:orphan:

Examples
========

Runnable, self-contained scripts demonstrating every chapter of
``physicskit.quantum`` -- from arbitrary-potential bound states through
wave-packet dynamics, entanglement, and Floquet driving.

Each script in this gallery is self-contained and can be run directly with
``python examples/quantum/<section>/<script>.py``. Every script also carries
an RST module docstring as its title/description and uses ``# %%`` markers to
split narrative text from code, which is exactly what Sphinx-Gallery renders
into the pages below -- the script *is* the source of truth for what you see,
not a copy of it.

Sections
--------

- **potentials** -- bound and scattering states in model potentials: the
  matrix Numerov solver for step and finite square wells and the
  gravitational "quantum bouncer" (checked against exact Airy-function
  zeros), real-time animated barrier tunneling and well scattering,
  double-well tunneling (including an animated complex-valued version),
  2D quantum boxes, and the Ramsauer-Townsend transmission resonance.
- **wave_packets** -- time-dependent dynamics: de Broglie's traveling matter
  wave, dispersion (with an animated view of the spreading), twin-slit
  interference and quantum revivals, Ehrenfest's theorem in an anharmonic
  well, and a phase-colored tunneling animation.
- **harmonic_oscillator** -- the harmonic oscillator suite of eigenstates and
  coherent/squeezed states, an animated Fock-state superposition, its Wigner
  phase-space quasi-probability distribution, and the raw ladder-operator
  algebra the whole suite is built on.
- **hydrogen** -- the hydrogen atom's orbitals in three dimensions, including
  an animated two-eigenstate superposition beating at the Bohr frequency.
- **entanglement** -- Bloch-sphere spin dynamics (Larmor precession and an
  animated Rabi oscillation), Bell correlations, the topological
  Aharonov-Bohm effect, and the dynamical generation of entanglement between
  two Ising-coupled qubits.
- **measurement** -- projective measurement and the Born rule, the
  position-momentum uncertainty bound traced to its operator-algebra root,
  and two measurement/interference demonstrations propagated genuinely in
  time: the double-slit experiment and Stern-Gerlach beam splitting.
- **perturbation** -- Stark/Zeeman splitting from perturbation theory and
  Floquet driving.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

.. thumbnail-parent-div-close

.. raw:: html

    </div>

Entanglement and topology
--------------------------

Bell/EPR spin correlations and the CHSH inequality, the Aharonov-Bohm
effect (a purely topological consequence of the vector potential),
Bloch-sphere qubit dynamics (Larmor precession and an animated Rabi
oscillation), the dynamical generation of entanglement between two
qubits coupled by an Ising interaction, and why no fixed unitary can
clone an arbitrary qubit state (the no-cloning theorem).


.. 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="A charged particle confined to a 1D ring of radius R threads a magnetic flux \Phi through its center; even though the field B=0 everywhere the particle can actually be, the eigenspectrum">

.. only:: html

  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_aharonov_bohm_ring_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_aharonov_bohm_ring`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Aharonov-Bohm effect</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A spin-1/2 (qubit) state evolves under H=-(\omega/2)\sigma_z (a static field along z) or H=-(\omega_R/2)\sigma_x (a resonant drive along x). Since \sigma_z^2=\sigma_x^2=I, the propagator has the closed form e^{-i\theta\sigma} = \cos\theta\, I - i\sin\theta\, \sigma, used directly here instead of a general matrix exponential. A final section drives the same kind of two-level system with Rabi&#x27;s own closed-form resonance formula (~physicskit.quantum.chapters.spin.RabiProblem), animating the Bloch vector itself as it nutates under the drive.">

.. only:: html

  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_bloch_sphere_spin_dynamics_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_bloch_sphere_spin_dynamics`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Bloch-sphere spin dynamics</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A textbook demonstration that quantum mechanics has no local, realistic description. Two spin-1/2 particles prepared in the entangled singlet state">

.. only:: html

  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_entanglement_and_topology_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_entanglement_and_topology`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Bell correlations and the CHSH inequality</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Two qubits, each individually prepared in the unbiased state \lvert+\rangle = (\lvert0\rangle+\lvert1\rangle)/\sqrt2 -- an unentangled product state -- are coupled by an Ising interaction \hat H = \hbar J\,\sigma_z^{(1)}\otimes\sigma_z^{(2)} and become genuinely entangled purely through that interaction&#x27;s time evolution, reaching a maximally entangled (Bell-equivalent) state at Jt=\pi/4. Unlike the static Bell/EPR correlations of /api/gallery/quantum/entanglement/plot_entanglement_and_topology (which start from an already-entangled state), this traces entanglement&#x27;s actual dynamical genesis.">

.. only:: html

  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_ising_entanglement_growth_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_ising_entanglement_growth`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Dynamical entanglement generation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Wootters, Zurek, and (independently) Dieks proved that no unitary process can take an arbitrary, unknown quantum state and produce two independent copies of it. The proof is a direct consequence of linearity: a unitary that faithfully clones two particular states must, by linearity, act on their superposition in a way that is not a faithful copy of that superposition. This example makes the failure completely explicit with the simplest possible candidate &quot;cloning machine,&quot; the CNOT gate: it successfully copies the computational basis states \lvert0\rangle and \lvert1\rangle, but applied to a superposition it produces exactly a Bell state -- physicskit.quantum.chapters.entanglement.bell_state -- rather than two independent copies, entangling the two qubits instead of cloning either one.">

