# Interleaving Theorems for the Rayleigh-Ritz Method in Quantum Mechanics

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The Rayleigh–Ritz variational method has been well known in mathematics for well over a century. Its application to quantum mechanics was definitively described by J. K. L. MacDonald in *Phys. Rev.* 43(10), 1933 pp. 830–833. The eigenfunctions of a quantum-mechanical Hamiltonian can be approximated by a linear combination of basis functions. This gives an secular equation with roots, approximating the lowest eigenvalues. Two interleaving theorems can be proven: (1) between each pair of successive roots of the secular equation, augmented by and , there occurs at least one exact eigenvalue; (2) if is increased to , then the new approximate roots will be interleaved by the previous ones. As a corollary to (1), often called simply "the" variational principle, the lowest approximate eigenvalue provides an upper bound to the exact ground-state eigenvalue.

Contributed by: S. M. Blinder (March 2011)

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