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.

 

[osilmag]

Correct me if I'm wrong, how I understand Dirac w.r.t. QM is that it combines the distribution of values of a quantum variable into a single value.

 

[TensorTronic-270]

It wasn’t until the 1940s–1950s that Laurent Schwartz developed the theory of distributions (also called generalized functions), which gave the Dirac delta a proper home. In distribution theory:

• The Dirac delta is not a function—it is a distribution (a continuous linear functional on the space of test functions).
• It is formally written δx = ϕ(0) for any test function ϕ.

This rigorously justifies everything Dirac was doing intuitively 20–25 years earlier.

 

[renormalize]

• It is formally written δx = ϕ(0) for any test function ϕ.

What is this notation? Shouldn't it be $\delta [\phi]=\phi (0)$?

 

[TensorTronic-270]

At the time (1930), this δ-function was not mathematically rigorous. Pure mathematicians were horrified because no ordinary function can be zero everywhere except at one point, blow up to infinity there, and still have integral 1. It violated the standard rules of functions and integrals.

Dirac was a physicist—he didn’t care. He said (paraphrased): “It’s a convenient notation, and it gives the right physical answers, so we’ll use it.” Engineers and physicists adopted it enthusiastically because it made calculations in quantum mechanics, electrodynamics, signal processing, etc., vastly simpler, lolz.

Physics Boy was thumbing his nose at Math Boy, I'm Theory Boy laughing.

 

[renormalize]

Physics Boy was thumbing his nose at Math Boy, I'm Theory Boy laughing.

But that doesn't address my questions. Could you please show me a simple curtesy by clarifying the meaning of your notation?

 

[TensorTronic-270]

But that doesn't address my questions. Could you please show me a simple curtesy by clarifying the meaning of your notation?

I did, the notation was Dirac's and its not used anymore.

 

[Dale]

I did, the notation was Dirac's and its not used anymore

You really didn’t clarify the notation. The fact that the notation is Dirac’s and is not used anymore is a bit of history, not a clarification about the notation. Similarly with the facts that it is not rigorous, that mathematicians were horrified, and all of the rest.

Of course, there is no requirement for you to actually engage in productive conversation with other people. If you just like writing to yourself, you are free to do that.

But @renormalize is right, you did not in fact clarify the notation.

 

[renormalize]

I did, the notation was Dirac's and its not used anymore.

Can you supply a reference in which Dirac explicitly writes the equation $\delta x=\phi (0)$? My objection to it is that $\phi$ appears on the right-side and yet is absent on the left.

 

[TensorTronic-270]

Oh I see, you want all of it.
In 1930, Paul Dirac published his influential book The Principles of Quantum Mechanics, where he needed a mathematical tool to elegantly describe certain physical situations in quantum theory—particularly the idea of a state with a perfectly sharp, definite position or momentum (something that doesn’t actually exist in reality but is extremely useful as an idealization).

To do this, he introduced what he called a “delta function”, denoted δ(x), with the following very unusual properties:

1. It is zero everywhere except at x = 0, where it is infinite.
2. Its integral over the entire real line is exactly 1:∫_{-∞}^{∞} δ(x) dx = 1
3. For any (sufficiently well-behaved) function f(x), it “picks out” the value of f at zero:∫_{-∞}^{∞} f(x) δ(x − a) dx = f(a)

Dirac explicitly described this δ as a continuum analogue of the Kronecker delta δ_{ij}, which you already mentioned is the components of the identity matrix:

• δ_{ij} = 1 if i = j, and 0 otherwise.In finite-dimensional linear algebra, multiplying a vector by the identity (or summing with Kronecker delta) just gives the vector back—i.e., it “picks out” components.

Dirac wanted the same kind of “sifting” or “selection” property, but in the continuous case. So the Dirac delta does for functions what the Kronecker delta does for vectors with discrete indices.

 

[renormalize]

Oh I see, you want all of it.

Yes I do and you still haven't directly answered my question: where in Principles of Quantum Mechanics (or elsewhere) does Dirac explicitly write $\delta x=\phi (0)$ (or its algebraic equivalent)? (Chapter, page, equation number, please.)
Or did you make up that equation?

 

[TensorTronic-270]

I dont write equations, I'll have to go find the paper, its somwhere around...

 

[TensorTronic-270]

Here it think, https://faculty.washington.edu/seattle/physics441/online/about Dirac.pdf Page 398 or hunt down that pesky P.A.M. Dirac, The Principles of Quantum Mechanics, First edition, Oxford University Press, 1930(Specifically §21 “The δ function”, pages 58–61 in the 1st edition; in the 4th edition (1958) it’s §15, pages 56–59) the location is allusive.

 

[renormalize]

the location is allusive.

It's elusive because neither of those citations support your claim.
Here's an excerpt from pg. 398 of your first reference:

These are the only mentions of $\delta$ on that page and clearly have nothing to do with your claimed equation in post #8.
Further, in my copy of Principles of Quantum Mechanics (4th ed. rev.) Dirac writes:

This is the only place in Dirac's $§15$ that displays an equation akin to your $\delta x=\phi (0)$. Using your notation, eq.(3) becomes$$\intop_{-\infty}^{\infty}\phi\left(x\right)\delta\left(x\right)dx=\phi\left(0\right)\tag{3*}$$Clearly the left side of eq.(3*) bears no resemblance to your "$\delta x$".
I assert therefore that the equation you've written in post #8 is simply wrong. That's why I've repeatedly asked you to either explain your notation or else cite references that use that notation. You've managed to do neither, which is a disservice to current and future readers of this thread who expect to find accurate content on Physics Forums.

 

[TensorTronic-270]

I cut and pasted, I dont write these sort equations. I'll attempt find the paper otherwise please ignore.

 

[TensorTronic-270]

I'm working on a Trinary Operation System with Dirac's work at the core of it. I saw your post, cut something from a paper I thought looked useful.

 

[renormalize]

I cut and pasted, I dont write these sort equations.

Hopefully you aren't just copying your equations and citations from AI responses.

 

[TensorTronic-270]

I deal in Papers, like: Brûlart de Léon’s Cipher Disk and Treatise on Cipher , S. TomokiyoVer. 1 posted at Academia.edu on 24 July 2021, Shape Analysis of Euclidean Curves under Frenet-Serret FrameworkPerrine Chassat11LaMME, University of Paris-Saclayperrine.chassat@univ-evry.frJuhyun Park1,22ENSIIE, Evryjuhyun.park@ensiie.frNicolas Brunel1,2,33Quantmetry, Parisnicolas.brunel@ensiie.fr

 

[TensorTronic-270]

FRENET-SERRET FORMULAS OF q-CURVESDENIZ ALTUN* and SALIM YÜCEDepartment of MathematicsFaculty of Arts and SciencesYildiz Technical University34220 IstanbulTurkeye-mail: deniz.altun@outlook.com, I could go on all day!

 

[renormalize]

I deal in Papers, like:

OK, I'll await your specific citation(s) supporting $\delta x=\phi(0)$. Until then I remain skeptical for reasons already explained.

 

[TensorTronic-270]

I will find the paper , You asked What Exactly is Dirac’s Delta Function? I was working on something similar and I cut and pasted, and action I'm now regretting. it won't happen again.

 

[TensorTronic-270]

Right, Théorie des distributions" (published in 1950) bugger if I know where my file is.

The requirements for the Dirac Delta function, δ(x), are that it satisfies the following two properties:
1. The Normalization Property (Area is 1)
∫−∞∞δ(x)dx=1

2. The Sifting Property (The Selection Rule)​

∫−∞∞f(x)δ(x−a)dx=f(a)