Talk:Perturbation theory (quantum mechanics)

Degenerate theory have false statement that first order correction in energy is the same

I will correct it, but I didn't use wiki formulas, so I apologize if output is not perfect. —Preceding unsigned comment added by 131.180.87.254 (talk) 13:05, 22 September 2008 (UTC)

Cancellation of Terms

The article claims:

The first-order equation is

$H_{0}|n^{(1)}\rangle +V|n^{(0)}\rangle =E_{n}^{(0)}|n^{(1)}\rangle +E_{n}^{(1)}|n^{(0)}\rangle$

Multiply through by <n⁽⁰⁾|. The first term on the left-hand side cancels with the first term on the right-hand side. (Recall, the unperturbed Hamiltonian is hermitian). This leads to the first-order energy shift:

(emphasis added) After multiplying through by <n⁽⁰⁾|, it looks like the first term on the left-hand side goes to zero on it's own (H_0 pops out of the bra-ket and turns into an E_0, leaving you with <n⁽⁰⁾| n⁽¹⁾> which is zero (orthogonality). (As opposed to canceling with the term on the right-hand side.) I believe the statement I've italicized is incorrect.--GameGod (talk) 18:22, 7 December 2007 (UTC)

• Who said that <n⁽⁰⁾| n⁽¹⁾> is zero? The <n⁽⁰⁾> and <n⁽¹⁾> don't belong to the same base (as opposed to, for example <n⁽⁰⁾> and <n+1⁽⁰⁾>) Gglanzani (talk) 16:44, 29 January 2010 (UTC)

NEEEEEEEDDDD EXAAAAAAMPLEEEEEEESSSSSSS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! —Preceding unsigned comment added by 213.64.27.237 (talk) 11:01, 19 September 2008 (UTC)

Nonperturbative?

Could someone highlight Nonperturbative and explain it more explicity ... the wlnk comes here ... but then it's not highlighted nor explained well ... thanks. J. D. Redding 16:24, 10 March 2006 (UTC)

Intermediate normalization

I get so tired of people getting their fingers into articles, while not knowing what they are doing. Now somebody removed the very well-known intermediate normalization condition and made an error that has the consequence that the first-order correction to the wave function is zero. Namely, (s)he writes (through first order)

$1=\langle n|n\rangle =\langle n^{(0)}+\lambda n^{(1)}|n^{(0)}+\lambda n^{(1)}\rangle =\langle n^{(0)}|n^{(0)}\rangle +\lambda \langle n^{(0)}|n^{(1)}\rangle +\lambda \langle n^{(1)}|n^{(0)}\rangle +\lambda ^{2}\langle n^{(1)}|n^{(1)}\rangle$

$=1+0+0+\lambda ^{2}\langle n^{(1)}|n^{(1)}\rangle$

so that, because a Hilbert space inner product is positive definite,

$\langle n^{(1)}|n^{(1)}\rangle =0\Longrightarrow |n^{(1)}\rangle =0$

Hence I reverted--P.wormer 14:41, 9 August 2007 (UTC)

at second order you are missing terms. you should have 2<n⁽⁰⁾|n⁽²⁾>+<n⁽¹⁾|n⁽¹⁾>. you can check this with the second order expressions actually given in the article!

the statement in the text regarding the intermediate normalization condition: "The same choice is made for all higher-order state corrections" is just wrong.

• The unit normalization of the exact wave function is not convenient. Already the second-order wave function $|n^{(2)}\rangle$ obtains the non-zero component $\langle n^{(0)}|n^{(2)}\rangle$ along the zeroth-order wave function $|n^{(0)}\rangle$ in unit normalization. This means that the weight (prefactor) of the zeroth-order wave function changes when the second-order correction is included. This continues for higher orders: every order included changes the weight of the zeroth-order wave function. The equations for higher order wave functions become very awkward in unit normalization. An elegant way out is to solve all the higher order contributions in the orthogonal complement of $|n^{(0)}\rangle$, so that $\langle n^{(k)}|n^{(0)}\rangle =0$ for all k > 0. As a consequence $\langle n|n^{(0)}\rangle =\langle n^{(0)}|n^{(0)}\rangle =1$. --P.wormer 13:05, 13 August 2007 (UTC).

