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Rayleigh-Ritz Variational Technique

A technique for computing Eigenfunctions and Eigenvalues. It proceeds by requiring

J=\int_a^b [p(x){y_x}^2-q(x)y^2]\,dx
\end{displaymath} (1)

to have a Stationary Value subject to the normalization condition
\int_a^b y^2 w(x)\,dx=1
\end{displaymath} (2)

and the boundary conditions
py_xy\vert _a^b=0.
\end{displaymath} (3)

This leads to the Sturm-Liouville Equation
{d\over dx}\left({p{dy\over dx}}\right)+qy+\lambda wy=0,
\end{displaymath} (4)

which gives the stationary values of
F[y(x)]={\int_a^b (p{y_x}^2-qy^2)\,dx\over\int_a^b y^2w\,dx}
\end{displaymath} (5)

\end{displaymath} (6)

where $\lambda_n$ are the Eigenvalues corresponding to the Eigenfunction $y_n$.


Arfken, G. ``Rayleigh-Ritz Variational Technique.'' §17.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 957-961, 1985.

© 1996-9 Eric W. Weisstein