The Theory that Solves "Unsolvable" Quantum Physics Problems - Perturbation Theory
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- Опубликовано: 4 июн 2024
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Sometimes, certain problems in quantum mechanics become unsolvable due to their mathematical complexity. But we still have techniques for approximating their solutions! One such technique is perturbation theory - let's see how we can use it. #perturbation #quantum #approximation
To begin this video, we will look at how we study quantum physics problems in the first place. We recall that every system has an associated wave function. For example if our system is an electron in space, then the wave function of that electron will give us the likelihood of finding the electron at different points in space. This is discussed in more detail in my wave functions video!
But how do we actually find the wave function of a system? Well, we have to solve the Schrodinger equation of course! This is the governing equation of the theory of quantum mechanics, and we plug in information about our system (such as kinetic energy and potential energy or potential well of the system), in order to solve for the allowed wave functions. Specifically, we plug the information about the system into the Hamiltonian of the Schrodinger Equation.
If we know how to solve the Schrodinger equation once we plug in the system's properties, then we can calculate the allowed wave functions (and energy levels) of the system. The energy levels are of course discrete rather than continuous, which is what is referred to as quantization.
But what happens when we cannot solve the Schrodinger equation for a given system? What if we don't have enough mathematical skills or techniques to solve a particular differential equation? One way to solve such problems is numerically, using a computer. And what about if we don't have a computer?
In such scenarios, physicists have developed some clever techniques to find approximate solutions to our equation. One such technique is perturbation theory. It works best for systems that are very close to other systems that we DO know the solutions for. In this scenario, the phrase "very close" means the new system can be described as the original system plus some small change. The example used in this video is the addition of a small dirac delta function (spike) in the middle of a square potential well.
Then, the new system's Hamiltonian can be written as the old system's Hamiltonian plus some small change. Usually we also multiply the new / added small change by a factor lambda, that helps us in our upcoming mathematical steps. Lambda takes values between 0 and 1 as we go from the unperturbed, original system (lambda = 0) to the perturbed, new system (lambda = 1).
We can then say that the new system's allowed wave functions are equal to the old system's wave functions plus a small term proportional to lambda, plus a smaller term proportional to lambda squared, and so on. This forms an infinite series of "corrections" to the original wave function. We don't have time to calculate infinitely many terms, but luckily for most situations just the first new term is enough. And exactly the same logic applies for energy levels.
Luckily, the first order correction just depends on the change between the old and new systems, and the wave functions of the old system. And nothing else. The first order energy level correction is something we know how to calculate, meaning we don't have to deal with an "impossible" differential equation whilst still getting a very good approximation.
And this is why perturbation theory is a very valuable technique for solving (or at least approximating) "impossible" to solve quantum mechanical systems.
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Timestamps:
0:00 - How Problems are Solved in Quantum Mechanics (Wave Functions, Schrodinger Eqn)
3:12 - Energy Levels and Wave Functions for Quantum Systems
4:53 - Perturbation Theory (for a Perturbed System)
6:30 - Sponsor Message (and magic trick!) - big thanks to Wondrium
8:55 - Approximating the new Wave Functions and Energy Levels
10:00 - First Order Approximation - EASY!
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As always, let me know what other topics to cover in future videos :)
This was a very good video. Perhaps you could do a follow-on video using the perturbation ideas on a concrete example. It would be cool to see how one could actually numerically solve one of these using only a scientific calculator. Thanks!
This is so wierd. I feel like we're telepathically linked since you keep making videos about physics phenomena like a week after I discover them and you've kept on doing that for about a year now.
Much love, Parth
It should be noted that this theory was NOT created to solve "Impossible" quantum physics problems without a computer. It was created to solve "impossible" three (and more) body problems in planetary physics by Laplace and Lagrange in the 18th and 19th centuries.
Also, an excellent introduction to Perturbation Theory, and other mathematical methods, can be found in Carl Bender's excellent videos on RUclips.
Carl Bender's lectures are great.
Thanks for info
Yes, please make the video about expectation values.
Yes please continue! I learn this in class in about 2 weeks and would love more of a head start with your explanations
Awesome! Will do :)
I'd love to see a worked out example of the process of approximating wavefunctions with perturbation.
Loved the video thank you so much for your content it is always really insightful and different from what a lot of other physics youtubers do.
What's most impressive to me with your magic trick. Is that you're able to shuffle the deck consistently perfectly.
Thanks for bringing a little bit of intuition back to a physics undergrad student, that quickly gets lost in the math :)
Thanks so much for this simple, understandable explanation.
