Wave eigenfunctions of quantum harmonic oscillator
==================================================

The time-independent Schrödinger equation describing the wave function of the quantum oscillator
can be solved to get the corresponding wave eigenfunctions and energy eigenvalues of the Hamiltonian
operator. Each eigenfunction describes a stationary state of the quantum mechanical system with the
corresponding energy value (eigenvalue of the Hamiltonian). The combination of all eigenfunctions and
eigenvalues represent the energy states allowed.

**Notation:**

#. :math:`\hbar` (:code:`hbar`) is :attr:`~symplyphysics.quantities.hbar`.
#. :math:`H_n` (:code:`hermite`) is the :math:`n`-th physicists' Hermite polynomial.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator#Hamiltonian_and_energy_eigenstates>`__.

.. py:currentmodule:: symplyphysics.laws.quantum_mechanics.harmonic_oscillator.wave_eigenfunctions

.. py:data:: wave_function

    :math:`n`-th wave eigenfunction (solution of the time-independent Schrödinger equation).
    See :attr:`~symplyphysics.symbols.quantum_mechanics.wave_function`.

Symbol:
    :code:`psi`

Latex:
    :math:`\psi`

Dimension:
    :code:`1/sqrt(length)`


.. py:data:: mode_number

    Mode number. See :attr:`~symplyphysics.symbols.basic.nonnegative_number`.

Symbol:
    :code:`N`

Latex:
    :math:`N`

Dimension:
    :code:`dimensionless`


.. py:data:: oscillator_mass

    :attr:`~symplyphysics.symbols.basic.mass` of the oscillator.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`mass`


.. py:data:: angular_frequency

    :attr:`~symplyphysics.symbols.classical_mechanics.angular_frequency` of the oscillator.

Symbol:
    :code:`w`

Latex:
    :math:`\omega`

Dimension:
    :code:`angle/time`


.. py:data:: position

    :attr:`~symplyphysics.symbols.classical_mechanics.position` of the oscillator.

Symbol:
    :code:`x`

Latex:
    :math:`x`

Dimension:
    :code:`length`


.. py:data:: law

    :code:`psi = (m * w / (pi * hbar))^(1/4) / sqrt(2^N * factorial(N)) * exp(-m * w / (2 * hbar) * x^2) * hermite(N, sqrt(m * w / hbar) * x)`


    Latex:
        .. math::
            \psi = \frac{\sqrt[4]{\frac{m \omega}{\pi \hbar}}}{\sqrt{2^{N} N!}} \exp{\left(- \frac{m \omega}{2 \hbar} x^{2} \right)} H_{N}\left(\sqrt{\frac{m \omega}{\hbar}} x\right)