Time dependent solution via time independent solution
=====================================================

When the potential energy is independent of time, the solution of the time-dependent Schrödinger
equation can be constructed using the solution of the time-independent equation.

**Notation:**

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

**Conditions:**

#. The potential energy is a function independent of time.
#. This law applies for 1-dimensional systems.

**Links:**

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

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

.. py:data:: position

    :attr:`~symplyphysics.symbols.classical_mechanics.position`, or spatial variable.

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:: time_dependent_wave_function

    Solution of the time-dependent Schrödinger equation. See :attr:`~symplyphysics.symbols.quantum_mechanics.wave_function`.

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

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

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


.. py:data:: time_independent_wave_function

    Solution of the time-independent Schrödinger equation. See :attr:`~symplyphysics.symbols.quantum_mechanics.wave_function`.

Symbol:
    :code:`psi(x)`

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

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


.. py:data:: particle_energy

    Particle :attr:`~symplyphysics.symbols.basic.energy`.

Symbol:
    :code:`E`

Latex:
    :math:`E`

Dimension:
    :code:`energy`


.. py:data:: law

    :code:`Psi(x, t) = psi(x) * exp(-I / hbar * E * t)`


    Latex:
        .. math::
            \Psi{\left(x,t \right)} = \psi{\left(x \right)} \exp{\left(- \frac{i}{\hbar} E t \right)}