• I see that an anonymous deleted again the part on intermediate normalization (including three references to reputable books!). I give up.--P.wormer 15:58, 13 August 2007 (UTC)

i reverted the article because as it stood it was wrong. according to the actual expressions given at second order $\langle n^{(k)}|n^{(0)}\rangle =0$ does not hold for k=2. feel free to use a more convenient normalization but you will need to correct the expressions given in the article. --Anonymous

I don't understand what P.wormer means by weight of zero-order changing when more orders are included. We have $\langle n|n\rangle =\langle n^{(0)}|n^{(0)}\rangle +\lambda (\ldots )+\lambda ^{2}(\ldots )+\ldots$, so choosing the normalization $\langle n|n\rangle =\langle n^{(0)}|n^{(0)}\rangle =1$ means that the rest of the expression must be equal to zero, and moreover all coefficients of $\lambda ^{k}$ must be zero independently. I don't see why the expressions should be terminated at some level; they must always be assumed to be there, even if you just want to calculate the first or second order expressions.

If the phase of the perturbed wave function is chosen so that the inner product $\langle n^{(0)}|n\rangle$ is real, then we get $\langle n^{(1)}|n^{(0)}\rangle =0$ in the first order. But I don't think it's acceptable to choose the perturbed wave function so that $\langle n^{(k)}|n^{(0)}\rangle =0$ for any order of $k$, since this restricts the perturbation corrections to be orthogonal to the unperturbed wave function, without any justification.

--Lehmus (talk) 13:58, 25 March 2014 (UTC)

The normalizations $\langle n|n\rangle =\langle n^{(0)}|n^{(0)}\rangle =1$ can only be correct if

$|n\rangle =c|n^{(0)}\rangle +|\Delta \Phi \rangle$,

where $|c|<1$ if $|\Delta \Phi |>0$. The weight $|c|$ of the zeroth-order contribution decreases if the norm of $|\Delta \Phi \rangle$ (containing higher and higher order corrections) increases. This is what I mean when I write: "the weight of zero-order [is] changing when more orders are included."

The justification of intermediate normalization is the fact that the operator

$H^{(0)}-E^{(0)}$

is singular on the whole Hilbert space and is regular on the orthogonal complement of $|n^{(0)}\rangle$.--P.wormer (talk) 16:28, 27 March 2014 (UTC)

Lambda dropped

$\lambda$ is lost in the final equation for the energy. HolIgor 20:29, 16 October 2007 (UTC)

Yes, thank you for the catch. But let me put it up for debate: Is it clearer to use the approach begun in the derivation here and used on the Perturbation theory page (which takes a formal mathematical standpoint), or would it be better to take the more utilitarian road and drop the lambda in favor of an alternative notation, such as the substitution $\lambda V\rightarrow H_{1}$ (as done by, among others, the standard textbook by Shankar)? Cgd8d (talk) 17:30, 8 May 2008 (UTC)

I also suggest to remove $\lambda$ since it was kept only for the purpose of bookkeeping. I also feel that the problem is more clear if is explained in terms of wave functions, opposed to the state vectors, since all the resulting formulas look cumbersome and long, overloaded with too many symbols (This is just an opinion). I refer Landau & Lifshitz, chapter VI, in that, the whole chapter is explained clearly and concisely. It is obviously much shorter than the wikipedia page. Also making the concepts concise allows space for examples.