Please more about this! Very good. Love from Portugal.
Dear Parth, thanks for explanation and i need more detail about the terme of expectation
Excellent video sir,!
So good explanation please continue
Great videos!! Question: how about the Dirac Equation?
Good work man🙌👊
That is some amazing explanation. This is coming from a machine learning perspective, the approx of wave and energy functions seems very similar to Taylor series. You think they are connected?
So cool! I'm actually carrying out a report on this exact topic as an assignment for my exams. The aim is to implement the variational principle in a computer program simulating a particle in an infinite potential well (a situation we do know how to solve analytically), starting with a finite basis set of functions for the Hilbert space. The goal is to estimate the relative error of the approximated energy levels with respect to the exact ones, and how does it change as we increase the dimension of the basis set (the actual Hilbert space would be infinite dimensional, so as we increase the dimension of our finite basis set the approximation should become arbitrarily precise). This is more or less the concept behind more complex simulations, such as molecular orbitals and so on. Sooo spicy :)
I always hit thumbs up button before watching the video . because I know video is going to be great!
Brilliant… thank you … expectation value yes please!
Hi Parth... is it not a good idea to make a video on virtual displacement?
Great video
Great video and I’m impressed you did a faro shuffle
Yes we would love to see it
yes please continue on.
Thank you 💖
A good extra thingh that I see in college is that exist a perturbation theory for degenerate an non degenerate systems (i.d. with different states which share energy) and another theory for time dependence potencials, as the radiation. In fact, with this last theory using the quantum theory of radiation we can obtain that te probability of spontaneous emission of a photon is different of zero, which is one of the phenomena that interview in the laser effect.
yes , please make a video about expectation values and let us know about it.
Thanks, yes definitely will do!
When are you going to come up with antigravity and quantum loop gravity? When are we going to have warp fields?
I definitely have to brush up on my math. It would seem my rudimentary Understanding of equations require some real help. These equations look ominous.
This was perfect
You could try to make your camera full screen when addressing the audience and no graphics are shown, first time viewer and I was a bit confused at the start of the video
I would like to listen to your explanation of expectation value of the perturbation
6:00 why a lambda and also little vague on lambda amd first order change, second order change and so on.. maybe I was lost in terminology. Also 11:40 I would like to know a lot more
Please more of perturbation theory thank you!
Is the Delta of (a/2) you are using just delta in the sense of a small infinitesimal change or is it actually the Dirac delta distribution? Also was a/2 just a random having it be looking at halfway within the well model?
Yes, that's comes from variational calculus so it represents a small variation. It has nothing to do with Dirac delta.
It's like adception. I got a YT ad in the middle of your sponsor ad.
nice video
Hello sir, Can you tell me about Vibraium
One usually gets their first exposure to perturbation theory in Classical Mechanics, but the basic idea is the same.
Expect to see other applications in Mathematics, Electromagnetism, Statistical Mechanics, in addition to QM.
What will be the wave function for our universe? At big bang? or now?
Why is the denominator of E1 not one? =1 or not ?
11:35 yes!
I love your videos 💓💜❤️♥️🧡
Nice video and good music tasty, can you, me path, recomend me some of your favority musics and bands
I always found this a very interesting topic. Unfortunately, in contrast to most of your other videos, this felt rather shallow. I would be very interested in a more depth video or video series, ideally with an on hands example.
Hi firstly thanks for this 🤗🤗😍😍
Can you plz tell about good book about physics like suject quantum mechanics , classical physics, thermodynamic physics, mathematics and nuclear physics
Plz give me suggestions for which book is the best for graduation lavel
😍😍
try the book "nouredine zettili" it would probably solve your most of the doubts with solved numericals based on every theories.
There is a fantastic 15 video lecture series on perturbation theory taught by the incredible Mr Carl Bender for free right here on RUclips. You’ll find it if you search for Mathematical Physics by Carl Bender. It was mind-blowing stuff and i ended up even taking notes as if I was in the class with them. Perturbation theory and asymptotic series are truly wild.
Can you talk about er=epr please?
Watch Lenord Suskind's Video on it He's one of the Founder of the Idea.
In a nutshell it kind off suggests that Wormholes might help in explaining the "Spooky Action at a distance" phenomena of Entanglement. In a way it provides a basis for the Non local phenomena for Entanglement using the features within a Wormhole.
Now if you already don't know about "Wormholes" basics of "GR" and "Principals in QM (Especially Superposition and Entanglement) then you first need to look into those before your jump into EPR = ER.
