Perturbation Theory in Quantum Mechanics - Cheat Sheet

[@aspiceoflife]

Why are corrections necessary? Corrections to what?

[@PrettyMuchPhysics]

The story usually goes like this: If you have a complicated system to solve, you can usually approximate it using a much simpler system, like a harmonic oscillator (HO) for example. The HO is pretty well known and there's no problem as to getting energies and other quantities. Now, although easy to obtain, those values are not the "real" solutions to the original system, since it was much more complicated. So using perturbation theory, we have a systematic way to "add difficulty" to the simple system (e.g. the HO), by doing this order-by-order. Here's a more concrete example: Suppose we have a potential V(x) = x^2 + 0.0001 x^4. Now this will be pretty similar to the HO, but not exactly. If we call the prefactor of x^4 lambda = 0.0001, then using perturbation theory, we can calculate the energy to be something like E = E_0 + lambda E_1 + lambda^2 E_2 + ... where E_0 is the (known) energy of a x^2 potential (= the HO), and E_1,2,3... are "corrections" to E_0. As you can see, since lambda is really small, E_1 will only change the energy a bit. And E_2 even less. So how do we calculate E_1, E_2 and so on? -> Using perturbation theory. I hope this answered your question! If now, please ask again!

[@xxdxma6700]

Pretty Much Physics This is very helpful , thanks man .

[@PrettyMuchPhysics]

Thanks! You’re very welcome 😊

[@stark5353]

It's beyond frustrating how I've spent many hours trying to follow my professor's explanations while barely understanding, only for a 7 minute video to put it all together and make it clear. Thank you so much!

[@PrettyMuchPhysics]

That‘s why we make these videos, thanks for your nice comment!

[@cleanestboi5854]

agreed, the chapter in sakurai is incomprehensible and my professor basically copied it onto the board

[@paugarciagulisano7967]

Your mini lectures are better than my 2 hours long lectures in university, thank you so much!

[@PrettyMuchPhysics]

That means a lot, thank you!

[@anando3992]

Exactly

[@SmajdalfFrogi12]

Great supplemental material to a rigorrous treatement of the topics. Well put together with correct pacing and thoughtout layouts. Keep up the amazing work. :)

[@PrettyMuchPhysics]

Thank you for your nice feedback! :D

[@danielcrakX]

Thanks for saving my extraordinary quantum mechanics test

[@ZaCharlemagne]

Nice explanation video. Correct me if I'm wrong, In a numerical setting, I believe once you've got your Hermitian (Hamiltonian) H_0 matrix of the unperturbed system (which will in most cases be a tridiagonal square matrix), the perturbation potential V'(x) would be a single diagonal square matrix which we can them multiply with \lambda and add to the original Hamiltonian H_0 to get H = H_0 + (\lambda . V'(x)), such that the new energy levels and wave functions would be the eigenvalues and eigenvectors of H (the perturbed Hamiltonian)

[@sebhudson1181]

in 7 minutes this video explains pertubation theory far better than my lecturer could in 70

[@PrettyMuchPhysics]

That’s the idea of this channel! Glad you liked it! :)

[@Usercehdbf]

Im thinking of doing physics in college i have no idea what this video is talking about lol

[@namesurname1040]

The video really helped me to learb about pertrubation theory But still I cant understand why the first correction term breaks the degeneracy of the states .Thank you very much for your time!

[@kankaiahk7197]

good explanation sir, please upload more videos about perturbation problems

[@bhoopendragupta4782]

Short and easy to follow up, thanks. 👍

[@prabhatp654]

There are a few questions: 1. If the corrections are linear to Hamiltonian, why do the new corrected energy and wave functions need to be linear to the original energy value and wave functions? 2. The corrections to be perpendicular to the original wave functions is related to the new Hilbert space we are in? 3. Is there any Physics in this perturbation theory, or is it just mathematical tricks for calculation? I am assuming yes because it's called a theory.

[@Niharikajain72828]

Very clear explanation ..Thank you very much

[@soumiksamanta10]

I was studying magnetic properties of soild (exchange interaction). It is really helpfull What is secular determinant? How does H'11=H'22 and H'12=H'21 hold for perturbation?

[@PrettyMuchPhysics]

The secular determinant (also known as the characteristic polynomial) helps us to determine the eigenvalues of V, which we called mu_1 and mu_2. Also, the procedure we discussed in this video is valid up to first order perturbation theory.

[@mehdaouinabil977]

we want more videos. thank's for your good job.