.. only:: html

  .. image:: /api/gallery/quantum/entanglement/images/thumb/sphx_glr_plot_no_cloning_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/entanglement/plot_no_cloning_theorem`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The no-cloning theorem: why a fixed unitary can copy a basis, not a state</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Harmonic oscillator
--------------------

The quantum harmonic oscillator suite: Fock states, Glauber coherent
states, squeezed states, thermal broadening, an animated Fock-state
superposition, and the Wigner quasi-probability distribution in phase
space.


.. 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="Four canonical states of the 1D quantum harmonic oscillator">

.. only:: html

  .. image:: /api/gallery/quantum/harmonic_oscillator/images/thumb/sphx_glr_plot_harmonic_oscillator_suite_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/harmonic_oscillator/plot_harmonic_oscillator_suite`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The harmonic oscillator suite</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Builds the harmonic oscillator&#x27;s raising/lowering operators \hat a^\dagger,\hat a in a truncated Fock basis with annihilation_operator and creation_operator, verifies the algebra [\hat a,\hat a^\dagger]=1, and reads the spectrum E_n=\hbar\omega(n+\tfrac12) directly off the number_operator -- reproducing the oscillator&#x27;s energy ladder without ever solving a differential equation.">

.. only:: html

  .. image:: /api/gallery/quantum/harmonic_oscillator/images/thumb/sphx_glr_plot_ladder_operators_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/harmonic_oscillator/plot_ladder_operators`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Dirac's ladder-operator method</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The Wigner function represents a quantum state as a quasi-probability distribution over classical phase space (x,p),">

.. only:: html

  .. image:: /api/gallery/quantum/harmonic_oscillator/images/thumb/sphx_glr_plot_wigner_phase_space_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/harmonic_oscillator/plot_wigner_phase_space`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Wigner phase-space quasi-probability</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Hydrogen orbitals
------------------

Radial wavefunctions, the hydrogen energy-level ladder, 3D volumetric
electron-density rendering, and an animated two-eigenstate superposition
beating at the Bohr frequency.


.. 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="The hydrogen-atom eigenstates factor into a radial part and a spherical harmonic,">

.. only:: html

  .. image:: /api/gallery/quantum/hydrogen/images/thumb/sphx_glr_plot_hydrogen_orbitals_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/hydrogen/plot_hydrogen_orbitals`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Hydrogen orbitals</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Measurement
-----------

The Born rule made concrete: simulated projective measurement outcomes
converging to the analytic probability density; the position-momentum
uncertainty bound; and two classic measurement/interference demonstrations
propagated genuinely in time -- the double-slit experiment and Stern-Gerlach
beam splitting.


.. 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="Draws simulated projective position measurements from a harmonic oscillator Fock state \lvert n{=}2\rangle via simulate_position_measurement (inverse-CDF sampling of the Born-rule density), and shows the sample histogram converging to the analytic \lvert\psi_2(x)\rvert^2 as the sample count grows.">

.. only:: html

  .. image:: /api/gallery/quantum/measurement/images/thumb/sphx_glr_plot_born_rule_measurement_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/measurement/plot_born_rule_measurement`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Born rule</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Propagates a genuine 2D Gaussian wavepacket -- elongated along y so it illuminates both slits coherently -- through an opaque wall pierced by two gaps (~physicskit.quantum.chapters.wave_packets.double_slit_potential) using the FFT split-operator method (~physicskit.quantum.core.solvers.SplitOperatorSolver2D). Unlike the far-field Fraunhofer formula used in TwinSlit (/api/gallery/quantum/wave_packets/plot_wave_packet_dynamics), the interference fringes here build up from genuine wave dynamics: the packet splits at the wall, the two emerging pieces spread and overlap downstream, and the familiar fringe pattern \lvert\psi_1+\psi_2\rvert^2 emerges directly from the time-dependent Schrodinger equation -- exactly the electron-at-a-time interference Jonsson observed in 1961.">

.. only:: html

  .. image:: /api/gallery/quantum/measurement/images/thumb/sphx_glr_plot_double_slit_propagation_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/measurement/plot_double_slit_propagation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The double-slit experiment, propagated in time</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Checks the Heisenberg bound \Delta x\,\Delta p \ge \hbar/2 directly from harmonic-oscillator wavefunctions with uncertainty, and verifies the underlying canonical commutation relation [\hat x,\hat p]=i\hbar in a truncated Fock basis with commutator, position_operator, and momentum_operator.">

