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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.

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    $\begingroup$ What matters is the fractional variation of the potential $\Delta V/V$, over the scale on which the wave function varies $\sim\lambda$. This is still not quite satisfactory, however, since the value of $V$ can be changed without changing the physics by shifting $V$ and $E$ simultaneously. It does deal with the units issue you are having though. $\endgroup$
    – Buzz
    Oct 8, 2022 at 23:51

2 Answers 2

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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.

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  • $\begingroup$ I understand it now for WKB approximation. But this naturally rises following attempt to generalize my original question: Is it possible to express the vague phrase that 'some quantity $A(x)$ varies slowly on certain scale $ds$ of parameter $x$' concisely in mathematical terms in similar spirit (ie using an apropriate " $blabla \ \ll \ blabla$" condition like you did it for WKB? $\endgroup$
    – user267839
    Nov 16, 2022 at 23:31
  • $\begingroup$ The point is that for WKB you used for the formulation of the condition as additional input the knowledge about the shape of the wave function. But say we want to express a precise mathematical condition to the phrase that some quantity $A(x)$ varies slowly on certain scale of variable $x$ assuming we have no additional information about $A(x)$. Trying to mimic your approach can this condition on slow variation be always expressed mathematically as $\left|\frac{A''(x) \cdot ds}{A'(x)}\right|\ll 1$ or is the question too broad? $\endgroup$
    – user267839
    Nov 16, 2022 at 23:31
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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; $\S46$.
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  • $\begingroup$ I have still a sneaky question: What about a generalization? Can in general the phrase that 'some value $A(x)$ varies slowly on certain scale $ds$ of argument $x$' be expressed precisely in mathematical terms in similar spirit like in WKB case? Note here we haven't any 'auxilary' constructions like the suitable generalization of the de Broglie wavelength function $\lambda(x) $ or the the wave function $\Psi(x)$ used in Hossein's answer in our quiver. My naive guess was that it can be $\endgroup$
    – user267839
    Nov 16, 2022 at 23:15
  • $\begingroup$ expressed like '$\vert \frac{A(x_0)-A(x)}{A(x_0)} \vert \ll 1$ for $x \in [x_0-ds, x_0+ds]$' but this would fail if eg $A(x_0) \approx 0$. So my question is if you know if it's possible to express the often used phrase in physics that some quantity $A(x)$ varies slowly on certain scale $ds$ of parameter $x$ can be formulated mathematically or is it in that generality not possible? $\endgroup$
    – user267839
    Nov 16, 2022 at 23:15

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