[@BackflipBeaver]

Very nice video, I've enjoyed several of them lately! Any chance you could do a video on the "2n+1" theorem? I can't seem to wrap my head around the derivation in my book :/

[@PrettyMuchPhysics]

Haven't heard of this theorem yet. What textbook are you reading?

[@BackflipBeaver]

@@PrettyMuchPhysics The book is "Electronic Structure" By R.M. Martin, in which ch. 3.7 concerns Perturbation theory and this theorem, both in the full many-body picture, as well as in the single-particle approximations of DFT or HF for example. I quote: "The '2n+1 theorem' states that knowledge of the wave function to all orders 0 through n determines the energy to order 2n+1." The problem is, the short derivations aren't very enlightening, nor can I access the sources he references (they're either in German, or books I don't have access to, for example the QM book by Landau & Lifshitz.) Cheers!

[@PrettyMuchPhysics]

Indeed, never heard this one before. Thanks for the reference, maybe it‘ll join our list of future video topics!

[@BackflipBeaver]

@@PrettyMuchPhysics Fingers crossed! It's of relatively minor importance personally all in all. Among other things, it is apparently crucial when justifying the "linearized" equations of the LAPW formalism in DFT, that's what lead me here ^^

[@theultimatereductionist7592]

6m44s I know all notation is "understood", but, I cringe at using "i" for both the index indicating "initial state" and the square root of -1 in the same expression.

[@PrettyMuchPhysics]

I feel the same way! Therefore I usually try to avoid this by using a curly "i" for indices and an upright "i" for the square root of (-1) (like this: imgur.com/a/ysdoP0s )

[@quark67000]

@@PrettyMuchPhysics You're right! By the way, can you add a space between the final s of the URL and the closed parenthesis? Because the KZbin parser takes the final parenthesis for the URL, which gives an incorrect URL: you will see a blank page instead of the image.

[@marioruiz3471]

Could you make a video about the theory of singular perturbations abstract linear operators? A good reference is Albeverio and Kurasov's book "Singular Perturbations of Differential Operators. Solvable Schrodinger Type Operators". Greetings.

[@chikououettar6531]

Thank you very much for this clear, simple and intelligent explanation. I just have one question to ask : what do you mean by diagonalized Hamiltonian ?

[@PrettyMuchPhysics]

The Hamiltonian is an operator. Therefore we can represent it as a matrix. But this matrix looks different for different basis systems. So a Hamiltonian in the basis ϕ1, ϕ2, ϕ3 will be a different 3x3 matrix than the same Hamiltonian in the basis ψ1, ψ2, ψ3. Now, if we use a basis system that consists of the eigenvectors (*) of the Hamiltonian, the corresponding matrix will be diagonal. (*) Since the Hamiltonian is a hermitian operator, its eigenvectors form an orthogonal basis system. Hope this helps!

[@chikououettar6531]

@@PrettyMuchPhysics Thank you very much for the explanation.

[@maryemelhayany5468]

Very clear explanation .thank's for your good job.

[@PrettyMuchPhysics]

Thank you very much! :D

[@Krypton92546]

An unnormalized wave function of the H atom is given by r²e^-r/3(3Cos²∅-1) then how to get it's l,m and n values?

[@PrettyMuchPhysics]

By taking a look at a table of hydrogen wave functions, i.e. the radial part and the spherical harmonics. Look for which radial part is quadratic with an exponential like this, and then which spherical harmonic goes like (3cos(t)²-1). In some cases you might have to assume a linear combination of several wave functions.

[@gerontius1726]

@ 3:10 the result for the first correction to the eigenfunction is unreadable because it is covered up by the subtitle.

[@PrettyMuchPhysics]

To disable subtitles, klick on the [CC] icon, or if you're on mobile, press the three vertical dots and then [CC]!

[@jogam2986]

Thank You! i just think that at 4:48 it should be lambda*u_1

[@Niharikajain72828]

How difficult or easy it is to give physical interpretation of mathematical aspects in quantum mechanics?

[@PrettyMuchPhysics]

The most common view is to see everything as a probability, particles are waves that can interact and interfere positively and negatively

[@kanakalakshmiy7450]

in the independent perturbute theory relation between perturbed hamiltonian and unperturbed Hamiltonian These two are equal or not equal

[@PrettyMuchPhysics]

The unperturbed Hamiltonian is H_0, and the perturbed Hamiltonian is H = H_0 + lambda V.

