Energy levels of harmonic oscillator
====================================

As opposed to the classical harmonic oscillator, the energy levels of a quantum harmonic oscillator
are quantized, meaning that its energy take a value out of a discrete range. These energy levels
are equidistant, i.e. the difference between successive energy levels is the same for all levels.

**Notation:**

#. :math:`\hbar` (:code:`hbar`) is :attr:`~symplyphysics.quantities.hbar`.

**Notes**

#. This means that the energy of a quantum oscillator cannot be zero and the lowest it can be
   is the zero-point energy :math:`E_0 = \hbar \omega / 2`.

**Links:**

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

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

.. py:data:: energy_level

    Energy of the level corresponding to the :attr:`~mode_number`.

Symbol:
    :code:`E_n`

Latex:
    :math:`E_{n}`

Dimension:
    :code:`energy`


.. py:data:: mode_number

    Quantum number of oscillator, which is any non-negative integer (:math:`0, 1, 2, \dots`).
    See :attr:`~symplyphysics.symbols.basic.nonnegative_number`.

Symbol:
    :code:`N`

Latex:
    :math:`N`

Dimension:
    :code:`dimensionless`


.. 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:: law

    :code:`E_n = (N + 1/2) * hbar * w`


    Latex:
        .. math::
            E_{n} = \left(N + \frac{1}{2}\right) \hbar \omega