.. only:: html

  .. image:: /api/gallery/quantum/measurement/images/thumb/sphx_glr_plot_heisenberg_uncertainty_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/measurement/plot_heisenberg_uncertainty`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Heisenberg's uncertainty principle</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A beam of spin-1/2 particles passes through an inhomogeneous magnetic field B_z(y) \approx B_0 + y\,\partial_y B_z. The spin-dependent force F_y = \pm\mu\,\partial_y B_z pushes the two spin-projection branches apart transversely while the beam drifts along x, splitting a single incoming beam into two spatially resolved lobes -- Stern and Gerlach&#x27;s 1922 demonstration that an atom&#x27;s magnetic moment (and hence angular momentum) is spatially quantized rather than continuously distributed. SternGerlach implements the standard textbook semiclassical treatment: each spin branch is an independent Gaussian wavepacket under a constant transverse force, exactly solvable via GaussianDispersion.">

.. only:: html

  .. image:: /api/gallery/quantum/measurement/images/thumb/sphx_glr_plot_stern_gerlach_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/measurement/plot_stern_gerlach`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Stern-Gerlach experiment</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Perturbation and Floquet driving
-----------------------------------

Static perturbation theory (linear Stark splitting and Zeeman sublevel
splitting) and time-dependent perturbation theory (multiphoton
transitions in an AC-driven infinite well, via Floquet analysis).


.. 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="Dirac&#x27;s time-dependent perturbation theory describes how a weak, time-dependent perturbation drives transitions between the unperturbed stationary states of a system; Fermi later gave the leading-order transition-rate result its &quot;golden rule&quot; name. The same machinery describes a system driven periodically in time, where transitions become resonant multiphoton processes whenever an integer number of drive quanta \hbar\omega bridges an energy gap.">

.. only:: html

  .. image:: /api/gallery/quantum/perturbation/images/thumb/sphx_glr_plot_floquet_driven_box_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/perturbation/plot_floquet_driven_box`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Floquet-driven box: time-dependent perturbation theory</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Two static, first-order perturbative effects on the hydrogen atom. In a magnetic field B, the weak-field Zeeman effect shifts each \lvert l, m_l; s, m_s\rangle sublevel by">

.. only:: html

  .. image:: /api/gallery/quantum/perturbation/images/thumb/sphx_glr_plot_perturbation_and_floquet_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/perturbation/plot_perturbation_and_floquet`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Stark and Zeeman splitting: static perturbation theory</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

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>


.. raw:: html

    <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>

Wave packets
------------

Non-stationary wave dynamics: free dispersion (with an animated view of
the spreading), twin-slit interference, quantum revivals in an infinite
well, Ehrenfest's theorem in an anharmonic potential, and a phase-colored
animation of double-well tunneling.


.. 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="Builds a free Gaussian wavepacket carrying de Broglie&#x27;s traveling-wave factor e^{ik_0 x} via free_gaussian_wavepacket, and verifies that its momentum-space density peaks exactly at p_0=\hbar k_0, so the associated wavelength is de Broglie&#x27;s \lambda = h/p.">

.. only:: html

  .. image:: /api/gallery/quantum/wave_packets/images/thumb/sphx_glr_plot_de_broglie_wavepacket_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/wave_packets/plot_de_broglie_wavepacket`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">De Broglie's matter wave</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Ehrenfest&#x27;s theorem states that quantum expectation values obey the classical equations of motion exactly:">

.. only:: html

  .. image:: /api/gallery/quantum/wave_packets/images/thumb/sphx_glr_plot_ehrenfest_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/wave_packets/plot_ehrenfest_theorem`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ehrenfest's theorem in an anharmonic well</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Propagates a wavepacket initially localized in the left well of a double well V(x) = \lambda(x^2-a^2)^2 with the real FFT split-operator solver (not the two-level analytic shortcut used in the double-well example), and renders it with animate_density&#x27;s HSV phase coloring: hue tracks the local complex phase \arg\psi(x,t), so the accumulating phase gradient as the packet tunnels back and forth is directly visible, not just the probability density.">

.. only:: html

  .. image:: /api/gallery/quantum/wave_packets/images/thumb/sphx_glr_plot_phase_colored_animation_thumb.gif
    :alt:

  :doc:`/api/gallery/quantum/wave_packets/plot_phase_colored_animation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">A phase-colored tunneling animation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Three non-stationary phenomena governed by the time-dependent Schrodinger equation, each solved via its own exact analytic route rather than direct numerical propagation. A free Gaussian wavepacket keeps its Gaussian shape but spreads as it travels, its width growing as">

.. only:: html

  .. image:: /api/gallery/quantum/wave_packets/images/thumb/sphx_glr_plot_wave_packet_dynamics_thumb.png
    :alt:

  :doc:`/api/gallery/quantum/wave_packets/plot_wave_packet_dynamics`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Dispersion, twin-slit interference, and quantum revivals</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/quantum/entanglement/index.rst
   /api/gallery/quantum/harmonic_oscillator/index.rst
   /api/gallery/quantum/hydrogen/index.rst
   /api/gallery/quantum/measurement/index.rst
   /api/gallery/quantum/perturbation/index.rst
   /api/gallery/quantum/potentials/index.rst
   /api/gallery/quantum/wave_packets/index.rst



.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_