# Weird Al Yankovic's Partial Differential Equation

In Weird Al Yankovic's music video "White and Nerdy" (1:20-1:36), there are flashes of a partial differential equation:

$$\left(-\frac{h^2}{2\mu}\nabla^2 - \frac{e^2}{r}\right)\psi(r)=E\psi(r)$$

Does anyone know what the name of this equation is?

This is the time independent Schrodinger equation for two elementary charges with a Coulomb interaction.

• Is this formulated as an eigenvalue problem?
– Paul
Mar 20, 2014 at 22:44
• Well the Schrödinger equation is a linear PDE so it always reduces to an eigenvalue problem. The fact that the equation has countably many solutions (eigenfunctions) so long as the electron is bounded to the proton in turn corresponds intuitively to the concept of quantization of energy $E$ and other physical properties. Mar 20, 2014 at 22:57

It's Schroedinger's equation for the Hydrogen atom with units chosen so that $4 \pi \epsilon_0 = 1$. See "Hydrogen atom" at the English Wikipedia's article on the Schrodinger Equation.

• Interesting. Does the expression $4\pi\epsilon_0=1$ mean anything, physically?
– Paul
Mar 20, 2014 at 23:16
• In the Coulomb force law, the constant is $k = \frac{1}{4 \pi \epsilon_0}$, where $\epsilon_0$ is the permittivity of free space. See en.wikipedia.org/wiki/Coulomb%27s_law#Coulomb.27s_constant and its referents. Mar 20, 2014 at 23:19