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I am reading Quantum Mechanics book by N. Zettili (2nd ed.) and encountered something confusing in the chapter of scattering theory, section 11.5: scattering of identical particles.

This is the problem:

First it is mentioned that for two classical identical particles, whose interaction potential is central the differential scattering cross section is given by (in center of mass frame),

$$ (d \sigma (\theta)/d \Omega )_{cl} = |f(\theta)|^2 + |f(\pi - \theta)|^2 $$

Here $f(\theta)$ is the scattering amplitude, which appears in the scattered wave function. Then after a few lines it is mentioned that for distinguishable particles when $ \theta = \pi/2 $, the differential cross section is $ |f(\pi/2)|^2 $.

I think classical identical particles are always distinguishable. So, it should be $ 2 |f(\pi/2)|^2 $, not $ |f(\pi/2)|^2 $, according to the 1st formula.

What am I missing here ?

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    $\begingroup$ "I think classical identical particles are always distinguishable." Why? If you have two identical classical particles and you shoot both of them into a box where they scatter off each other and then one of them comes out of the box at a certain angle, how are you going to tell "which one of them" it was? $\endgroup$
    – ACuriousMind
    Nov 10, 2023 at 12:06
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    $\begingroup$ @ACuriousMind I might be wrong, but I always thought that classical particle means a macroscopic particle. So even if they look identical, they might have some other character to distinguish them. Like some molecules of a gas, they are identical but they might have some internal modes such as vibrations and rotations etc, which we can use to distinguish them. $\endgroup$ Nov 10, 2023 at 12:41
  • $\begingroup$ @ACuriousMind In your experiment, suppose we have two baseballs and we put two different radioactive substances inside them, then they may look identical but they will radiate different particles. So we can distinguish them. I think classical identical particles means, they just look identical but internally they may differ. Correct me if I am wrong. $\endgroup$ Nov 10, 2023 at 12:48
  • $\begingroup$ What is $f$? How is it defined? There does appear to be an inconsistency given what you've described, but without more context it's impossible to say what it might be (or if it actually exists.) $\endgroup$ Nov 10, 2023 at 12:52
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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Nov 10, 2023 at 12:53

1 Answer 1

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Well, I think I have figured it out now. For two distinguishable particles the formula for differential scattering cross section will be (in center of mass frame),

$$( d \sigma (\theta)/d \Omega)_{distinguishable} = \frac{1}{4}(|f_1 (\theta)|^2 + |f_2 (\pi - \theta)|^2 + |f_2 (\theta)|^2 + |f_1 (\pi - \theta)|^2). $$

It's kind of obvious and the factor $1/4$ appears because all 4 possibilities are equally probable. So, when $\theta = \pi/2$,

$$ ( d \sigma (\theta)/d \Omega)_{distinguishable} = \frac{1}{4}(2|f_1 (\pi/2)|^2 + 2 |f_2 (\pi/2)|^2 ).$$

Now if the particles are identical, then $f_1 = f_2 = f $. Therefore the differential cross section would be $|f(\pi/2)|^2.$

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  • $\begingroup$ Books referred: Quantum Mechanics by N. Zettili $\endgroup$ Dec 4, 2023 at 13:36
  • $\begingroup$ J J Sakurai modern quantum mechanics was helpful too. $\endgroup$ Dec 4, 2023 at 13:37

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