the ground-state wavefunction. If is a normalized trial wavefunction that is orthogonal to (i.e., ) then, by repeating the previous analysis, we can easily demonstrate that

Thus, by varying until the expectation value of is minimized, we can obtain approximations to the wavefunction and the energy of the first excited state. Obviously, we can continue this process until we have approximations to all of the stationary eigenstates. Note, however, that the errors are clearly cumulative in this method, so that any approximations to highly excited states are unlikely to be very accurate. For this reason, the variational method is generally only used to calculate the ground-state and first few excited states of complicated quantum systems.

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)