Please sign your comments ! I strongly disagree with removing pedagogical parameters and seeking concision. This is a go-to review for students, as well, definitely not a rival to technical and hyper-competent specialized L&L. Recall WP is not for experts or specialists. Cuzkatzimhut (talk) 12:34, 14 April 2019 (UTC)

High, yes lambda is just introduced for what physicists like to call book keeping. Factually it does not make any sense to introduce it in the Ansatz. It originates from throwing an eigenstate from H0 onto the perturbed system. You get small contributions of other states of the "old" basis, as the state is not preserved. You take that changed state, you throw it again on the perturbed Hamiltonian, and the small contribution you got last time, will again be projected onto other states in the old basis, with an even smaller prefactor. and so on. Everytime you get smaller contributions, until that state does not change anymore...which then makes it an eigenstate. Something like taking the derivation with respect to lambda is wrong. This isn't possible, nor necessary, and should be skipped. Lambda is not a variable, but an arbitrary constant. I checked the source, Sakurai. In the subchapter "Formal Derivation of Pertubation Expansion" this does not occur either. But as long as you add a source, everything is ok! Source: "Trust me Dude" 2nd Edition (2020) -SomeGuy (not a user) — Preceding unsigned comment added by 129.69.48.35 (talk) 10:53, 9 April 2020 (UTC)

Rating and Mathematics

I have modified the rating of this article and linked it to mathematical project. It contains a lot of relevant material that is essential in the understanding of quantum mechanics and, mostly, of quantum field theory where perturbation theory is an essential tool. Pra1998 (talk) 08:26, 3 December 2008 (UTC)

Examples

I really think this article needs at least one example of perturbation theory at work. A quantum well in a weak electric field (the quantum-confined Stark effect) is about as simple as this gets. This also gives us the opportunity to discuss the limitations of the approximation: "new" behaviour caused by the electric field, such as Fang-Howard tunnelling is not accounted for by perturbation theory. Also, we can look at the range of perturbing electric field for which the model is valid: when the matrix element of the perturbing potential becomes comparable to the separation between the unperturbed states, the second-order perturbation terms fail to converge.

Does anyone else think such a section would be useful? If so, I'll start writing it! Papa November (talk) 09:30, 27 August 2009 (UTC)

Third- and possibly higher-order expressions are wrong

Hiya,

for a project I needed perturbation theory to third order in the states. I could not find anything going above second order in the states in any of the common QM textbooks I looked at. So I decided to re-derive it myself, and while of course I cannot be absolutely sure *my* result is correct, it does agree with the numerical solution I got, whereas the wikipedia version has a significant deviation from numerical & my perturbative solution. Therefore I strongly suspect the expression for third order of states, and possibly fourth order and higher in energy and states, to be wrong. I could exchange the third order in states with the version I have, but in any case I think it would be good to remove the wrong statements.

S.John 217.93.234.75 (talk) 17:49, 6 July 2010 (UTC)

I have restored the higher order corrections. Regarding the third order energy correction, I can not see any difference with "your" correct results. I don't know if the higher order formulas are right or not, but what I believe is if you can not prove that they are wrong, you should not simply delete them. If you do think there is a mistake, you should show it here, and correct it in the article.Everett (talk) 01:44, 17 February 2011 (UTC)

Hello,my name is Shimada.

I also think there is a mistake in the third order of state. I suspect that this auther forgot the sign of minus before the second term of the second order of states $(-{\frac {V_{nn}V_{k_{1}n}}{E_{nk_{1}}^{2}}})$,when he calculates the third order. Probably, $|n^{(3)}\rangle =\cdots {\dfrac {V_{nn}V_{k_{1}k_{2}}V_{k_{2}n}}{E_{k_{1}n}E_{nk_{2}}}}\left({\dfrac {1}{E_{nk_{1}}}}+{\dfrac {1}{E_{nk_{2}}}}\right)-{\dfrac {|V_{nn}|^{2}V_{k_{1}n}}{E_{k_{1}n}^{3}}}$is right,I think.I don't consider other parts yet.Please confirm this. — Preceding unsigned comment added by Y.Shimada1215 (talk • contribs) 15:49, 1 July 2011 (UTC)

