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

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

.. _sphx_glr_api_gallery_quantum_hydrogen_plot_hydrogen_orbitals.py:


Hydrogen orbitals
====================

The hydrogen-atom eigenstates factor into a radial part and a spherical
harmonic,

.. math::

    \psi_{nlm}(r,\theta,\phi) = R_{nl}(r)\, Y_l^m(\theta,\phi),
    \qquad E_n = -\frac{Z^2}{2n^2}\ \text{(Hartree)},

labeled by the principal, orbital, and magnetic quantum numbers
:math:`n, l, m` (with :math:`0\le l<n` and :math:`\lvert m\rvert\le l`);
:math:`E_n` depends on :math:`n` alone, so states sharing :math:`n` but
differing in :math:`l,m` are degenerate. This example plots the radial
probability density :math:`r^2\lvert R_{nl}(r)\rvert^2` for a few
:math:`(n,l,m)` combinations, the resulting energy ladder converging to the
ionization threshold :math:`E\to0` as :math:`n\to\infty`, a volumetric
:math:`\lvert\psi_{2,1,0}\rvert^2` electron-density cloud, and a coherent
superposition of two different eigenstates whose density beats in time
rather than staying static.

.. GENERATED FROM PYTHON SOURCE LINES 24-31

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.quantum.chapters.hydrogen_am import HydrogenOrbital
    from physicskit.quantum.visualizers.orbitals import animate_orbital_beating, plot_orbital_cloud








.. GENERATED FROM PYTHON SOURCE LINES 32-34

Radial densities and the energy-level ladder
-----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-62

.. code-block:: Python


    fig, (ax, ax_levels) = plt.subplots(1, 2, figsize=(12, 5), gridspec_kw={"width_ratios": [2, 1]})
    r = np.linspace(1e-6, 30, 2000)

    for n, l, m in [(1, 0, 0), (2, 0, 0), (2, 1, 0), (3, 2, 1)]:
        orb = HydrogenOrbital(n, l, m)
        ax.plot(r, orb.radial_density(r), label=f"n={n}, l={l}, m={m}  (E={orb.energy:.4f})")
        print(f"n={n} l={l} m={m}: normalization={orb.check_normalization():.6f}, most probable r={orb.most_probable_radius():.3f}")

    ax.set_xlabel("r (Bohr radii)")
    ax.set_ylabel(r"$r^2 |R_{nl}(r)|^2$")
    ax.set_title("Hydrogen radial probability densities")
    ax.legend(fontsize=8)

    # Energy-level ladder E_n = -1/(2n^2), converging to the n->infinity ionization limit
    for n in range(1, 6):
        E_n = HydrogenOrbital(n, 0, 0).energy
        ax_levels.axhline(E_n, xmin=0.1, xmax=0.9, color="C0", lw=2)
        ax_levels.text(0.92, E_n, f"n={n}", va="center", fontsize=8)
    ax_levels.axhline(0.0, color="gray", ls="--", lw=0.8)
    ax_levels.text(0.92, 0.0, "ionized", va="center", fontsize=8, color="gray")
    ax_levels.set_xlim(0, 1.3)
    ax_levels.set_xticks([])
    ax_levels.set_ylabel(r"$E_n = -Z^2/(2n^2)$  (Hartree)")
    ax_levels.set_title("Energy-level ladder")

    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/hydrogen/images/sphx_glr_plot_hydrogen_orbitals_001.png
   :alt: Hydrogen radial probability densities, Energy-level ladder
   :srcset: /api/gallery/quantum/hydrogen/images/sphx_glr_plot_hydrogen_orbitals_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    n=1 l=0 m=0: normalization=1.000000, most probable r=1.000
    n=2 l=0 m=0: normalization=1.000000, most probable r=5.237
    n=2 l=1 m=0: normalization=1.000000, most probable r=4.000
    n=3 l=2 m=1: normalization=1.000000, most probable r=9.000




.. GENERATED FROM PYTHON SOURCE LINES 63-69

A volumetric :math:`\lvert\psi_{2,1,0}\rvert^2` electron-density cloud
---------------------------------------------------------------------------

:func:`~physicskit.quantum.visualizers.orbitals.plot_orbital_cloud` returns an
interactive Plotly figure; call ``.show()`` or ``.write_html(...)`` on it
to view or export it.

.. GENERATED FROM PYTHON SOURCE LINES 69-73

.. code-block:: Python


    orb_2p = HydrogenOrbital(2, 1, 0)
    fig3d = plot_orbital_cloud(orb_2p, n_points=45)








.. GENERATED FROM PYTHON SOURCE LINES 74-84

A coherent superposition of two eigenstates, beating in time
--------------------------------------------------------------

A single orbital's density is static (an eigenstate of a time-independent
Hamiltonian), but a coherent superposition of *two* eigenstates is not:
:func:`~physicskit.quantum.chapters.hydrogen_am.orbital_superposition_density`
genuinely reshapes at the Bohr frequency :math:`\omega_{ab}=E_a-E_b`.
:func:`~physicskit.quantum.visualizers.orbitals.animate_orbital_beating`
renders this as a Plotly ``frames``-based isosurface animation with a Play
button, stepping through one full beat period.

.. GENERATED FROM PYTHON SOURCE LINES 84-91

.. code-block:: Python


    orb_a = HydrogenOrbital(2, 0, 0)
    orb_b = HydrogenOrbital(3, 1, 0)
    beat_period = 2 * np.pi / abs(orb_b.energy - orb_a.energy)
    times_beat = np.linspace(0, beat_period, 24)

    fig_beat = animate_orbital_beating(orb_a, orb_b, times_beat, n_points=35)








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

   **Total running time of the script:** (0 minutes 0.216 seconds)


.. _sphx_glr_download_api_gallery_quantum_hydrogen_plot_hydrogen_orbitals.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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