Slow variation of a quantity on certain scale

Question

The WKB approximation gives meaningful approximative solutions to the $1$-dimensional, time-independent Schroedinger equation

$$ -\frac{\hbar^{2}}{2m} \frac{d^{2}}{dx^{2}} \Psi(x) +V(x)\Psi(x)=E\Psi (x)$$

under the assumption that the potential $V(x)$ 'varies slowly' on the scale of the wavelength $\lambda$. I asked here about the precise way to express this vague formulation of 'slowly varying potential' $V(x)$ on wave length scale in mathematical terms.

Now my question is if it's possible to express in general context the vague phrase that 'some quantity $A(x)$ varies slowly on certain given scale $ ds $ (eg say $ds \approx 10^{-16}m$) of $x$' as a concise $ blabla(ds, A(x)) \ll 1 $ statement? To be more precise that blabla term should be an appropiate expression $ B(ds, A(x)) $ we are looking for, which as expected should depend on the scale $ ds $, the quantity $ A(x)$ and possibly it's derivatives and which should formally characterize the statement that $A(x)$ is assumed to vary slowly on the sclae $ds$.

How I think about this: let denote by $ds$ the 'scale' on which $A(x)$ should vary slowly. Then intuitively the statement that '$A(x)$ varies slowly on certain given scale of $x$' could be also roughly rephrased as if $x_0$ is any argument of $A(x)$ and we chose a $x \in [x_0-ds, x_0+ds]$ then the difference $\vert A(x_0)-A(x) \vert$ is 'small'.
By "small" I mean that in all consequent calculations we can set all as "small" considered terms to be zero and would obtain results consistent to experiments / effects performed / detected on the characteristic scale $ \sim ds$.

A naive guess: Can it be formulated like $\vert \frac{A(x_0)-A(x)}{A(x_0)} \vert \ll 1$ for any $x \in [x_0-ds, x_0+ds]$?

Alternatively, can it be expressed in terms of first and second derivatives like

$$\left| \frac{A''(x)}{A'(x)}\right| \cdot ds \ll 1$$

and if yes are these two ways to characterize that $A(x)$ varies slowly on scale $ ds $ of $x$ equivalent or is there a more standard way to formulate this condition in mathematical terms? So it's really a question only about to transform a sloppy statement to a precise one.

Answer 1

This is an incomplete answer but hopefully better than nothing. The way to quantify the degree of approximation in general is to give some estimate of the size of the quantity that has been neglected. So rather than saying 'blah $\ll 1$' you would write $$ y = f + \epsilon \simeq f \;\;\;\; \mbox{ where } |\epsilon| \simeq \mbox{blah}. $$ For many calculations it is, however, very difficult to provide an expression for this $\epsilon$ even though you are confident it will tend to zero in some limit. The WKB approximation is like this. It does indeed behave correctly in the appropriate limit, but it's hard to provide a single catch-all expression for how accurate it is. Broadly speaking what the WKB approximation requires, in order to be useful, is $$ \lambda \ll \left|\frac{V'}{V''}\right| $$ where $\lambda$ is the wavelength, but I don't have an expression (even an approximate one, which is all you need) for $\epsilon$ in terms of $\lambda V''/V'$.

Answer 2

First of all, a small but important correction: WKB approximation is not limited to 1-dimensional problems to give meaningful approximative solutions.

Second, the condition for the validity of WKB approximation should be provided in any decent QM textbook. In particular, you can check chapter 7 in Landau&Lifshits, vol.3, equation (46.6) (in the second (revised) edition from 1965): $$\left|\frac{d \lambda(x)}{d x} \right|\ll 2\pi$$ where $$\lambda(x)=\frac{h}{\sqrt{2m(E-V(x))}}$$