.. _sphx_glr_api_gallery_quantum_measurement:

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.


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


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

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    <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.">

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  .. 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`

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


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    <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.">

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  .. 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`

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


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    <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.">

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

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

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


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    <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.">

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

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

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


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/quantum/measurement/plot_born_rule_measurement
   /api/gallery/quantum/measurement/plot_double_slit_propagation
   /api/gallery/quantum/measurement/plot_heisenberg_uncertainty
   /api/gallery/quantum/measurement/plot_stern_gerlach