.. _sphx_glr_api_gallery_quantum_potentials:

Potentials
----------

Bound states and scattering in 1D potential wells, and 2D quantum boxes:
the matrix Numerov solver applied to asymmetric wells, the gravitational
"quantum bouncer", the symmetric double well (including an animated,
complex-valued tunneling oscillation), the finite square well (bound
states, resonant transmission, and real-time animated wavepacket
scattering/tunneling), and rectangular/circular/stadium 2D boxes.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Three infinite-wall (&quot;hard box&quot;) potentials in two dimensions, each solved for its stationary states \psi(x,y) with \psi=0 on the boundary. For the rectangular box [0,L_x]\times[0,L_y], separation of variables gives closed-form modes with energies">

.. only:: html

  .. image:: /api/gallery/quantum/potentials/images/thumb/sphx_glr_plot_2d_quantum_boxes_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/potentials/plot_2d_quantum_boxes`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">2D quantum boxes</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Propagates a Gaussian wavepacket incident on a finite rectangular barrier V(x) = V_0 for \lvert x\rvert &lt; \text{width}/2, using the FFT split-operator solver, and compares the numerically transmitted probability to the exact analytic transmission coefficient T(E) (reusing scattering with a negative &quot;depth&quot;, which turns the well into a barrier of the same \lvert V_0\rvert).">

.. only:: html

  .. image:: /api/gallery/quantum/potentials/images/thumb/sphx_glr_plot_barrier_tunneling_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/potentials/plot_barrier_tunneling`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Real-time barrier tunneling</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Solves and plots the low-lying eigenstates of an asymmetric step well, a finite square well (with its evanescent tails), and the gravitational &quot;quantum bouncer&quot; -- and cross-checks the bouncer&#x27;s numerically computed energies against the exact Airy-function zeros. Each panel shows the potential shape V(x) (black), the energy levels (dashed lines), and the eigenfunctions \psi_n(x) offset to sit at their own energy.">

.. only:: html

  .. image:: /api/gallery/quantum/potentials/images/thumb/sphx_glr_plot_bound_states_numerov_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/potentials/plot_bound_states_numerov`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Bound states via the matrix Numerov solver</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Builds the symmetric/antisymmetric doublet of V(x) = \lambda(x^2-a^2)^2, then follows a wavepacket initially localized in the left well as it tunnels back and forth to the right well with period 2\pi\hbar/\Delta E.">

.. only:: html

  .. image:: /api/gallery/quantum/potentials/images/thumb/sphx_glr_plot_double_well_tunneling_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/potentials/plot_double_well_tunneling`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Double-well tunneling</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots the exact transmission coefficient T(E) of a finite square well vs. incident energy. T(E)=1 exactly whenever k_2 a is a multiple of \pi -- the well becomes perfectly transparent, exactly as observed in low-energy electron scattering off noble-gas atoms.">

.. only:: html

  .. image:: /api/gallery/quantum/potentials/images/thumb/sphx_glr_plot_ramsauer_townsend_resonance_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/potentials/plot_ramsauer_townsend_resonance`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Ramsauer-Townsend effect</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/quantum/potentials/plot_2d_quantum_boxes
   /api/gallery/quantum/potentials/plot_barrier_tunneling
   /api/gallery/quantum/potentials/plot_bound_states_numerov
   /api/gallery/quantum/potentials/plot_double_well_tunneling
   /api/gallery/quantum/potentials/plot_ramsauer_townsend_resonance