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I am following Sakurai's Modern Quantum Mechanics, 3ed.

Define the scattering, or S-matrix elements as $$S_{ni} \equiv \delta_{ni} - 2\pi i\delta(E_n - E_i)T_{ni}.$$ We can then derive the differential scattering cross section $$\frac{d\sigma}{d\Omega} = \left(\frac{mL^3}{2\pi\hbar^2}\right)^2 \lvert T_{ni}\lvert^2.$$ How do I interpret the differential scattering cross section purely in quantum mechanical terms? For instance, "...the differential scattering cross section literally is defined to be the time derivative of the probability amplitude corresponding from a transition from an initial state to a final state per probability current per solid angle, which physically means..."

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    $\begingroup$ We need to know sufficient detail / clarity from you to know what kind of answer you would be expecting / accepting. $\endgroup$
    – naturallyInconsistent
    Commented Apr 10 at 5:38
  • $\begingroup$ @naturallyInconsistent I've tried to add clarity $\endgroup$
    – Silly Goose
    Commented Apr 10 at 5:45
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    $\begingroup$ Well, you said that you do not want a classical viewpoint of what this quantity is meant to be, but the whole point is that the quantum mechanical quantity that is so defined, corresponds to the quantum mechanical theoretical derivation, i.e. numerical value supplication, of the relevant classically interesting quantity. It is difficult to express it more than what is already being expressed in the quotation you have now inserted. $\endgroup$
    – naturallyInconsistent
    Commented Apr 10 at 5:49
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    $\begingroup$ Consulting a text-book dedicated to (quantum mechanical) scattering theory (e.g. Taylor) might give you a satisfactory answer with all necessary details. $\endgroup$
    – Hyperon
    Commented Apr 10 at 6:16

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