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Many different physics techniques for approximately solving differential equations seem to follow the same basic pattern. One starts with some differential equation $Df(x) = s(x)$ (or $s(x) f(x)$), where $D$ is some differential operator (not necessarily linear), $s$ is a known source function of one or several variables, and $f$ is an unknown function to be found in the same number of variables as $s$. One then solves the special case where $s(x) \equiv s$ is constant for the particular solution $g(s,x)$ such that $D_x g(s,x) \equiv s$ (or $s\, g(s,x)$). Then one assumes that, as long as $s(x)$ is "slowly varying", we can just locally use that constant-$s$ solution to get an approximate solution for varying $s$: $f(x) \approx g(s(x), x)$.

The textbook case of this technique is the WKB approximation in nonrelativistic 1D quantum mechanics, but I believe that similar ideas are used in the slowly-varying-envelope approximation in wave mechanics, quasi-FLRW spacetimes in general relativity with slowly varying matter distributions, the Born series, etc.

Is there a general math terminology/theory for this trick? I've seen variations on this idea used many time in different physics contexts, but I'm not sure if I've seen a more abstract or general unifying mathematical discussion of these special instances.

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  • $\begingroup$ Ask (also) at math.stackexchange perhaps? $\endgroup$ Feb 26, 2023 at 19:43
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    $\begingroup$ @MariusLadegårdMeyer Yeah, this is technically a math question, but since I'm fuzzily characterizating this technique in terms of examples from physics, I think it's just as much a physics question, and physicists might be more likely than mathematicians to understand what I'm asking. $\endgroup$
    – tparker
    Feb 26, 2023 at 19:46
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    $\begingroup$ Aren't they all called "WKB", as suggested by Wikipedia. Even more math-type people seem comfortable with this terminology. $\endgroup$ Feb 26, 2023 at 20:06
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    $\begingroup$ @tparker WKB-like stuff can be found in many math texts on ODEs. Of course it’s famous for its QM history but it does have very wide applications (to acoustics or fluids for instance) and there is an order by order scheme for higher WKB corrections. See for instance the chapter on WKB and Related Methods in Introduction to Perturbation Methods by Mark Holmes, or most other text of perturbative solutions to ODE’s $\endgroup$ Feb 27, 2023 at 0:34
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    $\begingroup$ @tparker still both references are valid and useful I would think. $\endgroup$ Feb 27, 2023 at 2:30

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To my knowledge, although there exist similar approximations in other fields, nowhere they are as developed as WKB in quantum mechanics.

Perhaps, adiabatic approximation is the most general term, as it is applied to WKB but also to some other problems (e.g., classical charges moving in slowly varying magnetic fields in synchrotrons, etc.)

Eikonal approximation is another method that is often compared to WKB, but which has its roots in more general wave theory (e.g., ray optics.) But the main idea has more to do with the method of characteristics than with the slow change (it is not even always approximate.) See also derivation of geometric optics by Sommerfeld-Runge method.

Finally, in studying wave propagation in random media one uses sometimes diffusion approximation to wave or Poisson equation - but this is a rather exotic example in my opinion (unless you work in the field.) See, e.g., On waves in random media in the diffusion-approximation regime

Update
This old question, raised today by the community bot, reminded me of another often overlooked example of slowly varying quantities: envelope function approximation. In crystals we will often consider only a single band, $\epsilon_c(\mathbf{k})$, treating the crystal quasimomentum $\mathbf{k}$, as if it were true momentum $\mathbf{p}$. In real space representation (rather than in quasimomentum representation) this means that we are working with wave-functions that are slowly varying over the extent of many unit cells. This is often accompanied by expanding the dispersion relation near its singular point, producing the effective mass approximation or Dirac cones in graphene/metallic carbon nanotubes.

The next layer in this approximation is imposing external fields, which often break the translational symmetry (see Volume 9 of Landau&Lifshitz for discussion of the magnetic field in crystal.) However, when these fields are slowly varying (i.e., adiabatically varying) over many lattice spacing, we can still treat the Hamitronian as that of a free particle, notably using Peierls approximation. $$ \epsilon_c(\hbar\mathbf{k})\rightarrow \epsilon_c(\mathbf{p})\rightarrow\epsilon_c(-i\hbar\nabla)\rightarrow \epsilon_c\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)$$

Another example, involving slowly varying order parameter (but often treated as a wave function) is Ginsburg-Landau theory for describing superconductivity.

This then branches onto the procedures associated with coarse-graning in order to introduce consistently the mean-field theory (see Lectures on phase transitions and renormalization group by Goldenfeld), and quasi-equilibrium approximation used to obtain hydrodynamic equations from the kinetic theory of gases, see this answer and this one.

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WKB is mostly called WKB. It is a well-established mathematical technique, discussed in several textbooks (usually beyond elementary level) on perturbation of differential equations.

For historical reasons lots of examples are lifted from quantum mechanics but nevertheless the general technique has wider applications.

Two reasonable examples of textbooks are

  1. Mark Holmes, Introduction to Perturbation methods, and
  2. Bender & Orszag, Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory.

Both texts have sections dedicated to WKB and related methods.

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