.. _sphx_glr_api_gallery_quantum_scattering:

Scattering theory
-----------------

Three-dimensional potential scattering: Rutherford's formula recovered
from the Born approximation, and the partial-wave phase shifts of the
hard sphere and the square well.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Max Born&#x27;s 1926 collision theory gives the scattering amplitude, to first order in the potential, as the Fourier transform of V(r) at the momentum transfer q = 2k\sin(\theta/2). For a screened Coulomb (Yukawa) potential V = g\,e^{-\mu r}/r it is f_B = -2mg/(\mu^2 + q^2). Removing the screening gives Rutherford&#x27;s classical 1911 cross section exactly. Here the numerical born_amplitude is compared with the closed form and with rutherford_dsigma_domega.">

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  .. image:: /api/gallery/quantum/scattering/images/thumb/sphx_glr_plot_born_approximation_rutherford_thumb.png
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  :doc:`/api/gallery/quantum/scattering/plot_born_approximation_rutherford`

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      <div class="sphx-glr-thumbnail-title">The Born approximation: Rutherford scattering from quantum mechanics</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Faxen and Holtsmark (1927) expanded scattering from a central potential in angular-momentum partial waves. Each wave \ell is changed only by a phase shift \delta_\ell, and \sigma = \frac{4\pi}{k^2}\sum_\ell(2\ell+1)\sin^2\delta_\ell. The hard sphere has exact phase shifts (~physicskit.quantum.chapters.scattering.hard_sphere_phase_shifts); its cross section goes from 4\pi a^2 at low energy to 2\pi a^2 at high energy, twice the geometric area. For an attractive square well, partial_wave_phase_shifts integrates Calogero&#x27;s variable-phase equation and finds the jumps of \delta_0 by \pi as bound states appear (Levinson&#x27;s theorem).">

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  .. image:: /api/gallery/quantum/scattering/images/thumb/sphx_glr_plot_partial_wave_phase_shifts_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/scattering/plot_partial_wave_phase_shifts`

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      <div class="sphx-glr-thumbnail-title">Partial-wave analysis: phase shifts of the hard sphere and the square well</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/quantum/scattering/plot_born_approximation_rutherford
   /api/gallery/quantum/scattering/plot_partial_wave_phase_shifts