.. _sphx_glr_api_gallery_particle_kinematics:

Relativistic kinematics
--------------------------

Mass-energy equivalence, Compton scattering, and Mandelstam variables --
the four-vector machinery in :mod:`physicskit.particle.kinematics` and
:mod:`physicskit.particle.scattering`.


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


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    <div class="sphx-glr-thumbcontainer" tooltip="In the last of his 1905 papers, Einstein showed that a body&#x27;s rest mass and rest energy are the same quantity, E_0=mc^2 -- the rest-frame special case of the fully relativistic relation E^2=(pc)^2+(mc^2)^2. FourVector builds exactly this object, with its .mass property returning the invariant \sqrt{E^2-|\vec p|^2}. This example checks that the invariant mass a moving observer computes agrees exactly with the rest energy measured in the particle&#x27;s own rest frame -- true at every boost velocity, however fast -- and that summing four-vectors before taking the mass, rather than after, is what makes invariant mass differ from simply adding up individual particle masses.">

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  .. image:: /api/gallery/particle/kinematics/images/thumb/sphx_glr_plot_01_mass_energy_equivalence_thumb.png
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  :doc:`/api/gallery/particle/kinematics/plot_01_mass_energy_equivalence`

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      <div class="sphx-glr-thumbnail-title">Einstein's mass-energy equivalence</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="X-rays scattered off light elements came back with a wavelength shift depending only on the scattering angle -- a result the classical wave picture of light could not explain. Compton (1923) treated the X-ray as a genuine relativistic particle, a photon of energy E=h\nu and momentum p=h\nu/c, colliding elastically with a free electron and imposing ordinary four-momentum conservation on the two-body collision \gamma+e^-\to\gamma&#x27;+e^{-\prime}. This example does not assume the textbook formula at all: it solves the conservation equations directly with FourVector arithmetic and a numerical root-find for the outgoing photon energy, then checks the result against the closed-form Compton formula.">

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  .. image:: /api/gallery/particle/kinematics/images/thumb/sphx_glr_plot_02_compton_scattering_thumb.png
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  :doc:`/api/gallery/particle/kinematics/plot_02_compton_scattering`

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      <div class="sphx-glr-thumbnail-title">Compton scattering: the photon as a relativistic particle</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Mandelstam (1958) introduced three Lorentz-invariant combinations of a 2-to-2 collision&#x27;s four-momenta, s=(p_1+p_2)^2, t=(p_1-p_3)^2, u=(p_1-p_4)^2, satisfying s+t+u=\sum_i m_i^2 for on-shell particles -- frame-independent coordinates for any scattering amplitude or cross section. This example computes all three with mandelstam_s, mandelstam_t, and mandelstam_u for an elastic 2\to2 scatter, checks the sum rule, and checks that s itself -- unlike the lab-frame energy of either particle -- comes out identical whether evaluated in the lab frame or after boosting the whole event into a new frame with boost_to_com.">

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  .. image:: /api/gallery/particle/kinematics/images/thumb/sphx_glr_plot_03_mandelstam_variables_thumb.png
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  :doc:`/api/gallery/particle/kinematics/plot_03_mandelstam_variables`

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      <div class="sphx-glr-thumbnail-title">Mandelstam variables and relativistic collision kinematics</div>
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.. toctree::
   :hidden:

   /api/gallery/particle/kinematics/plot_01_mass_energy_equivalence
   /api/gallery/particle/kinematics/plot_02_compton_scattering
   /api/gallery/particle/kinematics/plot_03_mandelstam_variables