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How to Relate the Scattering Amplitude and Differential Cross Section of Spinless Particles

The scattering amplitude of spinless particles is crucial to understanding scattering from the quantum physics point of view. To see that, take a look at the current densities, Jinc (the flux density of a given incident particle) and Jsc (the current density for a given scattered particle):

image0.png

(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)

Inserting your expressions for

image1.png

into these equations gives you the following, where

image2.png

is the scattering amplitude:

image3.png

Now in terms of the current density, the number of particles

image4.png

scattered into

image5.png

and passing through an area

image6.png

Plugging in

image7.png

into the preceding equation gives you

image8.png

Also, recall that

image9.png

You get

image10.png

And here's the trick — for elastic scattering, k = k0, which means that this is your final result:

image11.png

The problem of determining the differential cross section breaks down to determining the scattering amplitude.

To find the scattering amplitude — and therefore the differential cross section — of spinless particles, you work on solving the Schrödinger equation:

image12.png

You can also write this as

image13.png

You can express the solution to that differential equation as the sum of a homogeneous solution and a particular solution:

image14.png

The homogeneous solution satisfies this equation:

image15.png

And the homogeneous solution is a plane wave — that is, it corresponds to the incident plane wave:

image16.png

To take a look at the scattering that happens, you have to find the particular solution. You can do that in terms of Green's functions, so the solution to

image17.png

This integral breaks down to

image18.png

You can solve the preceding equation in terms of incoming and/or outgoing waves. Because the scattered particle is an outgoing wave, the Green's function takes this form:

image19.png

You already know that

image20.png

So substituting

image21.png

into the preceding equation gives you

image22.png
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