Dirac Delta Function vs probability distribution

[pibomb]

Hello,

What is the dirac delta function and how is it different from a probability distribution?

 

[leright]

A dirac delta function is a function with zero width and infinite height with an area of 1. A probability distribution also has an area of one but, unlike the delta function, it is finite and less than one everywhere.

 

[selfAdjoint]

However the Dirac delta is indeed a distribution, not a function as Dirac named it. In practice this sort of means that it lives inside the integral sign. For example $$\int_0^{\infty} \delta(x-a) dx = a$$ is perfectly well defined.

 

[ريمان]

It's can nomrmalize the wave Function
we write this Function by Fourier transform

 

[Rach3]

A Dirac Delta is a mathematical object; it's a legitimate probability distribution as well (just like a Gaussian is a function, but can also be applied as a probability distribution). It represents the trivial probability distribution, with a nonzero probability for exactly one outcome.

 

[leright]

A Dirac Delta is a mathematical object; it's a legitimate probability distribution as well (just like a Gaussian is a function, but can also be applied as a probability distribution). It represents the trivial probability distribution, with a nonzero probability for exactly one outcome.

it isn't a legitimate probability distribution because it has a value of infinity at the non zero point.

 

[Hurkyl]

it isn't a legitimate probability distribution because it has a value of infinity at the non zero point.

He said probability distribution, not probability density function.

 

[pibomb]

If the function has zero width than does that mean it yields only one probability? Doesn't this disagree with quantum laws?

 

[selfAdjoint]

If the function has zero width than does that mean it yields only one probability? Doesn't this disagree with quantum laws?

When you act on the state of a quantum observable with a Hermitian operator (representing an observation), the eigenvalue spectrum of the operator is a set of real numbers which can be normalized to represent probabilities, and the Dirac delta will have played its part in computing each of them.

The probabilities are interpeted to be the probability of each accompanyng eigenstate being actualized. Just one of these probabilities is actualized for observation. This is the 'state reduction" or "collapse of the wave function" that they talk about.

And BTW, there is some confusion in this thread about distributions. SOME distributions are of the form f(x)dm, for some measure dm, and SOME of these are probability distributions, and SOME have 0 < f(x) < 1 for all x. But there are cases of the last where f(x)dx has no moments of any order between 0 and infinity, and its integral from 0 to infinity, though positive and less than 1 for each x, does not exist. The Cauchy distribution is like that, in fact Cauchy thought it up just to confound sloppy thinking on these issues. It really is necessary to have at least a nodding acquaintance with measure theory, if not Schwartzian distribution theory, before we expound on them.

 

[selfAdjoint]

If the function has zero width than does that mean it yields only one probability? Doesn't this disagree with quantum laws?

When you act on the state of a quantum observable with a Hermitian operator (representing an observation), the eigenvalue spectrum of the operator is a set of real numbers which can be normalized to represent probabilities, and the Dirac delta will have played its part in computing each of them.

The probabilities are interpeted to be the probability of each accompanyng eigenstate being actualized. Just one of these probabilities is actualized for observation. This is the 'state reduction" or "collapse of the wave function" that they talk about.

And BTW, there is some confusion in this thread about distributions. SOME distributions are of the form f(x)dm, for some measure dm, and SOME of these are probability distributions, and SOME have 0 < f(x) < 1 for all x. But there are cases of the last where f(x)dx has no moments of any order between 0 and infinity, and its integral from 0 to infinity, though positive and less than 1 for each x, does not exist. The Cauchy distribution is like that, in fact Cauchy thought it up just to confound sloppy thinking on these issues. It really is necessary to have at least a nodding acquaintance with measure theory, if not Schwartzian distribution theory, before we expound on them.

 

[loom91]

Why can't it be a PDF? A PDF is defined as a function f(x) such that the probability of variate X lying between a and b is equal to $$P(a<=X<=b) = \int_a^b f(x) dx$$ and $$\int_{\infty}^{\infty} f(x)dx = 1$$. As was previously said, it lives inside integrals. Why doesn't the Dirac Delta satisfy this definition?

 

[Hurkyl]

Why can't it be a PDF?

You answered your own question: (I've added emphasis)

A PDF is defined as a function f(x) ...

 

[loom91]

You answered your own question: (I've added emphasis)

Why can't it be a function?

 

[Hurkyl]

Short answer: what is $\delta(0)$?

 

[loom91]

Short answer: what is $\delta(0)$?

Hmm, I get the picture. If it's a measure, what is the measure assigned to {0}?

 

[adjoint+]

Hello,

Can someone please explain to me how to do the following:

2/a * integrate sin^2(n*pi*x/a)*delta(x-a/2)dx from 0 to a.


The delta function is defined for the limit -infinity to infinity, right? How do I do this one? The limits of integration in this case is what I'm finding confusing.

I did this question assuming the result is the same as for the limits from -infinity to infinity but I don't understand the logic behind.

Can someone please help?