I don't see a citation for the high order expressions. I would think they should be deleted on that basis. As for third order, can someone check if that's what L&L actually have in their book? I would think some discussion of the high order degenerate case would make this more complete, if anyone knows a reference. Physicsjock (talk) 05:06, 13 February 2012 (UTC)

I computed these recently, the formula for the states starts being wrong at 3rd order, in that some of the coefficients are wrong. At 4th order it is completely wrong. The energy is right through 4th order, I didn't compute 5th order. (June 18th, 2012) — Preceding unsigned comment added by 171.66.175.8 (talk) 20:08, 18 June 2012 (UTC)

The higher order expressions are definitively wrong. Take the fourth order correction to the state $|n^{(4)}\rangle$, for example. The first term includes 'diagrams' like $V_{k_{1}k_{2}}V_{k_{2}k_{3}}V_{k_{3}k_{4}}V_{k_{4}k_{2}}$, which are not even connected to the state which is to be perturbed. If nobody can correct them, they should be removed. --Sharkdp (talk) 12:21, 21 January 2014 (UTC)

spelling, etc.

Please, please: WP:MOS and WP:MOSMATH both exist.

I found

• "Hellman–Feynman" appearing repeatedly with a hyphen rather than an en-dash;
• -1 instead of −1;
• Indiscriminate italicizing of everything in non-TeX math notation including even parentheses and digits;
• Numerous cases of initial letters capitalized merely because they were in section headings.
• Instances of </math>. or </math>, instead of .</math> or ,</math>. The punctuation often fails to get properly aligned when it's outside of the "math" tags;
• Failures to indent "displayed" things in math tags (via a preceding colon);
• etc., etc., ..... See my recent edit.

I've cleaned up a lot of this. Michael Hardy (talk) 23:09, 18 October 2012 (UTC)

Error in assumption

In the article it says: "Since the overall phase is not determined in quantum mechanics, without loss of generality, we may assume $\langle n^{(0)}|n\rangle$ is purely real." I'm not sure but this seems to be simply wrong. Of course you can choose your _overall_ phase at will in QM, because it will cancel out whenever calculating an expectation value. Therefore one cannot simply "make" or "assume" something like $\langle n^{(0)}|n\rangle$ to be real, a phase factor would have to be multiplied to both sides and would not change the result at all...

92.195.40.81 (talk) 12:51, 20 January 2013 (UTC)

This is not an error, it is true that the phase of the perturbed wave function $|n\rangle$ can be chosen so that the inner product with the unperturbed wave function $|n^{(0)}\rangle$ is real. If the inner product is initially complex, $\langle n^{(0)}|n\rangle =Re^{i\theta }$, changing the perturbed wave function by $|n\rangle \to e^{-i\theta }|n\rangle$ ensures that the inner product is real. This is convenient because it implies that $\langle n^{(1)}|n^{(0)}\rangle =0$ (using the normalization condition), making the calculations a bit simpler. This is explained well in Cohen-Tannoudji, Diu & Laloe.

--Lehmus (talk) 13:31, 25 March 2014 (UTC)

Edit war

I think that it is started and edit war on the section about "strong perturbation theory". The same anonymous from Boston (MA) deleted all of it twice. Maybe some Administrator can act to stop this? Thanks.--Pra1998 (talk) 19:47, 4 March 2014 (UTC)

He has switched IP, and is now 18.62.31.139, also from MIT, and I left a warning on that IP's Talk page. Any further peremptory editing without discussion on this page by him or his IP covers first should be treated as plain vandalism. Cuzkatzimhut (talk) 21:13, 10 December 2014 (UTC)

Fine. Thanks a lot.--Pra1998 (talk) 23:10, 10 December 2014 (UTC)

