Rayleigh-Ritz method

In quantum mechanics, the Rayleigh-Ritz method, also known as the linear variation method is a method to obtain (approximate) solutions of the time-independent Schrödinger equation. In numerical analysis, it is a method of solving differential equations with boundary conditions. In the latter field it is sometimes called the Rayleigh-Ritz-Galerkin procedure.

The method

The expression

$E[\Phi ]\equiv {\frac {\langle \Phi |H|\Phi \rangle }{\langle \Phi |\Phi \rangle }}$

is a functional (maps the function Φ onto the number E[Φ]). The bra-ket notation implies the integration over a configuration space, which very often is $\scriptstyle \mathbb {R} ^{3N}$ with N being the number of particles of the system described by the Hamiltonian H. We assume boundary conditions on Φ and integration limits such that E[Φ] is finite and the Hamiltonian H is Hermitian. In that case E[Φ] is a real number. The boundary condition is very often the vanishing of Φ at infinity, together with the integral over all of $\scriptstyle \mathbb {R} ^{3N}$. Bounded configuration spaces with periodic boundary conditions also occur in quantum mechanics, however.

Most Hamiltionians in quantum mechanics have a lower bound, that is there is a finite number E₀ such that

$E[\Phi ]\geq E_{0}$

for all admissible Φ. The function Φ is admissible when the functional E[Φ] is well-defined and Φ has the correct boundary conditions. We will show that the first variation of this functional leads to the eigenvalue equation

$\delta E[\Phi ]=0\Longleftrightarrow H\Phi =E\Phi \quad {\hbox{with}}\quad E={\frac {\langle \Phi |H|\Phi \rangle }{\langle \Phi |\Phi \rangle }}.$

In other words, the function Φ that gives a vanishing first variation is a solution of the eigenvalue equation (the time independent Schrödinger equation). Thus, the eigenvalue equation can be replaced by a variational equation. Ritz's idea was that a function, which minimizes the expectation value of H, simultaneously approximates the lowest eigenvector of H.

The Ritz method expands the unknown function in a set of admissible functions χ _i, which we first assume to be orthonormal (this can always be achieved by orthonormalizing a given linearly independent set)

$\Phi =\sum _{i=1}^{n}c_{i}\chi _{i}$

Inserting this expansion into the functional makes it a function of the expansion coefficients and differentiating ...

History

In the older quantum mechanics literature the method is known as the Ritz method, called after the mathematical physicist Walter Ritz, who first devised it. In prewar quantum mechanics texts it was customary to follow the highly influential book by Courant and Hilbert, who were contemporaries of Ritz and write of the Ritz procedure (Ritzsches Verfahren). It is parenthetically amusing to note that the majority of these old quantum mechanics texts quote the wrong year, 1909 instead of 1908, an error first made in the Courant-Hilbert treatise.

In the numerical analysis literature one usually prefixes the name of Lord Rayleigh to the method, and lately this has become common in quantum mechanics, too. Leissa recently became intrigued by the name and after reading the original sources discovered that the methods of the two workers differ considerably, although Rayleigh himself believed that the methods were very similar and that his own method predated the one of Ritz by several decades. However, according to Leissa's convincing conclusion, Rayleigh was mistaken and the method now known as Rayleigh-Ritz method is solely due to Ritz. Leissa states: Therefore, the present writer concludes that Rayleigh’s name should not be attached to the Ritz method; that is, the Rayleigh–Ritz method is an improper designation.

References

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