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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?

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  • $\begingroup$ @KP99 Why then the incident plane wave is $\psi = e^{ikz}$? It should be a more generic $e^{i \mathbf{k.r}}$ $\endgroup$ Oct 25, 2022 at 21:36
  • $\begingroup$ Is it a typo? Did you look up the errata for this edition of Griffiths? $\endgroup$
    – hft
    Oct 25, 2022 at 21:47
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    $\begingroup$ What edition are you looking at? $\endgroup$
    – hft
    Oct 25, 2022 at 21:47
  • $\begingroup$ 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. $\endgroup$
    – hft
    Oct 25, 2022 at 21:49

2 Answers 2

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. . . . . 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.

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  • $\begingroup$ 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. $\endgroup$ Oct 25, 2022 at 22:55
  • $\begingroup$ 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 $\endgroup$ Oct 25, 2022 at 22:55
  • $\begingroup$ @ClaudioSaspinski In my edition my quote and diagram from Griffiths is page 368, line following 11.69 $\endgroup$
    – Farcher
    Oct 25, 2022 at 22:59
  • $\begingroup$ Same as mine. what I've posted before is a text from an errata from the site vdoc.pub/download/… $\endgroup$ Oct 25, 2022 at 23:40
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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.

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