Abstract
Scattering is of crucial importance for investigating the structure of matter. It is no coincidence that the perhaps most expensive experiment on earth is the Large Hadron Collider (CERN), where the analysis of high-energy scattering processes has given information about the Higgs particle, whose existence was until 2012 only postulated.
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- 1.
We make this assumption for simplicity; the situation for long-range potentials can also be treated.
- 2.
The number \(dN\) of particles passing in a time \(dt\) through the area \(d \mathbf {A}\) is related to the current density \({\mathbf {j}}\) by \(dN=\) \({\mathbf {j}} \cdot d{\mathbf {A}}\,dt\).
- 3.
The surface area of a sphere of radius \(r\) is \(4\pi r^{2}\).
- 4.
We repeat the already familiar note: We refer to \(e^{ikz}\) etc. as waves, since we take tacitly into account the factor \(e^{-i\omega t}\).
- 5.
One uses wave packets, among other things, for the explicit graphical representation of scattering processes, as we have done in Chap. 15. For reasons of space, we omit this here and refer to other textbooks on quantum mechanics.
- 6.
For general potentials \(V\left( \mathbf {r}\right) \), we have no single radial equation as in the case \(V(r)\), but rather systems of coupled radial equations, which couple eigenfunctions of different angular momenta. The reason is that we need to expand the potential \(V\) and the wavefunction \( \varphi \) in terms of the angular momentum (multipole expansion). In the product \(V\varphi \), total angular momenta occur according to the laws of angular momentum addition. If we sort according to these total angular momenta, we obtain coupled systems of radial equations (see exercises).
- 7.
We note again that we assume an infinitely heavy scattering center, so that the mass of the scattered quantum object and not the reduced mass enters.
- 8.
These two conditions are necessary for the square integrability of the wavefunction, see Chap. 17.
- 9.
They can be calculated e.g. by means of the recursion relation \(j_{l}\left( x\right) =\left( -1\right) ^{l}x^{l}\left( \frac{1}{x}\frac{d}{dx}\right) ^{l}\frac{\sin x}{x}\).
- 10.
Note that due to the spherical symmetry of the potential, \(f\left( \vartheta \right) \) is independent of \(\varphi \). For general potentials, we have \( f=f\left( \vartheta ,\varphi \right) \).
- 11.
The point where \(E=V\); a classical object must reverse at this point, i.e. it is reflected. See also Chap. 15.
- 12.
In addition, there are solutions in the form of incoming spherical waves (mathematically on equal footing). But we can neglect them for physical reasons, because we want to describe scattering processes.
- 13.
This is none other than a somewhat technical formulation of Huygens’ principle.
- 14.
Of course it is not an exact explicit solution—this does not exist in general.
- 15.
In order to simplify the notation, we omit here the normalization factor \( \left( 2\pi \right) ^{-3/2}\), cf. Chap. 12, Vol. 1.
- 16.
More precisely: \(\left\| Gv\left| \psi \right\rangle \right\| \ll \left\| \left| \psi _{0}\right\rangle \right\| \).
- 17.
For the integration, we choose the \(\mathbf {q}\) axis as \(z\) axis; the integration then runs over the spherical coordinates \((r^{\prime },\vartheta ^{\prime },\varphi ^{\prime })\).
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Pade, J. (2014). Scattering. In: Quantum Mechanics for Pedestrians 2: Applications and Extensions. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00813-4_25
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DOI: https://doi.org/10.1007/978-3-319-00813-4_25
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