- #1
Manchot
- 473
- 4
A text I am reading has used the variational principle not only to find the ground state of a system, but also to find some higher order states. (Specifically, it was used to derive the Roothaan equations, which are ultimately related to the LCAO method of orbital calculations.) I don't see how this could be valid.
For finding the ground state energy of a system, it is obvious why minimizing the expected value of the Hamiltonian gets the best approximation to the ground state. But in what sense does the variational principle converge to the right result? I thought at first that it might minimize the variance of Hamiltonian in the trial wavefunction, but I do not believe that this is the case. So, does anyone know exactly in what precise sense the variational principle finds the "best" solutions to the eigenvalue problem?
For finding the ground state energy of a system, it is obvious why minimizing the expected value of the Hamiltonian gets the best approximation to the ground state. But in what sense does the variational principle converge to the right result? I thought at first that it might minimize the variance of Hamiltonian in the trial wavefunction, but I do not believe that this is the case. So, does anyone know exactly in what precise sense the variational principle finds the "best" solutions to the eigenvalue problem?