:orphan:

Examples
========

Runnable, self-contained scripts demonstrating every module of
``physicskit.semiclassical`` -- WKB/Bohr-Sommerfeld quantization,
semiclassical propagators, the Gutzwiller trace formula, and quantum
scars.

See also the narrative tutorials:

- :doc:`/tutorials/semiclassical_wavepacket_propagation`
- :doc:`/tutorials/gutzwiller_periodic_orbit_spectrum`

Each script in this gallery is self-contained and can be run directly with
``python examples/semiclassical/<section>/<script>.py``. Every script also
carries an RST module docstring as its title/description and uses ``# %%``
markers to split narrative text from code, which is exactly what
Sphinx-Gallery renders into the pages below -- the script *is* the source
of truth for what you see, not a copy of it.

Sections
--------

- **wkb** -- classical momentum, turning points, WKB wavefunctions, and
  Bohr-Sommerfeld (EBK) quantization, checked against the exact harmonic
  oscillator spectrum and eigenstates.
- **propagators** -- the Van Vleck-Morette single-trajectory propagator
  (with its Maslov-index caustic count) and the Herman-Kluk
  multi-trajectory frozen-Gaussian propagator, both checked against exact
  quantum evolution.
- **gutzwiller** -- the exact 1D Gutzwiller trace formula, reconstructing
  a spectrum from a single classical periodic orbit's action and period,
  and the general isolated-orbit stability amplitude.
- **scarring** -- quantum scars on the stadium billiard's bouncing-ball
  orbit family, and a Husimi phase-space view of a scarred eigenstate.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

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    </div>

The Gutzwiller trace formula
--------------------------------

The exact one-dimensional Gutzwiller trace formula, reconstructing a
spectrum purely from a single classical periodic orbit's action and
period, and the general isolated-orbit stability amplitude used beyond
the exactly-solvable 1D case.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Reconstructs the quantum density of states g(E)=\sum_n\delta(E-E_n) of a unit-mass, unit-frequency harmonic oscillator, V(x)=\tfrac12m\omega^2x^2 with m=\omega=\hbar=1, purely from its classical period T(E) and action S(E), with no reference to the quantum eigenstates themselves (~physicskit.semiclassical.core.gutzwiller.classical_period, gutzwiller_density_of_states). A bound 1D system has exactly one periodic orbit per energy, so Poisson-summing the discrete Bohr-Sommerfeld spectrum over its repetitions r gives an identity that is exact here -- not merely leading order in \hbar:">

.. only:: html

  .. image:: /api/gallery/semiclassical/gutzwiller/images/thumb/sphx_glr_plot_gutzwiller_trace_formula_thumb.png
    :alt:

  :doc:`/api/gallery/semiclassical/gutzwiller/plot_gutzwiller_trace_formula`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Gutzwiller's trace formula</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

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`.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

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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:">

.. only:: html

  .. image:: /api/gallery/semiclassical/path_integral/images/thumb/sphx_glr_plot_feynman_paths_classical_limit_thumb.png
    :alt:

  :doc:`/api/gallery/semiclassical/path_integral/plot_feynman_paths_classical_limit`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Feynman paths and the classical limit</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

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.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

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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,">

.. only:: html

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

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

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Herman-Kluk frozen-Gaussian wavepacket propagation</div>
    </div>


.. raw:: html

    <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.">

.. only:: html

  .. 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`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Maslov index: caustics and Keller's quantization condition</div>
    </div>


.. raw:: html

    <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.">

.. only:: html

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

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

.. raw:: html

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


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Quantum scars
-----------------

The stadium billiard's bouncing-ball scarred eigenstate family, the
scar-enhancement diagnostic, and a Husimi phase-space view of a scarred
state's wall slice.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Heller (1984) discovered that individual eigenstates of a classically chaotic system are not always the featureless, ergodically-spread blobs random-matrix intuition suggests: a fraction instead show enhanced probability density concentrated in a tube around one particular unstable classical periodic orbit -- a scar.">

.. only:: html

  .. image:: /api/gallery/semiclassical/scarring/images/thumb/sphx_glr_plot_quantum_scars_thumb.png
    :alt:

  :doc:`/api/gallery/semiclassical/scarring/plot_quantum_scars`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Quantum scars in a chaotic billiard</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

WKB and Bohr-Sommerfeld quantization
---------------------------------------

The classical momentum and turning points, the WKB wavefunction, and the
Bohr-Sommerfeld (EBK) quantization rule, checked against the exact
harmonic-oscillator spectrum and eigenstates.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Applies the WKB approximation to a unit-mass, unit-frequency harmonic oscillator, V(x)=\tfrac12m\omega^2x^2 with m=\omega=\hbar=1. Builds the classical momentum">

.. only:: html

  .. image:: /api/gallery/semiclassical/wkb/images/thumb/sphx_glr_plot_wkb_bohr_sommerfeld_thumb.png
    :alt:

  :doc:`/api/gallery/semiclassical/wkb/plot_wkb_bohr_sommerfeld`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">WKB and Bohr-Sommerfeld quantization</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/semiclassical/gutzwiller/index.rst
   /api/gallery/semiclassical/path_integral/index.rst
   /api/gallery/semiclassical/propagators/index.rst
   /api/gallery/semiclassical/scarring/index.rst
   /api/gallery/semiclassical/wkb/index.rst



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 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_