Chapter 9: Separation of Variables

### Important PDEs in Physics

On this page you will find a list of most of the important PDEs in physics with their names. Notice that some of the equation have no time dependence, some have a first order time derivative, and some have a second order time derivative. This difference is the foundation of an important classification scheme. Also notice that the spatial dependence always comes in the form of the laplacian $\nabla^2$. This particular spatial dependence occurs because space is rotationally invariant.

Laplace's Equation: Many time-independent problems are described by Laplace's equation. This is defined for $\psi=\psi(x,y,z)$ by:

$$\nabla^2\psi={\partial^2\psi\over \partial x^2}+{\partial^2\psi\over \partial y^2}+{\partial^2\psi\over\partial z^2}=0.$$

The differential operator, $\nabla^2$, defined by eq.(1) is called the Laplacian operator, or just the Laplacian for short. Some examples of Laplace's equation are the electrostatic potential in a charge-free region, the gravitational potential in a matter-free region, the steady-state temperature in a region with no heat source, the velocity potential for an incompressible fluid in a region with no vortices and no sources or sinks.

Poisson's Equation:

Poisson's equation is like Laplace's equation except that it allows an inhomogeneous term, $f(x,y,z)$, known as the source density. It has the form:

$$\nabla^2\psi=f(x,y,z)$$.

Schrödinger's Equation:

A great deal of quantum mechanics is devoted to the study of the solutions to the time-dependent Schrödinger equation:

$$-{\hbar^2\over 2m}\nabla^2\psi+V(x,y,z)\psi=i\hbar{\partial\psi\over\partial t}.$$

This equation governs the time dependence of the wave-function of a particle moving in a given potential, $V(x,y,z)$. A special role is played by solutions to (4) that have the simple form: $\psi=\phi(x,y,z)\exp(-iEt/\hbar)$. The function $\phi$ satisfies the time-independent Schrödinger equation or, more correctly, the Schrödinger eigenvalue equation:

$$-{\hbar^2\over 2m}\nabla^2\phi+V(x,y,z)\phi=E\phi.$$

In both of these equations $\hbar$ and $m$ represent real constants. $E$ is a constant that emerges during the separation of variables procedure. $i$, as usual, satisfies $i^2=-1$.

The Diffusion Equation:

The pde governing the concentration of a diffusing substance or the non-steady-state temperature in a region with no heat sources is the diffusion equation:

$${\partial\psi\over\partial t}-\kappa\nabla^2\psi=0.$$

$\kappa$ is a real constant called the diffusivity.

The Wave Equation:

Wave propagation, including waves on strings or membranes, pressure waves in gasses, liquids or solids, electromagnetic waves and gravitational waves, and the current or potential along a transmission line all satisfy the following wave equation:

$$-{1\over v^2}{\partial^2\psi\over\partial t^2}+\nabla^2\psi=0.$$

The real constant $v$ can be interpreted as the speed of the corresponding wave.

The Klein-Gordon Equation:

Disturbances traveling through fields that mediate forces with a finite range, satisfy a modification of the wave equation called the Klein-Gordon equation. It is given by:

$$-{1\over c^2}{\partial^2\psi\over\partial t^2}+\nabla^2\psi+{m^2c^2\over\hbar^2} \psi=0.$$

The coefficients $c$, $\hbar$ and $m$ all represent constants.

Helmholtz's Equation:

The equation:

$$\nabla^2\psi+k^2\psi=0$$

is known as Helmholtz's equation and arises as the time-independent part of the diffusion or wave equations. $k$ is a constant that emerges during the separation of variables procedure.