Separable partial differential equation

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A separable partial differential equation (PDE) is one that can be broken into a set of separate equations of lower dimensionality (fewer independent variables) by a method of separation of variables. This generally relies upon the problem having some special form or symmetry. In this way, the PDE can be solved by solving a set of simpler PDEs, or even ordinary differential equations (ODEs) if the problem can be broken down into one-dimensional equations.

(This should not be confused with the case of a separable ODE, which refers to a somewhat different class of problems that can be broken into a pair of integrals; see separation of variables.)

For example, consider the time-independent Schrödinger equation

[-\nabla^2 + V(\mathbf{x})]\psi(\mathbf{x}) = E\psi(\mathbf{x})

for the function \psi(\mathbf{x}) (in dimensionless units, for simplicity). (Equivalently, consider the inhomogeneous Helmholtz equation.) If the function V(\mathbf{x}) in three dimensions is of the form

V(x,y,z) = V1(x) + V2(y) + V3(z),

then it turns out that the problem can be separated in to three one-dimensional ODEs for functions ψ1(x), ψ2(x), and ψ3(x), and the final solution can be written as \psi(\mathbf{x}) = \psi_1(x) \cdot \psi_3(x) \cdot \psi_3(x). (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.[1])

[edit] References

  1. ^ L. P. Eisenhart, "Enumeration of potentials for which one-particle Schrodinger equations are separable," Phys. Rev. 74, 87-89 (1948).
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