.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/hydrogenlike/plot_01_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_hydrogenlike_plot_01_hydrogen_orbitals.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_quantum_hydrogenlike_plot_01_hydrogen_orbitals.py:


Schrodinger's hydrogen atom: radial wavefunctions and orbital energies
=========================================================================

Schrodinger's first 1926 paper solved his wave equation for the hydrogen
atom and recovered the Bohr energies :math:`E_n\propto-1/n^2` without any
quantization postulate.
:class:`~chemistrykit.quantum.systems.hydrogenlike.HydrogenLikeAtom`
reproduces the textbook 13.6 eV hydrogen ground-state energy exactly,
and its radial wavefunctions (built from the associated Laguerre
polynomials of :mod:`scipy.special`) are independently checked here by
numerically integrating them to a normalized probability of 1.

.. GENERATED FROM PYTHON SOURCE LINES 16-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import ELECTRONVOLT
    from chemistrykit.quantum.systems.hydrogenlike import HydrogenLikeAtom
    from chemistrykit.quantum.visualizers.quantum_plots import plot_radial_distribution

    h_atom = HydrogenLikeAtom(Z=1)
    print(f"Bohr radius a0 = {h_atom.bohr_radius:.6e} m")
    for n in (1, 2, 3, 4):
        print(f"E_{n} = {-h_atom.energy(n) / ELECTRONVOLT:.4f} eV (ionization energy from level n)")





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

 .. code-block:: none

    Bohr radius a0 = 5.291772e-11 m
    E_1 = 13.6057 eV (ionization energy from level n)
    E_2 = 3.4014 eV (ionization energy from level n)
    E_3 = 1.5117 eV (ionization energy from level n)
    E_4 = 0.8504 eV (ionization energy from level n)




.. GENERATED FROM PYTHON SOURCE LINES 29-32

Every radial wavefunction should integrate (as r^2 R^2) to exactly 1 --
a formula-agnostic sanity check independent of any particular
associated-Laguerre-polynomial normalization convention:

.. GENERATED FROM PYTHON SOURCE LINES 32-37

.. code-block:: Python


    for n, l in [(1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2)]:
        norm = h_atom.check_radial_normalization(n, l)
        print(f"n={n}, l={l}: integral of r^2 R_nl(r)^2 dr = {norm:.6f}")





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

 .. code-block:: none

    n=1, l=0: integral of r^2 R_nl(r)^2 dr = 1.000000
    n=2, l=0: integral of r^2 R_nl(r)^2 dr = 1.000000
    n=2, l=1: integral of r^2 R_nl(r)^2 dr = 1.000000
    n=3, l=0: integral of r^2 R_nl(r)^2 dr = 1.000000
    n=3, l=1: integral of r^2 R_nl(r)^2 dr = 1.000000
    n=3, l=2: integral of r^2 R_nl(r)^2 dr = 1.000000




.. GENERATED FROM PYTHON SOURCE LINES 38-45

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(7, 5))
    for n, l, style in [(1, 0, "-"), (2, 0, "--"), (2, 1, "-."), (3, 2, ":")]:
        plot_radial_distribution(h_atom, n, l, ax=ax, r_max_bohr_radii=25.0, linestyle=style, label=f"n={n}, l={l}")
    ax.legend()
    ax.set_title("Hydrogen radial distribution functions")
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/hydrogenlike/images/sphx_glr_plot_01_hydrogen_orbitals_001.png
   :alt: Hydrogen radial distribution functions
   :srcset: /api/gallery/quantum/hydrogenlike/images/sphx_glr_plot_01_hydrogen_orbitals_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 46-50

The 1s radial distribution function peaks at exactly r = a0 -- the
famous result that the Bohr-model "orbit radius" is really the *most
probable* radial distance in the full quantum treatment, not a literal
orbit:

.. GENERATED FROM PYTHON SOURCE LINES 50-58

.. code-block:: Python


    a0 = h_atom.bohr_radius
    r = np.linspace(1.0e-4 * a0, 6.0 * a0, 2000)
    P = h_atom.radial_distribution_function(r, n=1, l=0)
    r_peak = r[np.argmax(P)]
    print(f"1s radial distribution peaks at r = {r_peak / a0:.4f} * a0 (expected: 1.0)")

    plt.show()




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

 .. code-block:: none

    1s radial distribution peaks at r = 0.9996 * a0 (expected: 1.0)





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

   **Total running time of the script:** (0 minutes 0.061 seconds)


.. _sphx_glr_download_api_gallery_quantum_hydrogenlike_plot_01_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_01_hydrogen_orbitals.ipynb <plot_01_hydrogen_orbitals.ipynb>`

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

      :download:`Download Python source code: plot_01_hydrogen_orbitals.py <plot_01_hydrogen_orbitals.py>`

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

      :download:`Download zipped: plot_01_hydrogen_orbitals.zip <plot_01_hydrogen_orbitals.zip>`


.. only:: html

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

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