.. _sphx_glr_api_gallery_semiclassical_propagators:

Semiclassical propagators
-----------------------------

The Van Vleck-Morette single-trajectory propagator, the Maslov index's
topological caustic-counting correction, and the Herman-Kluk
multi-trajectory frozen-Gaussian propagator, each checked against exact
quantum evolution of the harmonic oscillator.


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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="Michael Herman and Edward Kluk&#x27;s 1984 method propagates a quantum wavepacket by summing many classical trajectories rather than following just one: launch one fixed-width (&quot;frozen&quot;) Gaussian wavepacket from every point of a phase-space grid under the initial state, propagate each one along its own classical trajectory, and sum the results with a monodromy-built prefactor and classical action phase,">

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  .. image:: /api/gallery/semiclassical/propagators/images/thumb/sphx_glr_plot_herman_kluk_wavepacket_thumb.png
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  :doc:`/api/gallery/semiclassical/propagators/plot_herman_kluk_wavepacket`

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      <div class="sphx-glr-thumbnail-title">Herman-Kluk frozen-Gaussian wavepacket propagation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The Van Vleck-Morette semiclassical propagator (/api/gallery/semiclassical/propagators/plot_semiclassical_propagators) needs a topological correction beyond its classical action and stability prefactor: every time a classical trajectory crosses a caustic (a focal point where \partial q_t/\partial p_0=0), the propagator picks up an extra phase of -\pi/2. Joseph Keller&#x27;s corrected Bohr-Sommerfeld (EBK) quantization condition built exactly this caustic-counting correction into the old quantum theory, and Viktor Maslov later showed that the resulting phase count -- the Maslov index \mu -- is a topological invariant of the trajectory itself, rather than an artifact of the coordinates used to compute it.">

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  .. image:: /api/gallery/semiclassical/propagators/images/thumb/sphx_glr_plot_maslov_index_caustics_thumb.png
    :alt:

  :doc:`/api/gallery/semiclassical/propagators/plot_maslov_index_caustics`

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      <div class="sphx-glr-thumbnail-title">The Maslov index: caustics and Keller's quantization condition</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Reconstructs the quantum propagator of a unit-mass, unit-frequency harmonic oscillator, V(q)=\tfrac12 m\omega^2q^2 with m=\omega=1, from a single classical trajectory alone -- no Schrodinger-equation solve involved.">

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  .. image:: /api/gallery/semiclassical/propagators/images/thumb/sphx_glr_plot_semiclassical_propagators_thumb.png
    :alt:

  :doc:`/api/gallery/semiclassical/propagators/plot_semiclassical_propagators`

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      <div class="sphx-glr-thumbnail-title">The Van Vleck-Morette semiclassical propagator</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/semiclassical/propagators/plot_herman_kluk_wavepacket
   /api/gallery/semiclassical/propagators/plot_maslov_index_caustics
   /api/gallery/semiclassical/propagators/plot_semiclassical_propagators