In the last video, we started exploring how to quantify the validity of the WKB approximation in terms of a slowly varying potential. And we arrived at an inequality that said that over this span of a De Broglie wavelength, the rate of change of the potential energy has to be much, much smaller than the local kinetic energy. And doing so, we found this inequality, which is at the right consequence of the assumptions that went into our derivation of the WKB solutions. And in this video, I'd like to take a different route and show that we can also derive an inequality for the validity of the WKB approximation completely in terms of the De Broglie wavelength. So we can rewrite this inequality by bringing over this piece square to this side. So that this has to be much, much smaller than one. This in turn can be rewritten as DDX of H bar over PX. H bar is just a constant, so we can bring it into the derivative. And by the chain rule, if you take the derivative of 1 over PX, you recover this term over here. This again has to be much, much smaller than one. We can recognize this as being proportional to the local De Broglie wavelength. So again, up to a factor of 2 pi, but our inequality still holds. So what this translates to is saying that our local De Broglie wavelength has to vary, very, very slowly in space. And we're still looking at absolute changes. We don't care if it's positive or negative. We can get a better physical interpretation of this or set a physical scale for this rate of change by multiplying both sides by lambda db. So what this inequality now says is the rate of change of the De Broglie wavelength over the span of the De Broglie wavelength has to be much, much smaller than the value of the wavelength at position X. So this sets a different scale. If you look at the rate of change over an entire De Broglie wavelength, that quantity has to be much, much smaller than the wavelength itself. So this is an alternative way of stating the conditions under which the wkb approximation is valid. Finally, there's one more situation that we have to be careful with, and that's at the classical turning points. This local momentum goes to a small value. So our wkb solutions will blow up at those turning points if we just take them at face value. So we need to develop a way to patch our solutions so that they remain valid close to the classical turning points. And we'll begin looking at that in the next video.