.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/rigid_rotor/plot_01_rigid_rotor.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_rigid_rotor_plot_01_rigid_rotor.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_quantum_rigid_rotor_plot_01_rigid_rotor.py:


Dennison's rigid rotor: rotational levels and the microwave spectrum
======================================================================

Dennison (1926) solved Schrodinger's equation for a rotating diatomic
molecule, giving levels :math:`E_J=J(J+1)\hbar^2/2I`.
Builds an HCl-like :class:`~chemistrykit.quantum.systems.rigid_rotor.RigidRotor`
from its atomic masses and bond length
(:meth:`~chemistrykit.quantum.systems.rigid_rotor.RigidRotor.from_diatomic`),
plots its energy-level diagram (each level's height showing its
:math:`(2J+1)`-fold degeneracy), and shows that the :math:`J\to J+1`
microwave absorption lines are evenly spaced by :math:`2B`.

.. GENERATED FROM PYTHON SOURCE LINES 16-32

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    import scipy.constants as sc

    from chemistrykit.quantum.systems.rigid_rotor import RigidRotor
    from chemistrykit.quantum.visualizers.quantum_plots import plot_energy_levels

    rotor = RigidRotor.from_diatomic(mass1=1.008 * sc.atomic_mass, mass2=34.97 * sc.atomic_mass, bond_length=127.5e-12)
    print(f"Moment of inertia I = {rotor.moment_of_inertia:.4e} kg m^2")
    print(f"Rotational constant B = {rotor.rotational_constant / sc.h / sc.c / 100.0:.4f} cm^-1")

    J_values = np.arange(0, 6)
    energies_cm1 = np.array([rotor.energy(J) for J in J_values]) / sc.h / sc.c / 100.0
    for J, e in zip(J_values, energies_cm1, strict=True):
        print(f"J={J}: E={e:.3f} cm^-1, degeneracy={rotor.degeneracy(J)}")





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

 .. code-block:: none

    Moment of inertia I = 2.6448e-47 kg m^2
    Rotational constant B = 10.5842 cm^-1
    J=0: E=0.000 cm^-1, degeneracy=1
    J=1: E=21.168 cm^-1, degeneracy=3
    J=2: E=63.505 cm^-1, degeneracy=5
    J=3: E=127.010 cm^-1, degeneracy=7
    J=4: E=211.683 cm^-1, degeneracy=9
    J=5: E=317.525 cm^-1, degeneracy=11




.. GENERATED FROM PYTHON SOURCE LINES 33-35

Each level is genuinely (2J+1)-fold degenerate -- drawn here as that
many short segments at the same height:

.. GENERATED FROM PYTHON SOURCE LINES 35-44

.. code-block:: Python


    all_energies_with_degeneracy = np.concatenate([np.full(rotor.degeneracy(J), rotor.energy(J)) for J in J_values]) / sc.h / sc.c / 100.0

    fig, ax = plt.subplots(figsize=(6, 5))
    plot_energy_levels(all_energies_with_degeneracy, ax=ax, color="steelblue")
    ax.set_ylabel("energy (cm^-1)")
    ax.set_title("Rigid-rotor levels (HCl-like), each drawn (2J+1)-fold degenerate")
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/rigid_rotor/images/sphx_glr_plot_01_rigid_rotor_001.png
   :alt: Rigid-rotor levels (HCl-like), each drawn (2J+1)-fold degenerate
   :srcset: /api/gallery/quantum/rigid_rotor/images/sphx_glr_plot_01_rigid_rotor_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 45-47

The J -> J+1 absorption transitions are evenly spaced by 2B -- the
defining signature of a rigid-rotor rotational spectrum:

.. GENERATED FROM PYTHON SOURCE LINES 47-61

.. code-block:: Python


    transition_J = np.arange(0, 5)
    transition_energies_cm1 = np.array([rotor.transition_energy(J) for J in transition_J]) / sc.h / sc.c / 100.0
    spacings = np.diff(transition_energies_cm1)
    print(f"Transition spacings (should all equal 2B): {spacings}")

    fig2, ax2 = plt.subplots(figsize=(6, 4))
    ax2.stem(transition_energies_cm1, np.ones_like(transition_energies_cm1))
    ax2.set_xlabel("wavenumber (cm^-1)")
    ax2.set_ylabel("absorption (arb. units)")
    ax2.set_title("Predicted rigid-rotor microwave spectrum")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/quantum/rigid_rotor/images/sphx_glr_plot_01_rigid_rotor_002.png
   :alt: Predicted rigid-rotor microwave spectrum
   :srcset: /api/gallery/quantum/rigid_rotor/images/sphx_glr_plot_01_rigid_rotor_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Transition spacings (should all equal 2B): [21.16834842 21.16834842 21.16834842 21.16834842]





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

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


.. _sphx_glr_download_api_gallery_quantum_rigid_rotor_plot_01_rigid_rotor.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_rigid_rotor.ipynb <plot_01_rigid_rotor.ipynb>`

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

      :download:`Download Python source code: plot_01_rigid_rotor.py <plot_01_rigid_rotor.py>`

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

      :download:`Download zipped: plot_01_rigid_rotor.zip <plot_01_rigid_rotor.zip>`


.. only:: html

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

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