Speaker
Description
In an infinite-dimensional Hilbert space it is possible to implement the Conjugate Gradient (CG) method to find the ground state eigenvectors and eigenvalues of a Hermitian operator H with sufficiently sparse matrix elements in a particular basis. Several different functions on the state vector can be minimized to achieve this goal.
To find excited eigenvectors and eigenvalues of operator H is non-trivial due to its infinite-dimensional representation. In this talk several approaches to determining excited state eigenvectors and eigenvalues of H using the CG method will be illustrated. The first approach will be based on the method's preservation of symmetry of the initial state chosen. The second approach will involve an iterative application of the CG method to operators of H that have been successively reduced using previous eigenvalue/vector determinations. Inspired by the properties of a multi-particle quantum state of identical fermions, the final approach will apply the CG method to a function acting on states in an antisymmetrized tensor power (n) of the original Hilbert space to simultaneously find the n lowest eigenvalues and eigenvectors of H with a single application of the CG method.
The approaches will be illustrated with one-dimensional quantum mechanical examples where H is a time-independent Hamiltonian of a single-particle system. Mitigation of the effects of numerical approximation in the CG method will be considered.
| Keyword-1 | Conjugate Gradient Method |
|---|---|
| Keyword-2 | Excited State Determination |
