Ch 5: What are Dirac deltas and wavefunction inner products? | Maths of Quantum Mechanics

Hello!

This is the fifth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll dig into what the Dirac delta is and how we should understand it as physicists. We'll then use it to show where the inner product for wavefunctions comes from.

If you have any questions or comments, shoot me an email at:
Thanks!

Animations:
All animations created by me within Python, using Manim. To learn more about Manim and to support the community, visit here:

Music:
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♪ 35 by HOME
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[quantumsensechannel]

By the way, if anyone would like to try and prove that sin(x/a)/(pi*x) is a Dirac delta in the limit, here’s some intuition into how we would prove that (and a quick mathstackexchange search reveals a multitude of rigorous proofs).

You might have seen that the Dirac delta is equal to the derivative of the Heaviside Step Function H(x), in the sense that it satisfies the integral property of picking out a certain function value. This is pretty easy to prove using integration by parts. So in other words, the *antiderivative* of a Dirac delta is a step function. Keep this in mind.

So here’s the idea: instead of proving that sin(x/a)/(pi*x) approaches a delta function in the limit, try proving that its antiderivative approaches a step function. Now you might notice that sin(x)/x has no elementary antiderivative, but that’s ok, just write the antiderivative as the integral from -inf to x of the limit (which you remember is the antiderivative via the fundamental theorem of calculus, right?). Now, show that taking the limit gives you a step function (this is the hardest part, but use the fact that the integral from -inf to +inf of sin(x)/x is equal to pi).

Great! Now to prove that sin(x/a)/(pi*x) is a Dirac delta: integrate a function against it, write the limit as the derivative of its anti derivative, you just showed that in the limit the antiderivative is a step function, so now you have the integral of the derivative of a step function times your function, which is easy to show gives you the Dirac property.

Now this isn’t the most rigorous proof, but it gives you an idea of what’s going on. In a way, what matters most about a Dirac delta is the fact that its integral equals the step function, even if the way we get to that step function is weird (like with sin(x/a)/(pi*x)). And a really cool way to “construct” Dirac deltas is to start with a function that approaches a step function in the limit, then take its derivative.

Anyway, this was a bit of a long comment, but I figured you might appreciate an outline of why this weird limit gives us Dirac behavior. Let me know if you have any questions!

See you all next chapter!
-QuantumSense

[BRORIGIN]

Honestly, I am shocked by how useful this series is for both experienced and inexperienced physicists.

Though I know all these individual aspects of QM which you are presenting, I have never seen them sown together to form such a seemless and compact narrative. I
It is also visually stunning! Great job! If you keep this up, I can see you becoming the "3Blue1Brown of theoretical physics"!

P.s. I like that your backgrounds are dark, because I often watch videos late at night 😉

[jamesbentonticer]

YouTube has lots of math channels but very little good physics channels. You and 'Physics with Elliott' are top shelf. I know the videos take a long time to make with such richness and quality, but please keep them coming!

[Kevinfreddo]

I’m so excited for this series to continue. Thanks for the rapid upload

[anumanful]

Glad to see that you are keeping this project alive!

[higjiidghjk]

There are channels on education with high quality content but no consistency and low quality with high consistency but both only I saw only with this channel

[it]

0:00-Recap
0:38-Dirac Delta extraction
2:06-Nature of Dirac Delta
4:33-Problems with spike interpretation
7:34-Continuous orthonormal basis inner product

[qtmeet]

Loving the quality, depth, and ofc upload interval. Very excited to see the rest of the series :)

[jaimelima]

I found this series yesterday, 4 AM, 3 F local temp in NY… Really usefully… Full of insights that may surprise many people studying this for years, including me.

[ChildOfTheLie]

This is series is really nice, I think its all starting to come together

[garibaymolinacarlos]

Awesome. I really look forward to the next video.

[zokalyx]

Everything in the video (content, clarity, audio, visuals) is perfect. Honestly I have no suggestions to make.

[yudoball]

Oh my God
After so many years it finally clicked in my head. (2:42)
Thank you for the awesome explanation and visuals

[jossarian]

The δ function is a functional which maps each function f(x) onto the number f(0).
The δ function is not an element of a Hilber Space but it may act on Hilbert Space vectors if the Hilbert Space is defined as a set of square integrable functions.

[anguskeesbury]

Great video! I've not seen much that aptly explains this, been brushing up on quantum mechanics recently and this is a real gem! Wish I had this when I was learning it...

[filipo]

I recommended this series to my professor :P
Very good work!

[quantumeveryone]

I think to be consistent with the notation you have been using you should have either put the psi(x) inside the bra, or, having it outside have made it a conjugate. The way you presented it, the conjugate appeared out of nowhere.

[bharatjoshi]

“If you can explain what you have learned …you are SPECIAL !! “That’s what you are brother

[firstlast-qyxn]

Im in the last semester of my bachelors in physics currently. Your video series gives many questions I had that I found some loose answers to a more well-formulated answer, and I truly appreciate it!
I have some more questions I asked myself, that would maybe be worth being turned into videos:
Why are commutators important?
Why can two commuting operators be diagonalized simultaneously?
Why should the angular momentum operator be defined the way it is?

[faisalsheikh]

Phenomenal explanation keep it up sir love you from India🇮🇳