.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/quantum/wave_packets/plot_twin_slit_interference.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_quantum_wave_packets_plot_twin_slit_interference.py: Jönsson's electron double slit: interference one electron at a time ====================================================================== In 1961 Claus Jönsson sent electrons through slits a micrometre apart in a thin foil and recorded the first two-slit interference fringes made by material particles. Each electron lands at one point, but the pattern they build up is :math:`|\psi_1+\psi_2|^2`, the interference of the amplitudes for going through either slit. This example follows the Gaussian electron wavepacket that :class:`~physicskit.quantum.chapters.wave_packets.GaussianDispersion` propagates exactly, computes the far-field two-slit pattern with :class:`~physicskit.quantum.chapters.wave_packets.TwinSlit`, and then draws individual electron arrivals from it to show the fringes emerging from single, point-like detections. Blocking one slit removes them. .. GENERATED FROM PYTHON SOURCE LINES 18-26 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from scipy.signal import find_peaks from physicskit.quantum.chapters.wave_packets import GaussianDispersion, TwinSlit from physicskit.quantum.visualizers.wavefunctions import animate_density .. GENERATED FROM PYTHON SOURCE LINES 27-32 The electron packet spreads on its way to the slits ------------------------------------------------------- A free Gaussian packet keeps its shape but widens as :math:`\sigma(t)=\sigma_0\sqrt{1+(\hbar t/2m\sigma_0^2)^2}`. The spreading is what lets a single electron's wave cover both slits. .. GENERATED FROM PYTHON SOURCE LINES 32-48 .. code-block:: Python gd = GaussianDispersion(x0=0.0, sigma0=1.0, k0=3.0) x1 = np.linspace(-20, 40, 1000) fig1, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4)) for t in [0, 2, 5, 10]: ax1.plot(x1, gd.density(x1, t), label=f"t = {t}") ax1.set_xlabel("x") ax1.set_title("Free electron wavepacket spreading") ax1.legend(fontsize=8) # The same exact solution, animated with its phase colour-coded: the de # Broglie phase :math:`e^{ik_0x}` rides along the advancing centre while # the envelope spreads. t_anim = np.linspace(0, 6.0, 120) x_anim = np.linspace(-20, 40, 600) anim = animate_density(x_anim, gd.trajectory(x_anim, t_anim), times=t_anim) .. rst-class:: sphx-glr-horizontal * .. image-sg:: /api/gallery/quantum/wave_packets/images/sphx_glr_plot_twin_slit_interference_001.png :alt: Free electron wavepacket spreading :srcset: /api/gallery/quantum/wave_packets/images/sphx_glr_plot_twin_slit_interference_001.png :class: sphx-glr-multi-img * .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 49-51 Two slits, and one ---------------------- .. GENERATED FROM PYTHON SOURCE LINES 51-66 .. code-block:: Python ts = TwinSlit(slit_separation=4.0, slit_width=0.4, k0=10.0) y = np.linspace(-10, 10, 2000) I_two = ts.intensity(y, screen_distance=50) # A single open slit: the two sources merged into one (zero separation). I_one = TwinSlit(slit_separation=0.0, slit_width=0.4, k0=10.0).intensity(y, screen_distance=50) ax2.plot(y, I_two / I_two.max(), color="steelblue", label=r"both slits: $|\psi_1+\psi_2|^2$") ax2.plot(y, I_one / I_one.max(), color="firebrick", ls="--", label="one slit") peaks, _ = find_peaks(I_two) print(f"bright fringes at y = {np.round(y[peaks], 2).tolist()}; expected spacing lambda L / d = {2 * np.pi / 10.0 * 50 / 4.0:.2f}") ax2.set_xlabel("position on the screen") ax2.set_ylabel("intensity (normalized)") ax2.set_title("Far-field two-slit pattern") ax2.legend(fontsize=8) fig1.tight_layout() .. rst-class:: sphx-glr-script-out .. code-block:: none bright fringes at y = [-7.79, -0.01, 7.79]; expected spacing lambda L / d = 7.85 .. GENERATED FROM PYTHON SOURCE LINES 67-73 Building the pattern one electron at a time ----------------------------------------------- Treat the normalized intensity as the probability of each landing position and draw electrons from it. A handful look random; thousands reproduce the fringes. With one slit blocked the same number of electrons gives a single smooth band. .. GENERATED FROM PYTHON SOURCE LINES 73-90 .. code-block:: Python rng = np.random.default_rng(1961) p_two = I_two / I_two.sum() p_one = I_one / I_one.sum() fig2, axes = plt.subplots(1, 4, figsize=(15, 3.5), sharey=False) for ax, n in zip(axes[:3], (50, 500, 20000)): hits = rng.choice(y, size=n, p=p_two) ax.hist(hits, bins=160, range=(-10, 10), color="steelblue") ax.set_title(f"{n} electrons, both slits open") ax.set_xlabel("screen position") hits_one = rng.choice(y, size=20000, p=p_one) axes[3].hist(hits_one, bins=160, range=(-10, 10), color="firebrick") axes[3].set_title("20000 electrons, one slit") axes[3].set_xlabel("screen position") fig2.tight_layout() plt.show() .. image-sg:: /api/gallery/quantum/wave_packets/images/sphx_glr_plot_twin_slit_interference_003.png :alt: 50 electrons, both slits open, 500 electrons, both slits open, 20000 electrons, both slits open, 20000 electrons, one slit :srcset: /api/gallery/quantum/wave_packets/images/sphx_glr_plot_twin_slit_interference_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 91-95 Check ----- Fringes spaced by lambda L / d, and a free packet's width grows as sigma(t) = sigma0 sqrt(1 + (t / 2 sigma0^2)^2) (hbar = m = 1). .. GENERATED FROM PYTHON SOURCE LINES 95-104 .. code-block:: Python spacing = np.diff(y[peaks]).mean() assert abs(spacing / (2 * np.pi / 10.0 * 50 / 4.0) - 1) < 0.02 x_wide = np.linspace(-30, 60, 6000) rho = gd.density(x_wide, 5.0) rho /= np.trapezoid(rho, x_wide) mean = np.trapezoid(x_wide * rho, x_wide) width = np.sqrt(np.trapezoid((x_wide - mean) ** 2 * rho, x_wide)) assert abs(mean - 3.0 * 5.0) < 1e-3 assert abs(width - np.sqrt(1 + (5.0 / 2) ** 2)) < 1e-3 .. rst-class:: sphx-glr-timing **Total running time of the script:** (1 minutes 6.835 seconds) .. _sphx_glr_download_api_gallery_quantum_wave_packets_plot_twin_slit_interference.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_twin_slit_interference.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_twin_slit_interference.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_twin_slit_interference.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_