.. _sphx_glr_api_gallery_chaos_quantum_chaos:

Quantum Chaos
-------------

Examples illustrating :mod:`physicskit.chaos.quantum`: quantized versions of physicskit.chaos's
own classical :class:`~physicskit.chaos.systems.maps.StandardMap` and
:class:`~physicskit.chaos.systems.maps.BakersMap`, plus the eigenstates of a quantum
particle confined to any of physicskit.chaos's classical billiard shapes.

.. note::
   Spectral *statistics* (level-spacing distributions, spectral rigidity, and
   the like) are intentionally not covered here -- that analysis belongs to
   the separate ``physicskit.rmt`` package, which can consume the eigenphases and
   wavenumbers these examples compute directly.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The quantum baker&#x27;s map is the quantization of physicskit.chaos&#x27;s classical BakersMap, T(x,y)=(x/\alpha,\ \alpha y) for x&lt;\alpha and T(x,y)=((x-\alpha)/(1-\alpha),\ \alpha+(1-\alpha)y) for x\ge\alpha. It is built (following Balazs-Voros/Saraceno) as a one-period unitary Floquet operator that applies the discrete Fourier transform separately to the q &lt; \alpha and q \ge \alpha halves of the (finite, dim-dimensional) position-basis Hilbert space, then transforms the result back to the full position representation -- the quantum-mechanical echo of the classical map&#x27;s &quot;stretch, cut, and stack&quot; mechanism. It is one of the simplest exactly-solvable models of quantum chaos: the underlying classical map is uniformly hyperbolic everywhere (see /api/gallery/chaos/maps/plot_bakers_map), with none of the mixed regular/chaotic phase space the kicked rotor has.">

.. only:: html

  .. image:: /api/gallery/chaos/quantum_chaos/images/thumb/sphx_glr_plot_quantum_bakers_map_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/quantum_chaos/plot_quantum_bakers_map`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Quantum Baker's Map</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A quantum particle confined to a 2D billiard -- &quot;particle in a box&quot; with a chaotic (or not) box shape -- is the other classic playground of quantum chaos, alongside quantized maps like the quantum kicked rotor &lt;plot_quantum_kicked_rotor&gt;. In units where \hbar^2/2m=1, its stationary states solve the Dirichlet Helmholtz eigenproblem">

.. only:: html

  .. image:: /api/gallery/chaos/quantum_chaos/images/thumb/sphx_glr_plot_quantum_billiard_eigenstates_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/quantum_chaos/plot_quantum_billiard_eigenstates`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Quantum Billiard Eigenstates</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The quantum kicked rotor is the quantization of physicskit.chaos&#x27;s classical StandardMap, p_{n+1} = p_n + k\sin\theta_n \pmod{2\pi}, \theta_{n+1} = \theta_n + p_{n+1} \pmod{2\pi}: a particle on a ring, kicked periodically by a potential k\cos\theta. Its one-period evolution is the unitary Floquet operator">

.. only:: html

  .. image:: /api/gallery/chaos/quantum_chaos/images/thumb/sphx_glr_plot_quantum_kicked_rotor_thumb.png
    :alt:

  :doc:`/api/gallery/chaos/quantum_chaos/plot_quantum_kicked_rotor`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Quantum Kicked Rotor</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/chaos/quantum_chaos/plot_quantum_bakers_map
   /api/gallery/chaos/quantum_chaos/plot_quantum_billiard_eigenstates
   /api/gallery/chaos/quantum_chaos/plot_quantum_kicked_rotor