What is the Interpretation of the Inverse Property of a Delta Function in QM?

In summary, the conversation discusses a property of the delta function as listed in a book on quantum mechanics. The property in question is x \delta^{-1}(x) = - \delta(x), which was initially interpreted as the inverse of the delta function but was later confirmed to be a typo and actually meant to be x \delta^\prime = -\delta. The conversation also mentions that the inverse of the delta function is not a meaningful operation in regular distribution theory and that the correct identity can be shown using integration by parts.
  • #1
Peeter
305
3
In a book on QM are listed a few properties of the delta function, one of which is:

[tex] x \delta^{-1}(x) = - \delta(x) [/tex]

I can't figure out how to interpret that? Putting the statement in integral form isn't particularily enlightening looking:

[tex]
f(x) = \int f(x-x') \delta(x') dx' =
\int -x' \delta^{-1}(x') f(x - x') dx'
[/tex]

any hint what this property is about or how one would show it?
 
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  • #2
I'm not aware of any meaningful operation corresponding to the "inverse of the delta function" in regular distribution theory and even intuitively speaking, it doesn't seem to be meaningful. Are you sure it's not actually:

[itex]x\delta^\prime = -\delta,[/itex]

which is a meaningful and true identity?
 
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  • #3
Thanks Preno. Your statement makes sense (ie: can show it with integration by parts).

I'm pretty sure it was listed as ^{-1}, but will have to wait til I'm home to verify. It's probably another typo in the text.

Confirmed. Just another typo in the text.
 
Last edited:

Related to What is the Interpretation of the Inverse Property of a Delta Function in QM?

What is the inverse of a delta function?

The inverse of a delta function is a function that, when convolved with the delta function, results in a Dirac delta function.

What is a Dirac delta function?

A Dirac delta function is a mathematical function that is zero everywhere except at one point, where it is infinite. It is often used to represent a point source in physics and engineering.

Why do we need the inverse of a delta function?

The inverse of a delta function is useful in solving convolution integrals, which are commonly used in signal processing, image processing, and other areas of science and engineering.

What is the mathematical representation of the inverse of a delta function?

The inverse of a delta function is represented as 1/δ(x), where δ(x) is the Dirac delta function.

Are there any practical applications of the inverse of a delta function?

Yes, the inverse of a delta function has practical applications in fields such as image deblurring, image reconstruction, and signal filtering.

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