The Sturm-Liouville Eigenvalue Problem

Many eigenvalue problems in quantum mechanics as well as classical physics fit into the class of DE called Sturm-Liouville equations:

Equ (1)

where *y(x)* is the quantum mechanical wave function or other physical quantity,
*f(x), g(x)*, and *h(x)* are real functions of *x* determined by the
nature of the system of interest, and *l* is a parameter
(perhaps the energy of the system) that adopts characteristic or eigen-values to satisfy
boundary conditions on *y(x)* at end points of an interval *a<x<b*.

**Exercise and Example**

Show that the vibrating string problem is an example of the S-L eigenvalue problem. do
this by deducing the form of the functions *f(x), g(x)* and *h(x)* in this
case. Other examples of Sturm-Liouville problems in physics abound.

An important example is the Legendre DE that gives rise to Legendre polynomials.

We now describe solutions of the Sturm-Liouville problem in those cases with *h(x) =
1*, and boundary conditions of the form *y(a)=y(b)=0*. These limitations are
appropriate for most quantum mechanics problems as well as many classical problems.
Solutions for these special S-L problems are described in several sources (e.g., H.
Margenau & G. M. Murphy, *The Mathematics of Physics and Chemistry* [D. van
Nostrand, 1956]). The properties of the general solutions are summarized in the following
few paragraphs.

*If f(x) and h(x) are positive in the interval a<x<b then the S-L equation is
satisfied only for a discrete set of eigenvalues l _{j}
with j=0,1, ..., and corresponding eigenfunctions y_{j}(x). Additionally, the
solution eigenvalues and eigenfunctions have the following properties.*

*The eigenvalues l*_{j}are countably infinite and may be ordered: l_{j}< l_{j}+1.*There is a smallest non-negative eigenvalue, l*_{0}> 0. There is no greatest eigenvalue.*Eigenfunctions, y*_{j}, posess nodes between a and b, the number of such nodes increases with increasing j. The eigenfunction y_{0}(x) has no nodes, y_{1}(x) has one node, and so forth.

For a complete solution of the Strum-Liouville problem, consult one the the references.

Show that the eigenfunctions {*y _{j}(x), j=0,1,...*} are orthogonal on
the interval (

To prove the desired property, begin by writing the eigenvalue equation:

. Now multiply by *y _{i}**
(we use both

By choosing

j = i, it follows thatlor that eigenvalues of the Sturm-Liouville problem are real._{j}= l_{j}*By choosing

jdiffers fromi, it follows that the integral of the productyover the interval is zero: the eigenfunctions are orthogonal._{i}(x)*y_{j}(x)

Name, notation |
Applications |
Formula or Property |

Legendre, P or _{n}(x)P_{n}(cosq) |
Laplace's equ. in spherical polar coordinates, multipole expansion, spherical harmonics, quantum rotator |
Rodrigues formula, Generating function, Examples, Recurrence Relations, Differential Equation, DE in Sturm-Liouville form, Orthonormality |

Hermite, H_{n}(x) |
quantum oscillator | Rodrigues formula, Generating function, Examples, Recurrence Relations, Differential Equation, DE in Sturm-Liouville form, Orthonormality |

Laguerre, L_{n}(r) |
hydrogen atom radial factor | Rodrigues formula, Generating function, Examples, Recurrence Relations, Differential Equation, DE in Sturm-Liouville form, Orthonormality |

Bessel's function of integral order, J_{n}(r) |
Laplace's equ. in cylindrical coordinates | Series Expansion, Generating function, Differential Equation, DE in Sturm-Liouville form, Orthonormality |

Circular functions, sin(nq), cos(nq) |
classical oscillator | Series Expansion, Generating function, Differential Equation, DE in Sturm-Liouville form, Orthonormality |

- F. W. Byron & R. W. Fuller,
*Mathematics of Classical and Quantum Physics*[Addison-Wesley, 1970] - J. D. Jackson,
*Mathematics for Quantum Mechanics*[W.A. Benjamin, 1962] - M. Abramowitz and I.A. Stegun,
*Handbook of Mathematical Functions*[Dover, 1965]

Created or up-dated 08/03/99
by R.D. Poshusta