.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/entanglement/plot_aharonov_bohm_ring.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_quantum_entanglement_plot_aharonov_bohm_ring.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_quantum_entanglement_plot_aharonov_bohm_ring.py:


The Aharonov-Bohm effect
============================

A charged particle confined to a 1D ring of radius :math:`R` threads a
magnetic flux :math:`\Phi` through its center; even though the field
:math:`B=0` everywhere the particle can actually be, the eigenspectrum

.. math::

    E_n(\Phi) = \frac{\hbar^2}{2mR^2}\left(n - \frac{\Phi}{\Phi_0}\right)^2,
    \qquad \Phi_0 = \frac{2\pi\hbar}{q},

still shifts periodically with :math:`\Phi/\Phi_0` -- the Aharonov-Bohm
effect, a purely topological/boundary-condition consequence of the vector
potential.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.quantum.chapters.entanglement import AharonovBohmRing








.. GENERATED FROM PYTHON SOURCE LINES 25-32

The ring's flux-periodic energy spectrum
--------------------------------------------------------------------------
The ring's energy levels :math:`E_n(\Phi)` for angular-momentum quantum
numbers :math:`n=-2,\dots,2`, plotted against the enclosed flux in units
of the flux quantum :math:`\Phi_0` -- each parabola is centered on the
:math:`n` that minimizes the shifted quantum number, giving the
sawtooth-like periodic ground-state energy.

.. GENERATED FROM PYTHON SOURCE LINES 32-48

.. code-block:: Python


    ring = AharonovBohmRing(R=1.0)
    Phi_over_Phi0 = np.linspace(-2, 2, 400)

    fig, ax2 = plt.subplots(figsize=(7, 4.5))
    for n in range(-2, 3):
        E_n = [ring.energy(n, f * ring.flux_quantum) for f in Phi_over_Phi0]
        ax2.plot(Phi_over_Phi0, E_n, label=f"n={n}")
    ax2.set_xlabel(r"$\Phi / \Phi_0$")
    ax2.set_ylabel("E_n(Phi)")
    ax2.set_title("Aharonov-Bohm ring: flux-periodic energy spectrum")
    ax2.legend(fontsize=7, ncol=2)
    fig.tight_layout()

    print(f"ground-state energy at Phi=0: {ring.energy(0, 0.0):.6f}")
    print(f"ground-state energy at Phi=Phi0/2: {min(ring.energy(n, 0.5 * ring.flux_quantum) for n in range(-2, 3)):.6f}")



.. image-sg:: /api/gallery/quantum/entanglement/images/sphx_glr_plot_aharonov_bohm_ring_001.png
   :alt: Aharonov-Bohm ring: flux-periodic energy spectrum
   :srcset: /api/gallery/quantum/entanglement/images/sphx_glr_plot_aharonov_bohm_ring_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ground-state energy at Phi=0: 0.000000
    ground-state energy at Phi=Phi0/2: 0.125000





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.050 seconds)


.. _sphx_glr_download_api_gallery_quantum_entanglement_plot_aharonov_bohm_ring.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_aharonov_bohm_ring.ipynb <plot_aharonov_bohm_ring.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_aharonov_bohm_ring.py <plot_aharonov_bohm_ring.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_aharonov_bohm_ring.zip <plot_aharonov_bohm_ring.zip>`


.. only:: html

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

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