This sounds very similar to reinforcement learning.
Reminds me of Taylor series
I am literally on my last semester and just got to perturbation theory in class. Please have my kids.
wow don't give up your kids
It's not long enough!
Physics and faro shuffles, who can ask for more!? Try the table faro shuffle if you need a reason to cry haha. Check out the concept of an anti-faro shuffle and I'll check out another of your videos!
Thanks for the recommendation, I will have to check these out! I can barely do the standard faro correctly though haha
What if we have several perturbations and the perturbations are significantly large?
and Time Travel
because I want to learn Quantum Physics and Time Travel and I Trust you
I'll leave square wave functions to electricians who design and fix wave detectors. They don't affect particle theory.
Are you confusing square waves with a “square” potential well? The two are unrelated.
@@drdca8263 Remember, I'm not an electrician. I was referring to waves, not capacitors.
I wrote that statement before coming back and watching it through, and I now understand how it applies to particles, but it is exactly what I thought it was, but a very specific and overly complicated application of it in regard to electron shells and potentials. It could only apply to a prediction that you can never predict because of quantum uncertainty. You can only use it to explain what has happened. It's novel at best; a trick for trick's sake. All that matters is whether negative electrical potential has been added to or subtracted from the electron. If you know that, then you know why and how it happened and what effect it has on the electron without any complex math that ultimately tells you nothing but that it moved to a different shell.
"We might end up with a differential equation that we have no techniques to solve." Can't any differential equation be solved numerically? Isn't that good enough?
I clicked because of the new tumbnail xd
I thought this is called calculus of variations ?
Calculus of variations is about like, functionals? And the variations of functionals when you vary a function...
Oh, I guess, if you take the wavefunctions to be the functions, and you use the calculus of variations to express the problem of finding the eigenfunctions of the Hamiltonian... uh... hm,
but then how does that relate to using an approximate version of the Hamiltonian?
So, I’m still not seeing a way to make this method an example of calculus of variations?
@@drdca8263 they are basically using the same principles to evaluate information . and al functions are functionals in some reference even one that you know an exact solution to, but i really dont know, i am asking not stating , even though i may have said it in an incorrect manner lol
@@CstriderNNS sure, all functions in a Hilbert space can be treated as a functional using the inner product..
And I guess all linear functionals on a Hilbert space can be represented with one of the functions, by the Riesz representation theorem...
... huh, that’s, making me a bit confused about something actually..
If I have L^2([0,1]) as my Hilbert space, then...
ah, well, I suppose that because functions in L^2 are defined only up to almost-everywhere equality, then I guess maybe there is no functional for “the function’s value at (1/2)”,
Even though there is a sequence of functionals that are “the average value of the function on [(1/2) - (1/n), (1/2)+(1/n)]”
Ok, so, if we instead look at Schwarz functions and tempered distributions..?
Uh, wait, does that form a Hilbert space though?
I am realizing that I’m a bit confused.
What is the appropriate Hilbert space?
I don’t think one can take an inner product of two general tempered distributions...
If you take a Fourier transform of something in the position basis to get something in the momentum basis, uh...
Well, I guess you would only get a Dirac delta if the function you had was like e^( i k x) which isn’t normalizable anyway, and so isn’t in L^2 ?
And also isn’t Schwarz.
The Schwarz functions can’t be all the things in the Hilbert space though, because the dual of the space of Schwarz functions is the space of tempered distributions, which isn’t (anti-)isomorphic to the space of Schwarz functions.. ..I’m pretty sure that it isn’t (anti-)isomorphic anyway...
So, uh, yeah, I’m still a bit confused.
I guess it could just be the space of L^2 functions?
... except not all L^2 functions are almost-everywhere differentiable , and I’d guess that not even all of them are, what was it called, weak-differentiable or something?
Uhh... I guess that’s because of differentiation not being a bounded operator..
So, I guess the domain of the Hamiltonian isn’t the entire Hilbert space?
????
@@drdca8263 ok...this will take me a lil to unpack but ill be back lol
@@drdca8263 hilbert space of e^ikx via euler's formula indicate that it in fact is normaliable via R ^(n+1) , meaning e^ikx is looking at the function through a dimension 1 step lower then cosx+sinx , and vis the Crouchee Riemann equality indicating that evaluating a wave equation purely by one form or another loses a certain amount of information .example looking at e^ikx one loosest the sense of "spatial" distance and looking at cos+sin losses the sense of continuity through to infinity ?
How do you manage your hair without getting dandruff?
9th comment!
Nice video. But where we practically use this theory?