.. _sphx_glr_api_gallery_semiclassical_path_integral:

Feynman paths and the classical limit
------------------------------------------

Feynman's sum-over-histories path integral, sampled by Monte Carlo for the
exactly-solvable free particle and harmonic oscillator, showing the
phasor sum concentrate onto the classical (stationary-action) trajectory
as :math:`\hbar\to 0`.


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    <div class="sphx-glr-thumbcontainer" tooltip="Feynman&#x27;s path-integral formulation (Feynman &amp; Hibbs 1965, building on Dirac&#x27;s 1933 &quot;The Lagrangian in Quantum Mechanics&quot;) writes the quantum propagator as a sum over every continuous path x(t) connecting fixed endpoints, each weighted by a phase built from its classical action S[x(t)]. The same construction also applies to the harmonic oscillator, whose classical path curves rather than running straight between the fixed endpoints (~physicskit.semiclassical.core.path_integral.harmonic_oscillator_classical_path, harmonic_oscillator_classical_action), showing that the same closest-to-classical-first phasor concentration at small \hbar is not an artifact of the free particle&#x27;s trivially straight classical trajectory:">

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  .. image:: /api/gallery/semiclassical/path_integral/images/thumb/sphx_glr_plot_feynman_paths_classical_limit_thumb.png
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  :doc:`/api/gallery/semiclassical/path_integral/plot_feynman_paths_classical_limit`

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      <div class="sphx-glr-thumbnail-title">Feynman paths and the classical limit</div>
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.. toctree::
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   /api/gallery/semiclassical/path_integral/plot_feynman_paths_classical_limit