'Slow variation' in WKB approximation as precise mathematical condition The WKB approximation provides an approximative solution
to the one-dimensional, time-independent Schrödinger 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 $\lambda$.
Can this sloppy assumption be turned in precise mathematical
condition?
My unsuccessful attempt to express it as the
condition $\frac{\Delta V}{\Delta \lambda} 
\ll 1 $ runs into troubles for dimensional reasons, since
$V$ doesn’t have the dimension of a length.
Even more confusing in this approach above is that
$\Delta V = V'(\xi) \Delta \lambda $ for $\xi \in 
[\lambda_0, \lambda_0 + \Delta \lambda] $ by the mean value theorem
would lead to $V'(\xi) \ll 1$.
But the derivative $V'(x)$ of the potential is not necessarily
almost zero globally, so something is wrong in my considerations.
Could somebody explain how the 'slow variation' of $V(X)$ on wavelength scale of $\lambda$ should be expressed mathematically here? My intention is to prove that if this 'slow variation' assumption is posed, then the WKB approximation approaches a solution. But in order to do it, I need to translate the condition on 'slow variation' into a strict mathematical form.
 A: The basic assumption for the WKB approximation is that the potential $V(x)$ should vary slowly in space, compared to the wave function $\Psi(x)$. The more precise way to express this is
$$\left| \frac{V''(x)}{V'(x)}\right| \ll \left|\frac{\Psi''(x)}{\Psi'(x)}\right|.$$
Usually, the wave function has a  wave-like profile of the form $\sim e^{ikx}$, where $k$ is the wave vector defined locally and is related to wavelength through $\lambda=\frac{2\pi}{k}$, and therefore
$$\left|\frac{\Psi''(x)}{\Psi'(x)}\right| \sim \frac{1}{\lambda}.$$
So
$$\left|\frac{V''(x)}{V'(x)}\right|\ll \frac{1}{\lambda},$$
which is required for WKB to be valid. Approximately we can write,
$$k\approx \sqrt{2m[E-V(x)]}$$
At points where $V(x)=E$ (the classical turning points) the local wavelength becomes large, and the inequality cannot be satisfied. So special care should be given to these points and matching the WKB solutions on both sides of these points is required.
A: It should be mentioned that the mathematical condition for the validity of the WKB approximation for the 1D TISE can be neatly formulated as
$$\left| \frac{d\lambda(x)}{dx} \right|~\ll~2\pi, \tag{46.6}$$
where
$$ \lambda(x) ~=~\frac{h}{p(x)}, \qquad p(x)~=~\sqrt{2m|E-V(x)|} ,$$
is the de Broglie wavelength, cf. Ref. [LL].
References:

*

*[LL] L.D. Landau & E.M. Lifshitz, QM, Vol. 3, 2nd & 3rd ed, 1981; $\S46$.

