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A naive question about WKB approach. It is dubbed to be a "semiclassical" method. What is precisely mean in quantum mechanical context to be "semiclassical"? Wikipedia states that generally semiclassical methods just mean that one part of system is described classically, another quantum mechanically. But what is in WKB precisely the classical part?

Isn't it naively thinking purely non-classical since we expand the solution ansatz in $\hbar$? Where sits the precise reason to call it nevetheless "semiclassical" approach?

Can from this example an abstract reason or indication be extracted when an approximation method applied to a quantum mechanical system should be called to be "semiclassical"?

Addendum/ sidemark: I read (eg wiki again) that the "One- Loop Approximation" is also regarded in certain sense as semiclassical. (namely the latter mean the breaking of the expansions of entities in powers of coupling constant after terms corresponding to a simple loop wrt Feynman graphs) Why? After all, what is "classical" component there?

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  • $\begingroup$ More here: physics.stackexchange.com/q/417877/226902 (the idea is that $h$ is treated as a small parameter) $\endgroup$
    – Quillo
    Commented Sep 16, 2023 at 15:55
  • $\begingroup$ yes sure, but what does it mean here to be "semiclassical"? I' m not sure if calling an approach to be semiclassical is really equivalent to the possibility to expand a system ( more precisely the action S) in Taylor series in $/hbar$... $\endgroup$
    – user267839
    Commented Sep 17, 2023 at 17:22
  • $\begingroup$ Taylor around $h=0$ is an expansion around the classical limit, hence the name "semiclassical". $\endgroup$
    – Quillo
    Commented Sep 17, 2023 at 18:56
  • $\begingroup$ @Quillo: ok, but the Schrödinger equation which we try to solve there is intrinsically " non classical". So say we solve it using WKB and we let then $h \to 0$. Which pure classical result(s) we would then recover from this? ( if I understand your concern correctly then "semiclassical" means in this approach that performing $h \to 0$ provides something what we know from classical mechanics, but which result is it?) $\endgroup$
    – user267839
    Commented Sep 17, 2023 at 19:15

2 Answers 2

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What is precisely mean in quantum mechanical context to be "semiclassical"? Wikipedia states that generally semiclassical methods just mean that one part of system is described classically, another quantum mechanically. But what is in WKB precisely the classical part?

Isn't it naively thinking purely non-classical since we expand the solution ansatz in ℏ ? Where sits the precise reason to call it nevetheless "semiclassical" approach?

I have never heard term semiclassical to be applied to the situations where one part is described classically and another in quantum way. Indeed, Fermi Golden rule, Rabi oscillations and many other effects are described as interaction of quantum systems with a classical electromagnetic field, but no one calls them "semiclassical".

What we mean by "semiclassical" when talking about WKB is that we consider quantum fluctuations around a classical trajectory, as if describing a motion of a wavepacket with some uncertainty of momentum and position. Although tunneling, an essentially quantum effect, can be also described using WKB, the approximations involved are essentially the same as for the allowed trajectories - fluctuations about the trajectory maximizing the action (in Eucledean space - the most coherent derivation of WKB and its applications beyond literal particles trajectories are obtained in path integral formalism.)

Another well-known "semiclassical" effect, closely related with WKB (and sometimes derived from it), are Bohr-Sommerfeld quantization rules.

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It is semiclassical since you use a gaussian wavepacket. The central point of such a wavepacket follows exactly the classical equation of motion.

It is then called semiclassical since the central part moves like a classical point object, but you still have a quantistical wave function.

Tell me in the comments if you want me to expand into mathematical details

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  • $\begingroup$ thank you, yes, I would be very grateful if you would elaborate the details $\endgroup$
    – user267839
    Commented Sep 16, 2023 at 10:45
  • $\begingroup$ Ok i'll do it as soon as possible $\endgroup$
    – LolloBoldo
    Commented Sep 16, 2023 at 10:50
  • $\begingroup$ what I' m not sure about, the Schrödinger equation in en.m.wikipedia.org/wiki/… which one in WKB wants to approximate is purely from QM. You said that the center of wave packet follows the classical equation of motion. Yes, but the WKB approximation itself approaches / is designed for a solution of the linked Schrödinger equation, which is purely QMechanical, isn' t? So the crucial point is, where the WKB approach method " involves" the impact of this classical equation of motion of the center of the wavepacket precisely? $\endgroup$
    – user267839
    Commented Sep 16, 2023 at 10:57
  • $\begingroup$ That is the topic of my undergrad thesis, i'll try to make a summary :) $\endgroup$
    – LolloBoldo
    Commented Sep 16, 2023 at 11:29

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