Ritz method

The Ritz method is a direct method to find an approximate solution for boundary value problems. The method is named after Walther Ritz. The Ritz method is a direct method to find an approximate solution for boundary value problems. The method is named after Walther Ritz. In quantum mechanics, a system of particles can be described in terms of an 'energy functional' or Hamiltonian, which will measure the energy of any proposed configuration of said particles. It turns out that certain privileged configurations are more likely than other configurations, and this has to do with the eigenanalysis ('analysis of characteristics') of this Hamiltonian system. Because it is often impossible to analyze all of the infinite configurations of particles to find the one with the least amount of energy, it becomes essential to be able to approximate this Hamiltonian in some way for the purpose of numerical computations. The Ritz method can be used to achieve this goal. In the language of mathematics, it is exactly the finite element method used to compute the eigenvectors and eigenvalues of a Hamiltonian system. As with other variational methods, a trial wave function, Ψ {displaystyle Psi } , is tested on the system. This trial function is selected to meet boundary conditions (and any other physical constraints). The exact function is not known; the trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration. It can be shown that the ground state energy, E 0 {displaystyle E_{0}} , satisfies an inequality: