Time dependent Schrödinger equation in one dimension
====================================================

The Schrödinger equation is a linear partial differential equation that governs the wave
function of a quantum-mechanical system. This law describes the general case of a time-dependent
potential and a time-dependent wave function.

**Notation:**

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

**Notes:**

#. This law works in the case of a single spatial dimension. To use it for the :math:`3`-dimensional
   space, replace the spatial second derivative with the Laplace operator.

**Links:**

#. Wikipedia <https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Separation_of_variables>

.. py:currentmodule:: symplyphysics.laws.quantum_mechanics.schrodinger.time_dependent_equation_in_one_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

    Time-dependent :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:: potential_energy

    Time-independent :attr:`~symplyphysics.symbols.classical_mechanics.potential_energy` as a function of :attr:`~position`.

Symbol:
    :code:`U(x)`

Latex:
    :math:`U{\left(x \right)}`

Dimension:
    :code:`energy`


.. py:data:: particle_mass

    :attr:`~symplyphysics.symbols.basic.mass` of the quantum particle.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`mass`


.. py:data:: law

    :code:`-hbar^2 / (2 * m) * Derivative(psi(x, t), (x, 2)) + U(x) * psi(x, t) = I * hbar * Derivative(psi(x, t), t)`


    Latex:
        .. math::
            - \frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} \psi{\left(x,t \right)} + U{\left(x \right)} \psi{\left(x,t \right)} = i \hbar \frac{\partial}{\partial t} \psi{\left(x,t \right)}