Rename the article into Quantum Perturbation Theories

I'm very sorry for some mistakes in the page Perturbation theory but I think the original purpose is rather correct: I wish to specify that these ones are applications of analytical perturbation methods to QM, while many other fields have seen perturbation methods like neutronics basing on linear Boltzmann equation, viscoelasticity and so on. That is also the reason I have just suggested in the discussion of [Mathematical] Perturbation Theory page to solve the ambiguity on one hand by inverting the redirect with perturbation methods, and on the other hand to rename the Perturbation Theory (QM) page into Quantum perturbation Theories, (since there are more than one QPT).95.238.49.157 (talk) 18:14, 7 October 2014 (UTC)

As indicated on my talk page plea, I have no interest in said wildcat relabelling stunts, but the loss of the explanatory sentence "The idea is to start with a simple system for which a mathematical solution is known, and add an additional perturbing Hamiltonian representing a weak disturbance to the system." from the introduction is manifestly regrettable. Cuzkatzimhut (talk) 13:17, 8 October 2014 (UTC)

Source for experimental and theoretical value precision

From the article: "the calculation of the electron's magnetic moment has been found to agree with experiment to eleven decimal places." http://www.oberlin.edu/physics/dstyer/StrangeQM/Moment.pdf Is a viewable discussion of the context with it's citation to the data being Physical Review D 85 (2012) 033007

I'm much out of my depth in both this topic and in Wikipedia editing however, so I will leave any changes to the good people with higher capabilities. 98.230.96.47 (talk) 19:31, 5 April 2015 (UTC)

Though I'm only a moderately good person and have few capabilities, I did add the Phys. Rev. D paper. The other pdf, while a good read and undoubtedly reliable, does not pass as a reference unfortunately. Next time, add it yourself! YohanN7 (talk) 22:59, 5 April 2015 (UTC)

Quantum correction

Can quantum correction please be made to point to this article? 96.40.48.159 (talk) 18:17, 14 October 2019 (UTC)

Bad idea. A quantum correction is a correction of $O(\hbar)$ to a classical expression, and not a small coupling correction to a quantum expression as detailed here. Why on earth would one imagine this could be otherwise? Cuzkatzimhut (talk) 21:15, 14 October 2019 (UTC)

Ok, guess I should apologize for not being an expert, asshole? Where should quantum correction redirect, then? I see it used all the time here, but there's not a proper redirect. 96.40.48.159 (talk) 22:37, 14 October 2019 (UTC)

Correction to "Time-independent perturbation theory"

First order:

"Since the overall phase is not determined in quantum mechanics, without loss of generality, in time independent theory we may assume {\displaystyle \langle n^{(0)}|n^{(1)}\rangle }{\displaystyle \langle n^{(0)}|n^{(1)}\rangle } is purely real. Therefore,

{\displaystyle \left\langle n^{(0)}\right|\left.n^{(1)}\right\rangle =\left\langle n^{(1)}\right|\left.n^{(0)}\right\rangle ,}"

There is a minus sign missing. Don't want to edit the article myself, because someone will again demand a citation for basic math.

Edit: Well I don't know how to Wikipedia, so excuse me for the botched formula

Please sign your comments! Further, read all three formulas involved, and make a minimal effort to understand the logic flow of what is being asserted: It is because both the sum and difference of two numbers vanish that both numbers vanish. Cuzkatzimhut (talk) 16:49, 9 April 2020 (UTC)

I get it now. It read like a=a* implies a=0, since it said "therefore a=a* leads to a=0" where 'a' is the term in question. But well... 129.69.48.35 (talk) 13:50, 10 April 2020 (UTC)SomeGuy ...let's hope that is how I sign?

5th oder Energy correction, time indepandent

I have implemented the Energy corrections with some minimal example. After spending many ours I came to the suspicion that the 5th oder energy correction might contain some error. Can someone check that? 185.58.28.55 (talk) 21:48, 4 December 2022 (UTC)

eigenvector perturbation to first order, when degenerate energy NOT lifted to first order

Can someone post what this looks like? I.e. Vnk is the same for all k in the degenerate space, so the expression for first order correction of eigenvectors goes to infinity. Iuvalclejan (talk) 23:27, 10 September 2023 (UTC)