[@kanakalakshmiy7450]

Thanq so much

[@PrettyMuchPhysics]

You’re welcome!

[@0777coco]

i was so thrown off by you suddenly saying _Eigenfunction_ and _Ansatz_ lmao

[@PrettyMuchPhysics]

Bless the German language for bringing us such useful words :P

[@ecuanam2288]

great video. Thank you

[@PrettyMuchPhysics]

Thank you!

[@nellvincervantes6233]

I have a question about bra-ket notation sir. How would you know if a state is in bra or ket? In what situation will you use bra or ket in real life? I thought ket is the "initial" state? Sorry for this question. I got confuse on how you use the bra-ket in the time dependent perturbation in this vid. .

[@nellvincervantes6233]

Why psi_initial is in bra and psi_final is in ket?

[@PrettyMuchPhysics]

Usually, you use a ket for wave functions, and a bra for their complex conjugate. In this case, we want to project one wave function onto the other, which mathematically corresponds to taking the inner product. In principal, and are not the same (they are the complex conjugates of each other), but since we will take an absolute square, the order does not matter: ||² = ||²

[@beleggo4532]

Hello, great video!

[@PrettyMuchPhysics]

Thanks!

[@devsikarmur8465]

Dear, What is need for perturbation to system? What's need for correction?

[@PrettyMuchPhysics]

For perturbation theory to work, you need to be able to separate your Hamiltonian into a part that you can solve (usually called H_0) and a part that involves a small parameter (usually called lambda).

[@peasantarcher2486]

You are a fucking hero!!!

[@MushiMe100]

This video is explaining in a different way than i am used to, but I am trying to understand, @4:23, when you say we need to solve the eigenvalue problem, what is written inside the brackets for the determinant, i read V tilde, mu, and what is next to it?

[@PrettyMuchPhysics]

Inside the determinant, we have (V-tilde minus µ times the identity matrix). We usually denote it using two "1" that are very close together.

[@suhail1696]

Which graphic tablet are you using? And writing software?

[@PrettyMuchPhysics]

We're using an iPad with an app called "Explain Everything"!

[@ckb1137]

4:44 once you have determined A and B how do you determine psi 1and psi 2 ? ( in the relation psi = A psi 1 + B psi 2)

[@PrettyMuchPhysics]

They are those wave functions that previously had the same, degenerate eigenvalue E_1=E_2=epsilon. So they are already known!

[@philippwachter8748]

Amazing!

[@PrettyMuchPhysics]

Thank you very much! :)

[@MushiMe100]

@1.:26 what is being said about the subscript n, I cant quite understand what he is saying, is it eigenfunction? And could someone explain what is meant by the theta looking symbol in the power series of E and Psi at 1:48?, I am following along with my own university book, but we do not have these in our equations

[@PrettyMuchPhysics]

The subscripts indicate which eigenfunction/value we are considering and the superscript tells us the order of correction. PS: we have subtitles for all our videos!

[@MushiMe100]

@@PrettyMuchPhysics awesome thank you!

[@PrettyMuchPhysics]

@@MushiMe100 As for your second question, that's called big-O notation (en.wikipedia.org/wiki/Big_O_notation#Infinitesimal_asymptotics ). It basically says that the terms that come after O(x^3) behave like x^3 for small x.

[@phyzics4all343]

Hi there! How would we define ψ_k and E_k for the non-degenerate state case in an infinite 1d well?

[@PrettyMuchPhysics]

You would start with the wave functions and energies of the infinite well, and then calculate the corrections according to our video.

[@clemenstombozichama645]

Perfect

[@PrettyMuchPhysics]

Thank you!

[@tensorfeld295]

I am no expert in perturbation theory, but maybe the book 'Perturbation Theory for Linear Operators' by Tosio Kato can help to derive these things.

[@PrettyMuchPhysics]

Many textbooks on quantum mechanics deal with these formulas, I guess one of the more popular ones is the one by Griffiths!

[@tensorfeld295]

@@PrettyMuchPhysics Yeah, but for spectral theory and operator theory, i guess, one needs more functional analysis.

[@surendrasaini3656]

why we want to learn perturbation theory

[@PrettyMuchPhysics]

Good question! There are actually only a few problems in quantum mechanics that we can solve by hand. For more difficult problems, we either have to use numerical methods (=computers) or use perturbation theory to get an approximate result.

[@RaniShrividya]

Thank you so much for your content. I don't understand everything. But I love trying to get Physics, and you make the process so interesting. Love you.

