.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples_gallery/simple_pdes/pde_schroedinger.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_gallery_simple_pdes_pde_schroedinger.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_gallery_simple_pdes_pde_schroedinger.py:


Schrödinger's Equation
======================

This example implements a complex PDE using the :class:`~pde.pdes.pde.PDE`. We here
chose the `Schrödinger equation <https://en.wikipedia.org/wiki/Schrödinger_equation>`_
without a spatial potential in non-dimensional form:

.. math::
    i \partial_t \psi = -\nabla^2 \psi

Note that the example imposes Neumann conditions at the wall, so the wave packet is
expected to reflect off the wall.

.. GENERATED FROM PYTHON SOURCE LINES 15-34



.. image-sg:: /examples_gallery/simple_pdes/images/sphx_glr_pde_schroedinger_001.png
   :alt: pde schroedinger
   :srcset: /examples_gallery/simple_pdes/images/sphx_glr_pde_schroedinger_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    from math import sqrt

    from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph

    grid = CartesianGrid([[0, 20]], 128, periodic=False)  # generate grid

    # create a (normalized) wave packet with a certain form as an initial condition
    initial_state = ScalarField.from_expression(grid, "exp(I * 5 * x) * exp(-(x - 10)**2)")
    initial_state /= sqrt(initial_state.to_scalar("norm_squared").integral.real)

    eq = PDE({"ψ": "I * laplace(ψ)"})  # define the pde

    # solve the pde and store intermediate data
    storage = MemoryStorage()
    eq.solve(initial_state, t_range=2.5, dt=1e-5, tracker=[storage.tracker(0.02)])

    # visualize the results as a space-time plot
    plot_kymograph(storage, scalar="norm_squared")


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 8.055 seconds)


.. _sphx_glr_download_examples_gallery_simple_pdes_pde_schroedinger.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: pde_schroedinger.ipynb <pde_schroedinger.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: pde_schroedinger.py <pde_schroedinger.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: pde_schroedinger.zip <pde_schroedinger.zip>`