.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/orthogonal_polynomials/plot_02_hermite_laguerre.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_special_functions_orthogonal_polynomials_plot_02_hermite_laguerre.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_special_functions_orthogonal_polynomials_plot_02_hermite_laguerre.py:


Hermite and Laguerre polynomials on infinite domains
==========================================================

Plots the first Hermite polynomials H_n (weight e^{-x^2} on the whole
line) and Laguerre polynomials L_n (weight e^{-x} on the half-line),
checks their orthogonality over infinite intervals, and draws the
quantum harmonic oscillator states H_n(x) e^{-x^2/2} they describe.

.. GENERATED FROM PYTHON SOURCE LINES 12-20

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import hermite_polynomial, inner_product, laguerre_polynomial
    from mathematicskit.special_functions.visualizers.plots import plot_polynomial_family








.. GENERATED FROM PYTHON SOURCE LINES 21-23

The two families
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-31

.. code-block:: Python


    fig, (ax_h, ax_l) = plt.subplots(1, 2, figsize=(10, 4))
    plot_polynomial_family(hermite_polynomial, degrees=[0, 1, 2, 3], x_range=(-2.0, 2.0), ax=ax_h)
    plot_polynomial_family(laguerre_polynomial, degrees=[0, 1, 2, 3], x_range=(0.0, 8.0), ax=ax_l)
    ax_h.set_title(r"Hermite $H_n(x)$, weight $e^{-x^2}$ on $(-\infty, \infty)$")
    ax_l.set_title(r"Laguerre $L_n(x)$, weight $e^{-x}$ on $[0, \infty)$")
    fig.tight_layout()




.. image-sg:: /api/gallery/special_functions/orthogonal_polynomials/images/sphx_glr_plot_02_hermite_laguerre_001.png
   :alt: Hermite $H_n(x)$, weight $e^{-x^2}$ on $(-\infty, \infty)$, Laguerre $L_n(x)$, weight $e^{-x}$ on $[0, \infty)$
   :srcset: /api/gallery/special_functions/orthogonal_polynomials/images/sphx_glr_plot_02_hermite_laguerre_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 32-34

Orthogonality over infinite intervals
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-43

.. code-block:: Python


    hermite_ip = inner_product(lambda x: hermite_polynomial(1, x), lambda x: hermite_polynomial(3, x), lambda x: np.exp(-(x**2)), -np.inf, np.inf)
    hermite_norm = inner_product(lambda x: hermite_polynomial(3, x), lambda x: hermite_polynomial(3, x), lambda x: np.exp(-(x**2)), -np.inf, np.inf)
    laguerre_ip = inner_product(lambda x: laguerre_polynomial(1, x), lambda x: laguerre_polynomial(3, x), lambda x: np.exp(-x), 0.0, np.inf)
    laguerre_norm = inner_product(lambda x: laguerre_polynomial(3, x), lambda x: laguerre_polynomial(3, x), lambda x: np.exp(-x), 0.0, np.inf)

    print(f"<H_1, H_3> = {hermite_ip:.2e}, <H_3, H_3> = {hermite_norm:.6f} (2^3 3! sqrt(pi) = {8 * 6 * math.sqrt(math.pi):.6f})")
    print(f"<L_1, L_3> = {laguerre_ip:.2e}, <L_3, L_3> = {laguerre_norm:.6f} (expected 1)")





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

 .. code-block:: none

    <H_1, H_3> = 3.12e-16, <H_3, H_3> = 85.077785 (2^3 3! sqrt(pi) = 85.077785)
    <L_1, L_3> = 7.89e-17, <L_3, L_3> = 1.000000 (expected 1)




.. GENERATED FROM PYTHON SOURCE LINES 44-48

Hermite functions: the quantum harmonic oscillator
-----------------------------------------------------
The normalized states psi_n(x) = H_n(x) e^{-x^2/2} / sqrt(2^n n! sqrt(pi)),
drawn at their energy levels E_n = n + 1/2.

.. GENERATED FROM PYTHON SOURCE LINES 48-60

.. code-block:: Python


    x = np.linspace(-5, 5, 500)
    fig, ax = plt.subplots()
    ax.plot(x, x**2 / 2, color="gray", lw=1)
    for n in range(4):
        psi = hermite_polynomial(n, x) * np.exp(-(x**2) / 2) / math.sqrt(2**n * math.factorial(n) * math.sqrt(math.pi))
        ax.plot(x, n + 0.5 + 0.8 * psi, label=rf"$\psi_{n}$, $E = {n}+\frac{{1}}{{2}}$")
    ax.set_ylim(0, 5)
    ax.set_xlabel("x")
    ax.set_ylabel("energy / wavefunction (offset)")
    ax.set_title("Harmonic oscillator states built from Hermite polynomials")
    ax.legend(fontsize="small")



.. image-sg:: /api/gallery/special_functions/orthogonal_polynomials/images/sphx_glr_plot_02_hermite_laguerre_002.png
   :alt: Harmonic oscillator states built from Hermite polynomials
   :srcset: /api/gallery/special_functions/orthogonal_polynomials/images/sphx_glr_plot_02_hermite_laguerre_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11cf701a0>




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

   **Total running time of the script:** (0 minutes 0.110 seconds)


.. _sphx_glr_download_api_gallery_special_functions_orthogonal_polynomials_plot_02_hermite_laguerre.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_hermite_laguerre.ipynb <plot_02_hermite_laguerre.ipynb>`

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

      :download:`Download Python source code: plot_02_hermite_laguerre.py <plot_02_hermite_laguerre.py>`

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

      :download:`Download zipped: plot_02_hermite_laguerre.zip <plot_02_hermite_laguerre.zip>`


.. only:: html

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

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