Integral Involving the Dirac Delta Function

[gabriellelee]

Homework Statement

What is the integral of the following equation?

Relevant Equations

See below for the equation.

 

[Abhishek11235]

What are your limits of integral?

 

[gabriellelee]

What are your limits of integral?

-∞ to +∞

 

[Abhishek11235]

-∞ to +∞

So what do you think?

 

[gabriellelee]

So what do you think?

1?

 

[Abhishek11235]

1?

Yes

 

[gabriellelee]

Yes

I know intuitively that the integral is 1, can you explain it to me mathematically?

 

[PeroK]

I know intuitively that the integral is 1, can you explain it to me mathematically?

By definition:
$$\int_{-\infty}^{+\infty}\delta(x) f(x) dx = f(0)$$

 

[wrobel]

Actually it is not an integral it just a symbol. As well as $\delta-$function is not a function in usual sense. These things are clarified in functional analysis

 

[vanhees71]

Hm, is this an exercise in a physics course or a trick question in a mathematics course? In the latter case, I think, the answer is that the integral is undefined, because the exponential function is not a proper test function in the domain (of, e.g., Schwartz functions), the Dirac distribution is defined on.

 

[wrobel]

There is no sense to speak about integrals here. It is nothing more than symbol in this context.
A space of test functions is a conditional question. In different problems this space is introduced differently this depends of problem we consider . Loosely speaking, the smaller space of test functions the bigger space of generalized functions.

For example, consider a subspace of $\mathcal D'(\mathbb{R})$ that consists of generalized functions with compact support. Such generalized functions are naturally defined on $C^\infty(\mathbb{R})$.

 

[vanhees71]

I'm only a bit familiar with the formalism, but I don't think that $\exp(\mathrm{i} x)$ is a proper test function in any case. I'm a bit unsure, whether the Dirac $\delta$ function (which has compact support, namely only 1 point) is defined on the entire $C^{\infty}(\mathbb{R})$. Isn't it usually either $C_0^{\infty}(\mathbb{R})$ or the "Schwartz space of quickly falling functions"?

 

[wrobel]

What is the problem to define the Dirac δ function on $C^\infty(\mathbb{R})$?
Just put
$$(\delta, \varphi):=\varphi(0),\quad \varphi\in C^\infty(\mathbb{R})$$
This is a continuous linear functional with respect the standard topology in $C^\infty(\mathbb{R})$

 

[vanhees71]

I don't know. Maybe you can do that, but in quantum mechanics you often have confused students, because they think $\exp(\mathrm{i} p x)$ is a "eigenstate of the momentum operator", but it is not square integrable nor in the domain of the momentum operator as an essentially self-adjoint operator. Here you need the rigged-Hilbert space description!