.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/particle_in_box/plot_01_particle_in_box.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_particle_in_box_plot_01_particle_in_box.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_quantum_particle_in_box_plot_01_particle_in_box.py:


Solving Schrodinger's equation for a particle in a box
========================================================

Plots the first few 1D particle-in-a-box wavefunctions
(:class:`~chemistrykit.quantum.systems.particle_in_box.ParticleInBox1D`),
each offset by its own energy level, then checks the textbook 3D
cubic-box degeneracy pattern
(:class:`~chemistrykit.quantum.systems.particle_in_box.ParticleInBox3D`).
These are the simplest exact solutions of Schrodinger's 1926
time-independent equation :math:`-\frac{\hbar^2}{2m}\nabla^2\psi=E\psi`
inside hard walls: quantized energies :math:`E_n=n^2h^2/8mL^2` appear
from the boundary conditions alone, with no quantization postulate.

.. GENERATED FROM PYTHON SOURCE LINES 17-32

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.quantum.systems.particle_in_box import ParticleInBox1D, ParticleInBox3D
    from chemistrykit.quantum.visualizers.quantum_plots import plot_particle_in_box_wavefunctions

    box = ParticleInBox1D(length=1.0e-9)
    for n in (1, 2, 3, 4):
        print(f"E_{n} = {box.energy(n) / 1.602176634e-19:.4f} eV, E_{n}/E_1 = {box.energy(n) / box.energy(1):.4f} (n^2 = {n**2})")

    fig, ax = plt.subplots(figsize=(7, 5))
    plot_particle_in_box_wavefunctions(box, [1, 2, 3, 4], ax=ax, scale=2.0e-20)
    ax.set_title("1D particle-in-a-box wavefunctions (1 nm box)")
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/particle_in_box/images/sphx_glr_plot_01_particle_in_box_001.png
   :alt: 1D particle-in-a-box wavefunctions (1 nm box)
   :srcset: /api/gallery/quantum/particle_in_box/images/sphx_glr_plot_01_particle_in_box_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    E_1 = 0.3760 eV, E_1/E_1 = 1.0000 (n^2 = 1)
    E_2 = 1.5041 eV, E_2/E_1 = 4.0000 (n^2 = 4)
    E_3 = 3.3843 eV, E_3/E_1 = 9.0000 (n^2 = 9)
    E_4 = 6.0165 eV, E_4/E_1 = 16.0000 (n^2 = 16)




.. GENERATED FROM PYTHON SOURCE LINES 33-35

A cubic 3D box shows genuine degeneracy from symmetry alone: the three
permutations of (2, 1, 1) share exactly the same energy.

.. GENERATED FROM PYTHON SOURCE LINES 35-53

.. code-block:: Python


    cubic_box = ParticleInBox3D(Lx=1.0e-9, Ly=1.0e-9, Lz=1.0e-9)
    e211 = cubic_box.energy(2, 1, 1)
    e121 = cubic_box.energy(1, 2, 1)
    e112 = cubic_box.energy(1, 1, 2)
    print(f"E(2,1,1) = E(1,2,1) = E(1,1,2)? {np.allclose([e211, e121, e112], e211)}")

    degeneracies = cubic_box.degeneracy(n_max=3)
    fig2, ax2 = plt.subplots(figsize=(6, 4))
    energies_ev = np.array(sorted(degeneracies.keys())) / 1.602176634e-19
    counts = [degeneracies[e] for e in sorted(degeneracies.keys())]
    ax2.bar(range(len(energies_ev)), counts, tick_label=[f"{e:.2f}" for e in energies_ev])
    ax2.set_xlabel("energy (eV)")
    ax2.set_ylabel("degeneracy")
    ax2.set_title("Cubic-box level degeneracies")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/quantum/particle_in_box/images/sphx_glr_plot_01_particle_in_box_002.png
   :alt: Cubic-box level degeneracies
   :srcset: /api/gallery/quantum/particle_in_box/images/sphx_glr_plot_01_particle_in_box_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    E(2,1,1) = E(1,2,1) = E(1,1,2)? True





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

   **Total running time of the script:** (0 minutes 0.070 seconds)


.. _sphx_glr_download_api_gallery_quantum_particle_in_box_plot_01_particle_in_box.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_particle_in_box.ipynb <plot_01_particle_in_box.ipynb>`

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

      :download:`Download Python source code: plot_01_particle_in_box.py <plot_01_particle_in_box.py>`

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

      :download:`Download zipped: plot_01_particle_in_box.zip <plot_01_particle_in_box.zip>`


.. only:: html

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

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