Talk:Adiabatic theorem

Article importance
I have listed the article as being of high importance within physics because it certainly does more that "fill in more minor details". My field is Atomic and molecular physics, in which we run accross this concept regularly in describing the behaviour of a system around an avoided energy level crossing. I would be interested to know whether other people agree with this rating. I'm working on an expanded version of the article.--DJIndica (talk) 02:27, 18 November 2007 (UTC)

I think it's more of a mid-level entry. Check out WikiProject Physics/Projects of the WeekHeadbomb (ταλκ · κοντριβς) 21:20, 14 June 2008 (UTC)

Removing scriptstyle in math
Why are all the inline maths equations in the article in ? They shouldn't be. Can someone with enough regex skills remove that? Using the find and replace feature, I am able to remove  but not the bracket at the last.--ApoorvPotnis2000 (talk) 19:20, 1 May 2021 (UTC)

Possible typo and simple relativity
The word "slowly" in the italicized definition seems like it should be "quickly", no? Also I did not find a relativistic treatment of the issue in the article, but that's another matter. -Ahmed

errors in the technical section
This section is at best very misleading. The treatment given not appropriate in general - you cannot use that method when the Hamiltonian is varying slowly. Zeta is NOT in general the probability of finding the state in a state other than the one it started in. As an example, consider an initial state which is the sum of two energy eigenstates. Even if the Hamiltonian is independent of time, Delta \bar H will be non-zero.

Messiah, which is given as a reference, has a similar analysis in the section on the sudden approximation. A different analysis is needed for adiabatic changes, which is what this article is supposed to be about. I don't have time to fix this now, but maybe someone else can. —Preceding unsigned comment added by 96.246.177.88 (talk) 13:18, 26 November 2008 (UTC)

Also, the last comment on the energy-time uncertainty principle is bogus--it gets the inequality backwards. --128.12.252.4 (talk) 23:21, 24 January 2016 (UTC)

"the probability density remains unchanged"
There should probably be more explanation of what this phrase means, or the phrase replaced by something a layman can understand. My (poor) explanation: if the system is in a state where repeated measurement would yield various outcomes each with probabilities, changing the environment quickly (non-adiabatically) will result in the probability distribution of outcomes unchanged. The system is not in the same state - it is in partially collapsed state (as if partially measured?) - but but it has the same probability distribution. —Preceding unsigned comment added by 64.180.21.140 (talk) 20:41, 24 January 2009 (UTC)

Analogy to entropy and quantum number
There seems to be an argument taking place on the article page over the correct analogy to make between quantum mechanical and thermodynamical adiabaticity. My guess is that this was introduced to point out the possibly confusing point that "adiabatic" has a different definition in the two fields. Not only is the argument inappropriate for the article page (should take place on the discussion page), but the analogy is actually distracting, making the explanation more complicated than the original concept. For these reasons, I'm going to keep the clarifying point and cut the elaborating discussion out. Maniacmagee (talk) 17:04, 23 December 2010 (UTC)

Article needs expansion - this is not just in quantum mechanics
Like many things in quantum mechanics, this is not really a statement limited to quantum mechanics per se but is true in general for any linear equations of the form &part;u/&part;t=Au where A is slowly varying in t.

The article already gives a pendulum example, but that one is a bit awkward because, in the ordinary conception of a pendulum, a single pendulum only has a single mode of oscillation, so there is no possibility of switching "modes" even if the pendulum is changed rapidly. A clearer example would probably be two coupled pendula, e.g. I used one such example for a class at MIT here: http://math.mit.edu/~stevenj/coupling.pdf

Moreover, this extend to problems where "t" is space rather than time, i.e. to a system that is slowly varying in space. For example, if you have waves propagating through an electromagnetic waveguide (e.g. a metal tube) and slowly change the waveguide's cross-section shape over some distance (a taper transition), then the mode amplitudes change adiabatically (i.e. if you start with one mode then it "adiabatically" transforms into the the corresponding mode as the waveguide changes shape). Mathematically, the analysis is essentially identical to the QM case. See e.g. this paper in Physical Review E: http://math.mit.edu/~stevenj/papers/JohnsonBi02.pdf

— Steven G. Johnson (talk) 19:49, 21 September 2011 (UTC)

Error in the proof of the adiabatic theorem
I think the proof of the adiabatic theorem as presented is wrong. It currently claims "For the adiabatic approximation, which says that the time derivative of the Hamiltonian $$\dot{\hat{H}}$$ is extremely small as a long time is taken, the last term will drop out". However, in the limit that the speed of change tends to 0, the other two terms in the differential equation vanish just as quickly as H does, so we can't just say that it is "small".

More concretely, if the total time taken for the change is $$\tau$$, then as we take $$\tau \to \infty$$ so that the change becomes slower and slower, all of $$\dot{c}_m$$, $$\dot{\psi}_m$$ and $$\dot{\hat{H}}$$ fall as $$1/\tau$$. Since we need to integrate over a period of time $$\tau$$ to get the final state, this has no effect.

The actual reason why the second term becomes insignificant should be the phase factor $$e^{i(\theta_n - \theta_m)}$$. As $$\tau \to \infty$$, the exponent $$i(\theta_n - \theta_m)$$ grows linearly with $$\tau$$, and some Riemann-Lebesgue like result should suggest that the term becomes negligible. However, turning this into an actual proof seems entirely non-trivial. 223.19.95.149 (talk) 03:42, 27 September 2017 (UTC)

Hi,

You are correct that the previous "proof" was very wrong. I inserted the minimal amount of information needed to make this correct, or at least a lot less misleading. This article definitely needs some love from someone knowledgeable.

--Smeuuh (talk) 09:52, 12 December 2018 (UTC)


 * Regarding the third proof of the adiabatic theorem, I think the bracket in the last line depends on t' rather than t. In this case, rewriting the geometric phase as a line integral needs one more step of explanation. 207.251.102.116 (talk) 13:34, 25 October 2022 (UTC)

Quote from the paper
The theorem statement is attributed to Born and Fock, yet their paper does not contain that quote. (Of course the paper is written in German, but the current text is not a *direct* translation of their statements) I think it should be made clearer that this is paraphrasing what they said. — Preceding unsigned comment added by 205.210.143.170 (talk) 20:37, 24 September 2020 (UTC)


 * Changing the quote into a reported speech and put it into the first paragraph would be appropriate. I also looked up and cannot find any statement in the original text that is directly translated into what the wiki page claimed. (I use Google Translate, I am not a German speaker.) Rumor Tray (talk) 10:15, 12 January 2023 (UTC)