.. _sphx_glr_api_gallery_quantum_relativistic:

Relativistic quantum mechanics
------------------------------

The Dirac equation's exact hydrogen levels and their fine structure, and
Klein's paradox: how Dirac and Klein-Gordon particles scatter off a
potential step taller than twice their rest energy.


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    <div class="sphx-glr-thumbcontainer" tooltip="Dirac&#x27;s 1928 equation predicts the hydrogen levels exactly (apart from the tiny QED Lamb shift). Its energies, from dirac_hydrogen_energy, depend on n and j only: each Bohr level n splits into n fine-structure sublevels, with 2S_{1/2} and 2P_{1/2} still degenerate. The splittings grow as (Z\alpha)^4, so they become large in heavy hydrogen-like ions.">

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  .. image:: /api/gallery/quantum/relativistic/images/thumb/sphx_glr_plot_dirac_hydrogen_fine_structure_thumb.png
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  :doc:`/api/gallery/quantum/relativistic/plot_dirac_hydrogen_fine_structure`

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      <div class="sphx-glr-thumbnail-title">The Dirac equation and the fine structure of hydrogen</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="In 1929 Oskar Klein found that a Dirac electron hitting a step V_0 higher than E + mc^2 is partly transmitted, however tall the step. The transmitted wave is a negative-energy state: the step bends the antiparticle continuum down to the particle&#x27;s energy. dirac_step_scattering and klein_gordon_step_scattering show the three regimes: ordinary transmission, total reflection, and the Klein zone. There, fermions have R &lt; 1 but spin-0 bosons have R &gt; 1.">

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  .. image:: /api/gallery/quantum/relativistic/images/thumb/sphx_glr_plot_klein_paradox_thumb.png
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  :doc:`/api/gallery/quantum/relativistic/plot_klein_paradox`

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      <div class="sphx-glr-thumbnail-title">Klein's paradox: relativistic particles at a tall potential step</div>
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.. toctree::
   :hidden:

   /api/gallery/quantum/relativistic/plot_dirac_hydrogen_fine_structure
   /api/gallery/quantum/relativistic/plot_klein_paradox