Insights What Exactly is Dirac’s Delta Function? - Insight

[Greg Bernhardt]

TL;DR

Dirac introduced the delta function in 1930 as a continuum analog to the Kronecker delta.

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles.

In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra

Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/

by @jambaugh

 

[zerodish]

TL;DR Summary: Dirac introduced the delta function in 1930 as a continuum analog to the Kronecker delta.

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles.

In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra

Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/

by @jambaugh

Viewed as a mathematical object it is simply a function where the value is 1 at 0 and 0 every where else. The idea that you can take the derivative or intergal of this monster is interesting. I would say it is a degenerate function like a line segment could be viewed as a degenerate ellipse where one of the axis is 0. As to how it is used in quantum mechanics I have no idea.

 

[renormalize]

Viewed as a mathematical object it is simply a function where the value is 1 at 0 and 0 every where else.

At what value of $x$ does the Dirac delta function $\delta\left(x\right)$ equal to $1\,$?

 

[weirdoguy]

Viewed as a mathematical object it is simply a function

No it is not (it's functional) and I don't see why you try to make up your own definitions when this is a completly understood and formalized topic. I haven't read the insights but I guess it goes into details.

 

[DEvens]

Oh, I didn't notice the insite article. How do I delete a posting?

 

[QuarkyMeson]

Viewed as a mathematical object it is simply a function where the value is 1 at 0 and 0 every where else. The idea that you can take the derivative or intergal of this monster is interesting. I would say it is a degenerate function like a line segment could be viewed as a degenerate ellipse where one of the axis is 0. As to how it is used in quantum mechanics I have no idea.

It's a distribution, or continuous linear functional.

You're quoting the insight but didn't actually read it. The insight itself would shed light on why what you're saying is misguided and the wrong way to think about things. Also I don't know what a degenerate function is, but it is a degenerate probability distribution.

The biggest use cases in QM for the new student (like me) are that continuous eigenstates aren’t normalizable with Kronecker deltas, so $<x|x'> = \delta(x-x')$, which is the continuous analogue of orthonormality (same idea in other continuous bases like p) and for completeness $\int |x><x| dx = \mathbb{1}$. There are others, but these are probably the two most common use cases.

Most students will have seen the delta function before QM in EM, where it has many uses in dipoles, line/surface charges/currents/ and more.