Expectation value of observable
===============================

Roughly speaking, an observable is a measurable property of a physical system, such as spin, position,
energy, momentum, etc. In terms of Quantum Mechanics, every physical observable corresponds to an operator
which acts on the wave function.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Operator_(physics)#Expectation_values_of_operators_on_%CE%A8>`__.

.. py:currentmodule:: symplyphysics.laws.quantum_mechanics.expectation_value_of_observable

.. py:data:: mean_observable_value

    The mean value of the :attr:`~observable` operator. See :attr:`~symplyphysics.symbols.basic.any_quantity`.

Symbol:
    :code:`avg(O)`

Latex:
    :math:`\langle O \rangle`

Dimension:
    :code:`any_dimension`


.. py:data:: position

    :attr:`~symplyphysics.symbols.classical_mechanics.position`.

Symbol:
    :code:`x`

Latex:
    :math:`x`

Dimension:
    :code:`length`


.. py:data:: time

    :attr:`~symplyphysics.symbols.basic.time`.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: wave_function

    :attr:`~symplyphysics.symbols.quantum_mechanics.wave_function` as a function of :attr:`~position` and :attr:`~time`.

Symbol:
    :code:`psi(x, t)`

Latex:
    :math:`\psi{\left(x,t \right)}`

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


.. py:data:: observable

    Observable operator as a function of :attr:`~wave_function`. See :attr:`~symplyphysics.symbols.basic.any_quantity`.

Symbol:
    :code:`O(psi)`

Latex:
    :math:`O{\left(\psi \right)}`

Dimension:
    :code:`any_dimension`


.. py:data:: law

    :code:`avg(O) = Integral(conjugate(psi(x, t)) * O(psi(x, t)), (x, -oo, oo))`


    ..
        The conjugation operator isn't recognized yet by the LaTeX printer.

    .. math::
        
        \langle O \rangle = \int \limits_{-\infty}^{\infty} \psi^* (x, t) O[\psi](x, t) dx