Understanding Dirac Delta Squares: Clarifying Doubts

[hermitian]

hi,

may someone help me to clarify my doubts...

in my work, i encounter diracdelta square $$\delta(x-x_1)\delta(x-x_2)$$ i am not sure what it means... it seems if i integrate it

$$\int dx \;\delta(x-x_1)\delta(x-x_2) = \delta(x_1-x_2)$$ is either zero of infinity.

is this correct?

thanks

 

[CompuChip]

Technically, saying that it has "value zero or infinity" doesn't make sense. Any Dirac delta only makes sense under an integral sign (although in physics, we tend to think of it as being an "infinite spike with a finite area").

It is correct that
$$\int dx \;\delta(x-x_1)\delta(x-x_2) = \delta(x_1-x_2)$$

So, again, this expression again only makes sense inside an integral, like
$$\int dx_1 \int dx \; \delta(x - x_1) \delta(x - x_2) = \int dx_1 \; \delta(x_1 - x_2)$$
which is one or zero (depending on whether or not x₂ lies in the integration interval of the x₁ integral).

 

[centry57]

Here are some more information：http://en.wikipedia.org/wiki/Dirac_delta_function