Wave function of a scattered particle and cross section

[dRic2]

In Landau-Lifsits's book about non relativistic QM it is said that if I have a particle described by a plane wave $\phi = e^{ikz}$ (I think he choses the $z$ direction for simplicity) the wave function after the scattering event is (far from the scattering event)
$$\psi \approx e^{ikz} + f(\theta) \frac{e^{ikr}} r$$
But then he said that the square value of $\psi$ is just $|f(\theta)|^2$ because we neglect the contribution of the first term (that is, we neglect some sort of interference phenomena). My first question is: then why bother says $\psi = e^{ikz} + f(\theta) \frac{e^{ikr}} r$ instead of simply putting $\psi = f(\theta) \frac{e^{ikr}} r$ ?

Then he says that the differential cross section is just $|f(\theta)|^2$ but I don't really follow the given explanation.

Can someone provide any hint?

Thanks
Ric

 

[Charles Link]

In most scattering experiments, there is the main beam, often almost as strong as the incident beam, that passes through unscattered. This term, represented by $\psi=e^{ikz}$ is discarded when computing $\frac{d \sigma}{d \Omega}$. In computing $\psi^* \psi$, there is a middle term consisting of products of $e^{ikz}$ and $\frac{f(\theta)e^{ikr}}{r}$ (a sum of two terms with the appropriate complex conjugates, etc.) . I think that term is being called the interference term, and it generally gets discarded, presumably because it is small. I'm no expert on this particular topic, but I believe my inputs here are accurate. Perhaps someone with expertise in this area can concur. $\\$ Edit: A text that covers this topic reasonably well is Squires' Introduction to the theory of Thermal Neutron Scattering. He shows how the differential cross section results from the scattered part of the wave function: $\frac{d \sigma}{d \Omega}=\frac{I_{scattered}}{E_{incident}}$, where $E_{incident}$ is watts/m^2 and $E_{scattered}=\frac{I_{scattered}}{r^2}$. Thereby, the $\frac{1}{r}$ of the scattered wave function isn't used in computing the cross section. (Note that irradiance $E$ is proportional to $\psi^* \psi$ for both the incident and the scattered beams). In Squires' text, he does the computation by using $\psi_{scattered}=\frac{f(\theta)e^{ikr}}{r}$, rather than using the complete wave function, with $E_{scattered}$ being proportional to $\psi^*_{scattered} \psi_{scattered}$, with the result that $I_{scattered}$ is proportional to $|f(\theta)|^2$. $\\$ And I gave your question some additional thought: The incident beam is usually somewhat localized, so that $\psi_{incident}=e^{ikz}$ can be a good description of the incident beam in the region where the incident beam has an approximately uniform flux, but it doesn't adequately describe the beam outside of this region. The question you ask is a good one=why not discard the $e^{ikz}$ part in treating $\psi_{scattered}$? Squires basically discards that part, and works with $\psi_{scattered}$.

 

[vanhees71]

This becomes much clearer, when you think about physical states rather than plane-wave representations. You have an asymptotic free incoming particle, described by a localized wave packet running towards the place where the potential is effective and you observe the scattered particle at a far distance from this place as an asymptotic outgoing state.

You get this description in terms of wave packets from the plane-wave scattering state by Fourier transformation. The plane-wave piece translates into a wave packet that moves as if it were unaffected by the potential. This is pretty much peaked in forward direction (i.e., in direction of the incoming particle). The spherical-wave part refers to the really scattered particles, and that's what's of main interest in terms of the cross section. Thus cutting out some small solid angle around the direction of the incoming particle leads to the conclusion of Landau Lifshitz, i.e., that the appropriate matrix element for the cross section is simply $|f(\theta)|^2$.

For a good discussion of scattering along these arguments, see the textbook by Messiah.

 

[dRic2]

Sorry for my late reply. Basically they include in "scattered" wave function the non scattering event, right ? Sounds reasonable, but a bit counterintuitive to me... at least I still don't get the point of adding that contribution if it will be discarded lately (it represents a result I'm not interested in).

I'll check the books you both recommended.

Thank you very much.

 

[vanhees71]

You need it to derive the "optical theorem", which is very important too!

 

[dRic2]

You need it to derive the "optical theorem"

it -> the book, or was it a typo, meaning I should derive it as an exercise ?

 

[Charles Link]

@dRic2 Here is a thread that you might find of interest in regard to the differential cross section: https://www.physicsforums.com/threa...a-rutherfords-experiment.965947/#post-6131309 I think it is relevant in that some of the scattering theory is not written up as well as it could be, for the quantum mechanical case, or otherwise, and it helps to have a good handle on what the differential cross section refers to.

 

[vanhees71]

it -> the book, or was it a typo, meaning I should derive it as an exercise ?

I mean the complete asymptic wave function, including the incoming asymptotic free wave. As I said, a very thorough discussion of scattering theory (using the traditional wave-mechanics approach, which of course is particularly intuitive for scattering theory to begin with) is the famous textbook by Messiah:

Here's a Dover reprint, combining the two volumes of the original edition into one volume:

A. Messiah, Quantum Mechanics, Dover Publications, New York (1999).

 

[dRic2]

Thanks to both for the replies