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## The variational principle

Suppose that we wish to solve the time-independent Schrödinger equation
 (1146)

where is a known (presumably complicated) time-independent Hamiltonian. Let be a normalized trial solution to the above 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 wave-function: i.e.,
 (1147)

Thus, by varying until the expectation value of is minimized, we can obtain an approximation to the wave-function and energy of the ground-state.

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

 (1148)

Furthermore, let
 (1149)

so that is the ground-state, the first excited state, etc. The are assumed to be orthonormal: i.e.,
 (1150)

If our trial wave-function is properly normalized then we can write
 (1151)

where
 (1152)

Now, the expectation value of , calculated with , takes the form
 (1153)

where use has been made of Eqs. (1148) and (1150). So, we can write
 (1154)

However, Eq. (1152) can be rearranged to give
 (1155)

Combining the previous two equations, we obtain
 (1156)

Now, the second term on the right-hand side of the above expression is positive definite, since for all [see (1149)]. Hence, we obtain the desired result
 (1157)

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

 (1158)

Thus, by varying until the expectation value of is minimized, we can obtain an approximation to the wave-function and 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.

Next: The helium atom Up: Variational methods Previous: Introduction   Contents
Richard Fitzpatrick 2006-12-12