[@PrettyMuchPhysics]

Thank you very much! We‘re glad you like our videos! :D

[@ianlee5812]

Is the potential another name for the perturbation?

[@gogatesudeep]

What is exactly meant by 'correction' herein?

[@PrettyMuchPhysics]

If the unperturbed energies are E and the new, perturbed energies are E’, we can write E’ = E + something This “something” is assumed to be small, because the perturbation is also small. So it cannot change much. And this “something” is what is called correction in this video!

[@gogatesudeep]

@@PrettyMuchPhysics Hey, really appreciate the videos and the content. And thanks for responding so quickly ! It's extremely helpful and thank you for posting it (y) Also, if i.e x cap| psi(t) implies the operator x cap acting on the state vector psi(t), what does the notation later used in perturbation theory, imply? What is acting on what and what is it that we are looking for

[@PrettyMuchPhysics]

Let's summarize: = psi(x), where is a ket vector x-hat |psi> is the position *operator* acting on the state psi is the expectation value of the operator A if the system is in the state psi, therefore

[@TheRossJenkinson]

@@PrettyMuchPhysics Hello! What does the notation < psi k | V | psi n > mean? How do you calculate this value? Thank you

[@PrettyMuchPhysics]

Good question. is called a matrix element. If you look at the operator A as a matrix, then this would be A_ij. As for how to calculate it, this depends on the special case. Usually, you know the functions, e.g. in the harmonic oscillator |0>, |1>, |2>, ... then the operator in the middle can be represented by some combination of the raising and lowering operators a and a*. And since it is known how they act on |n>, you can calculate this matrix element.

[@mohibmohib6442]

Bonjoure Svp réponde pleas Dans une perturbation stationnaire Un oscillateur harmonique perturbé par V={Lampda×hw×racin(mw/h)×X] A quoi est egale les enrgiers exacte de Hamiltoniane ToTale (H= H°+V) H° non perturbé

[@PrettyMuchPhysics]

My French is a bit rusty, but in that case you can complete the square: try to write a x² + b x as c (x+d)² + e, and then shift x → x-d. This special case does not require perturbation theory.

[@luisgarabito8805]

this was not a video on how to use it smh. it's quite literally a brief derivation of the different scenarios.

[@inshaikhlaq376]

Hiii :) only consider subspace of V mean?

[@W4TzReAdY]

Consider the part of the V matrix related to the degenerated eigenfunctions. I imagine you know that the matrix coefficients are representing the decomposition of a vector according to a certain basis, This is what it is about

[@StefSubZero270]

Because you are considering the degenerate subspace of the Hamiltonian since you are interested in the degenerate eigenvalue

[@RB-mm7ce]

cool stuff, but I gotta learn the underlying maths first, hahah

[@paulgets2737]

Just 7 minutes to explain PT?

[@PrettyMuchPhysics]

7 minutes to explain how to use the important equations in perturbation theory!

[@moritzfinke4518]

His "Ansatz" was too smooth to not be a German native speaker.

[@PrettyMuchPhysics]

🤔

[@lebachelorsahraoui2688]

I have only one question .. i discovered that THEORY watching an episod of X-FILES .. When Mulder and Scully were talking about The Liberty Bell .. there were some numbers written on there numberplate's car i noticed them and went To google .. and then i felt on that DRESSED PERTURBATION THEORY .. So please tell me why .. i apologize for my english maybe not very clear .. thanks .. and good job ..

[@lebachelorsahraoui2688]

Thank you so much .. now i see .. 🤞

[@DOUBLE0SEVUN]

How did you discover a theory just by googling a license plate number? That's strange

[@adamkloc9518]

i am too dumb to understand this

[@PrettyMuchPhysics]

If you have any questions, let us know!

[@adamkloc9518]

@@PrettyMuchPhysics i just don't have the background knowledge to understand this, so i guess i have a lot of learning ahead of me

[@adamkloc9518]

but i must say, this looks like a pretty cool channel

[@PrettyMuchPhysics]

Thank you very much! The videos on this channel are aimed at university-level physics students, so there is definitely background knowledge necessary!

[@slimjim7329]

This shit makes no sense

[@2psnopod]

Lmao i done looked up a headace 😂

[@Niharikajain72828]

How difficult or easy it is to give physical interpretation of mathematical aspects in quantum mechanics?

[@sumsar01]

Sometimes very easy, sometimes not so much.