# How can $k$ and $k'$ be vectors of different directions on Griffiths chapter on scattering?

The last chapter of the Griffiths book on quantum mechanics is about scattering. When dealing with the Born approximation, he says that the scattering wave has a direction $$\mathbf k'$$, different from the incident wave, with direction $$\mathbf k$$. And that $$\mathbf k' = k\hat z$$, that means it has $$z$$ direction and the same magnitude of $$\mathbf k$$.

But the incident wave is $$\psi(\mathbf r) = e^{ikz}$$. I understand this equation as: $$\psi(\mathbf r) = e^{i\mathbf{k.r}}$$, where $$\mathbf k = (0,0,k)$$ and $$\mathbf r = (x,y,z)$$. So the incident wave has also $$z$$ direction.

What am I missing?

• @KP99 Why then the incident plane wave is $\psi = e^{ikz}$? It should be a more generic $e^{i \mathbf{k.r}}$ Oct 25, 2022 at 21:36
• Is it a typo? Did you look up the errata for this edition of Griffiths?
– hft
Oct 25, 2022 at 21:47
• What edition are you looking at?
– hft
Oct 25, 2022 at 21:47
• But anyways, you are correct, trivially, that if both $\vec k$ and $\vec k'$ are in the $\hat z$ direction then they are both in the same direction. So Grifftihs' words and symbols are not consistent if what you are saying about his book is true.
– hft
Oct 25, 2022 at 21:49

. . . . . he says that the scattering wave has a direction $$k′$$, different from the incident wave, with direction $$k$$ . . . .
Griffiths writes In case you have lost track of the definitions of $${\bf k}[= k\hat r]$$ and $${\bf k'}[=k\hat z]$$, they both have magnitude $$k$$, but the former points in the direction of the incident beam, while the latter points towards the detector - see Figure 11.10.

• hft suggested look for an errata. I found one in the web, and it seems that k is the scattered and k' the incident wave. Oct 25, 2022 at 22:55
• page 368, line following 11.69: take the prime off the second k and put it on the first one; Fig. 11.10: remove arrows over all bold letters (5 times); in the caption take the prime off the second k and put it on the first one Oct 25, 2022 at 22:55
• @ClaudioSaspinski In my edition my quote and diagram from Griffiths is page 368, line following 11.69 Oct 25, 2022 at 22:59
• Same as mine. what I've posted before is a text from an errata from the site vdoc.pub/download/… Oct 25, 2022 at 23:40

As I said in a comment, I found an errata in the web saying that $$\mathbf k$$ and $$\mathbf k'$$ are mistakenly identified as incident wave and scattering wave respectively in the first edition.

In reality, $$\mathbf k$$ is previously defined as $$\mathbf k = k\hat r$$, where $$\hat r$$ is the unit vector that defines the orientation of the wave.

$$\mathbf k'$$ is defined as $$\mathbf k' = k\hat z$$, where $$\hat z$$ is the unit vector in the z-direction, which was by hypothesis the incident wave.

So, the correct information is: $$\mathbf k'$$ refers to the incident wave and $$\mathbf k$$ to the scattered wave.

That definitions are used in the Born approximation, that supposes that the incident wave is not much modified by the potencial $$V(\mathbf {r_0})$$ around $$\mathbf {r_0} = 0$$, where the target particule is located.