Time independent solution in one dimension
==========================================

The Schrödinger equation is a linear partial differential equation that governs the wave
function of a quantum-mechanical system.

**Notation:**

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

**Condition:**

#. This law works in the case of a single spatial dimension. To use it for the 3-dimensional space
   replace the spatial second derivative with the Laplace operator.
#. The wave function is independent of time.

**Links:**

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

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

.. py:data:: position

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

Symbol:
    :code:`x`

Latex:
    :math:`x`

Dimension:
    :code:`length`


.. py:data:: wave_function

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

Symbol:
    :code:`psi(x)`

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

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


.. py:data:: potential_energy

    :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`.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`mass`


.. py:data:: particle_energy

    :attr:`~symplyphysics.symbols.basic.energy` of the particle.

Symbol:
    :code:`E`

Latex:
    :math:`E`

Dimension:
    :code:`energy`


.. py:data:: law

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


    Latex:
        .. math::
            - \frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} \psi{\left(x \right)} + U{\left(x \right)} \psi{\left(x \right)} = E \psi{\left(x \right)}