.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/perturbation/plot_perturbation_and_floquet.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_perturbation_plot_perturbation_and_floquet.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_quantum_perturbation_plot_perturbation_and_floquet.py:


Stark and Zeeman splitting: static perturbation theory
==============================================================

Two static, first-order perturbative effects on the hydrogen atom. In a
magnetic field :math:`B`, the weak-field Zeeman effect shifts each
:math:`\lvert l, m_l; s, m_s\rangle` sublevel by

.. math::

    \Delta E = \mu_B B\,(m_l + g_s m_s);

in a uniform electric field :math:`F`, the linear Stark effect splits
hydrogen's 4-fold degenerate :math:`n=2` shell (using parabolic quantum
numbers :math:`n_1, n_2, m` with :math:`n_1+n_2+\lvert m\rvert+1=n`) by

.. math::

    \Delta E = \tfrac{3}{2}\, n\, (n_1 - n_2)\, a_0 F.

The Zeeman effect is the textbook example of non-degenerate perturbation
theory; the linear Stark effect of hydrogen specifically is the
*degenerate* variety, made possible only by hydrogen's accidental level
degeneracy in :math:`l`.

.. GENERATED FROM PYTHON SOURCE LINES 26-32

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.quantum.chapters.perturbation import stark_n2_quartet, zeeman_spectrum








.. GENERATED FROM PYTHON SOURCE LINES 33-39

The Stark quartet and the Zeeman splitting
--------------------------------------------------------------------------
Left: the three Stark-shifted energies of the :math:`n=2` quartet vs.
field :math:`F` (the :math:`m=\pm1` sublevels stay unshifted since only
the :math:`m=0` pair couples to the field). Right: the Zeeman sublevel
shifts vs. :math:`B` for :math:`l=0,1,2`.

.. GENERATED FROM PYTHON SOURCE LINES 39-66

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))

    # Linear Stark effect: hydrogen n=2 quartet
    F_values = np.linspace(-0.02, 0.02, 50)
    shifts = np.array([[lv.shift for lv in stark_n2_quartet(F)] for F in F_values])
    for i in range(shifts.shape[1]):
        axes[0].plot(F_values, shifts[:, i])
    axes[0].set_xlabel("Field F")
    axes[0].set_ylabel("Energy shift")
    axes[0].set_title("Linear Stark effect: hydrogen n=2 quartet")

    # Zeeman splitting
    B_values = np.linspace(0, 2, 40)
    for l in [0, 1, 2]:
        levels = np.array([zeeman_spectrum(l, B) for B in B_values])
        for i in range(levels.shape[1]):
            axes[1].plot(B_values, levels[:, i], color=f"C{l}", alpha=0.6, label=(f"l={l}" if i == 0 else None))
    axes[1].set_xlabel("Magnetic field B")
    axes[1].set_ylabel("Energy shift")
    axes[1].set_title("Zeeman sublevel splitting")
    axes[1].legend(fontsize=8)

    fig.tight_layout()

    print(f"Stark n=2 shifts at F={F_values[-1]:.3f}: {shifts[-1]}")
    print(f"Zeeman l=1 shifts at B={B_values[-1]:.3f}: {zeeman_spectrum(1, B_values[-1])}")



.. image-sg:: /api/gallery/quantum/perturbation/images/sphx_glr_plot_perturbation_and_floquet_001.png
   :alt: Linear Stark effect: hydrogen n=2 quartet, Zeeman sublevel splitting
   :srcset: /api/gallery/quantum/perturbation/images/sphx_glr_plot_perturbation_and_floquet_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Stark n=2 shifts at F=0.020: [-0.06  0.    0.    0.06]
    Zeeman l=1 shifts at B=2.000: [-4. -2.  0.  0.  2.  4.]





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

   **Total running time of the script:** (0 minutes 0.067 seconds)


.. _sphx_glr_download_api_gallery_quantum_perturbation_plot_perturbation_and_floquet.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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