Compared to perturbation theory, the variational method can be more robust in situations where it's hard to determine a good unperturbed Hamiltonian i.
26 - The Variational Method.pdf
On the other hand, in cases where there is a good unperturbed Hamiltonian, perturbation theory can be more efficient than the variational method. The resulting trial wavefunction and its corresponding energy are variational method approximations to the exact wavefunction and energy.
Why would it make sense that the best approximate trial wavefunction is the one with the lowest energy? This results from the Variational Theorem, which states that the energy of any trial wavefunction is always an upper bound to the exact ground state energy. This can be proven easily. Let the trial wavefunction be denoted. Any trial function can formally be expanded as a linear combination of the exact eigenfunctions. Nevertheless, that does not prevent us from using the exact eigenfunctions in our proof, since they certainly exist and form a complete set, even if we do not happen to know them: We are asuming that the physical states are normalized, i.
Let us assume that we have a candidate wavefunction to describe the ground-state, that we call , and that this function deppends on a set of parameters , that we call variational parameters and are complex numbers.
Catalog Record: Variational methods for eigenvalue problems; | HathiTrust Digital Library
Ignoring complications involved with a continuous spectrum of H, suppose that the spectrum is bounded from below and that its greatest lower bound is. In other words, the energy of any approximate wavefunction is always greater than or equal to the exact ground state energy.
This explains the strategy of the variational method: since the energy of any approximate trial function is always above the true energy, then any variations in the trial function which lower its energy are necessarily making the approximate energy closer to the exact answer. The trial wavefunction is also a better approximation to the true ground state wavefunction as the energy is lowered, although not necessarily in every possible sense unless the limit is reached. Frequently, the trial function is written as a linear combination of basis functions, such as.
Variational methods and optimization in imaging
This leads to the linear variation method, and the variational parameters are the expansion coefficients. We shall assume that the possible solutions are restricted to a subspace of the Hilbert space, and we shall seek the best possible solution in this subspace. The energy for this approximate wavefunction is just
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