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{{Short description|Method for approximating eigenvalues}}
{{Additional citations}}
The '''Rayleigh–Ritz method''' is a direct numerical method of approximating [[eigenvalues and eigenvectors|eigenvalues]], originated in the context of solving physical [[boundary value problem]]s and named after [[Lord Rayleigh]] and [[Walther Ritz]].
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=== In quantum physics ===
In quantum physics, where the spectrum of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is the set of discrete energy levels allowed by a quantum mechanical system, the Rayleigh–Ritz method is used to approximate the energy states and wavefunctions of a complicated atomic or nuclear system.<ref name=arfkenweber /> In fact, for any system more complicated than a single hydrogen atom, there is no known exact solution for the spectrum of the Hamiltonian.<ref name=pryce />
In this case, an [[ansatz|trial wave function]], <math>\Psi</math>, 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.
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