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

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

.. _sphx_glr_api_gallery_quantum_harmonic_oscillator_plot_ladder_operators.py:


Dirac's ladder-operator method
==================================

Builds the harmonic oscillator's raising/lowering operators
:math:`\hat a^\dagger,\hat a` in a truncated Fock basis with
:func:`~physicskit.quantum.core.operators.annihilation_operator` and
:func:`~physicskit.quantum.core.operators.creation_operator`, verifies the
algebra :math:`[\hat a,\hat a^\dagger]=1`, and reads the spectrum
:math:`E_n=\hbar\omega(n+\tfrac12)` directly off the
:func:`~physicskit.quantum.core.operators.number_operator` -- reproducing
the oscillator's energy ladder without ever solving a differential
equation.

.. GENERATED FROM PYTHON SOURCE LINES 15-49

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.quantum.chapters.harmonic_spin import HarmonicOscillator
    from physicskit.quantum.core.operators import (
        annihilation_operator,
        commutator,
        creation_operator,
        is_hermitian,
        number_operator,
    )

    ho = HarmonicOscillator()
    n_max = 12

    a = annihilation_operator(n_max)
    a_dag = creation_operator(n_max)
    N = number_operator(n_max)

    # The bulk of [a, a^dagger] is 1 (truncation only breaks the top level).
    comm = commutator(a, a_dag)
    print("[a, a^dagger] diagonal (should be 1, except the truncated top level):")
    print(np.round(np.real(np.diag(comm)), 6))
    print("N = a^dagger a is Hermitian:", is_hermitian(N))

    # Build the spectrum by repeated raising from the vacuum |0>, annihilated by a.
    vacuum = np.zeros(n_max, dtype=complex)
    vacuum[0] = 1.0
    print("a|0> = 0:", np.allclose(a @ vacuum, 0.0))

    E_ladder = ho.hbar * ho.omega * (np.diag(N).real + 0.5)
    E_exact = ho.energy(np.arange(n_max))





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

 .. code-block:: none

    [a, a^dagger] diagonal (should be 1, except the truncated top level):
    [  1.   1.   1.   1.   1.   1.   1.   1.   1.   1.   1. -11.]
    N = a^dagger a is Hermitian: True
    a|0> = 0: True




.. GENERATED FROM PYTHON SOURCE LINES 50-53

The ladder-operator spectrum :math:`E_n=\hbar\omega(n+1/2)` against the
exact eigenfunction energies, and the operator matrices themselves
--------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 53-73

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))

    axes[0].plot(np.arange(n_max), E_exact, "o", label=r"$E_n$ from HarmonicOscillator.energy")
    axes[0].plot(np.arange(n_max), E_ladder, "x", label=r"$\hbar\omega(N+1/2)$ from the ladder operators")
    axes[0].set_xlabel("n")
    axes[0].set_ylabel("E")
    axes[0].set_title("Energy ladder built purely from operator algebra")
    axes[0].legend(fontsize=8)

    im = axes[1].imshow(np.abs(a), cmap="viridis")
    axes[1].set_title(
        r"$|\hat a|$ in the truncated Fock basis"
        "\n(nonzero only one step below the diagonal)"
    )
    fig.colorbar(im, ax=axes[1], fraction=0.046)

    fig.tight_layout()

    print(f"max |E_ladder - E_exact| (excluding the truncated top level): {np.max(np.abs(E_ladder[:-1] - E_exact[:-1])):.2e}")



.. image-sg:: /api/gallery/quantum/harmonic_oscillator/images/sphx_glr_plot_ladder_operators_001.png
   :alt: Energy ladder built purely from operator algebra, $|\hat a|$ in the truncated Fock basis (nonzero only one step below the diagonal)
   :srcset: /api/gallery/quantum/harmonic_oscillator/images/sphx_glr_plot_ladder_operators_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max |E_ladder - E_exact| (excluding the truncated top level): 1.78e-15





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

   **Total running time of the script:** (0 minutes 0.071 seconds)


.. _sphx_glr_download_api_gallery_quantum_harmonic_oscillator_plot_ladder_operators.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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