Understanding the Role of the Dirac Delta Function in Multipolar Polarization

[scarecrow]

Homework Statement

Using the explicit expression for the mulitpolar polarization, find the matrix elements <2s | P(r) | 1s> of the microscopic polarization between the 1s and 2s states of the hydrogen atom.

Homework Equations

P(r) = \int dr&#039; r&#039; \rho(r&#039;) \delta(r-r&#039;)

I don't understand how the Dirac Delta Function is supposed work? What it is and how does it operate in the intergral?

The Attempt at a Solution

&lt;\psi(2s) | P(r) | \psi(1s) = \int dr&#039; r&#039; \rho(r&#039;) \delta(r-r&#039;)

\psi(1s), \psi(2s) are just H-atom wavefunctions.

 

[JeffKoch]

I'm not familiar with this operator, but just looking at the equation for P(r) I would read it as rho(r)*r. The delta function restricts the integrand to the point where r' = r, and it's equal to zero for other values of r'.

 

[VChernyak]

This is easy!