Variational method (quantum mechanics)

The variation method in quantum mechanics is an approximation method to find an upper bound for the eigenvalues ​​of a quantum mechanical observables with discrete spectrum. A generalization of the method leads to the min-max principle.

Method

Ground state

The method is based on the eigenvalue of the ground state is a lower bound for the expected value of the measurement of observables: Is the degeneracy of an eigenvalue, it can be any condition as

Writing, which form a complete orthonormal system. Then for the expectation value of the state at measurement of an observable with eigenvalues

It can therefore find an upper bound for when calculating the expected value for a bevy of conditions and the infimum are looking for:

Excited States

If the eigenfunction of a ( non-degenerate ) ground state with eigenvalue, so can be written for an arbitrary state

Where. If we decompose above in eigenstates, obtained under the constraint

Which was lacking in the sum of the value.

The search for other eigenstates is analogous, then being to minimize orthogonality among several compartments that span the lower eigenvalues ​​.