Breakthroughs in Quantum Mechanics ================================== .. include:: /_generated/nav/quantum.rst .. epigraph:: "I think I can safely say that nobody understands quantum mechanics." -- Richard Feynman, *The Character of Physical Law*, 1965 Quantum mechanics replaced a world of definite trajectories with one of wavefunctions, probabilities, and operators -- and did so not as a single discovery but as a three-decade argument among the physicists who built it, from Planck's reluctant quantum in 1900 to Bell's 1964 proof that no "common-sense" theory could ever reproduce its predictions. The chronology behind :mod:`physicskit.quantum` follows that argument in order: the old quantum theory of Bohr, the two rival (and ultimately equivalent) mechanics of Heisenberg and Schrodinger, the interpretive battles over what the wavefunction even *means*, and the exactly solvable model problems -- the hydrogen atom, the harmonic oscillator, the square well -- that remain the field's working vocabulary today. This chronology traces that thread, with a pointer to the corresponding implementation in this package at each stop. .. contents:: Timeline :local: :depth: 1 1900 -- Planck's Quantum Hypothesis ----------------------------------- To fit the observed spectrum of blackbody radiation -- which classical electrodynamics predicted should diverge at short wavelengths (the "ultraviolet catastrophe") -- Max Planck proposed, in a December 1900 paper to the German Physical Society, that the energy exchanged between radiation and matter is not continuous but comes in discrete quanta, .. math:: E_n = n h \nu, \qquad n = 0, 1, 2, \dots, for an oscillator of frequency :math:`\nu`. Planck himself considered this a mathematical device rather than a physical claim -- he later called it "an act of desperation" -- but the idea that a harmonic oscillator's energy spectrum is discrete, evenly spaced by :math:`h\nu` (or, in the full quantum treatment developed a generation later, by :math:`\hbar\omega`), turned out to be exactly right, and became the seed of everything that follows in this chronology. *Connection:* :meth:`physicskit.quantum.chapters.harmonic_spin.HarmonicOscillator.energy` returns exactly this evenly spaced spectrum, :math:`E_n = \hbar\omega(n + \tfrac12)`, for the quantum harmonic oscillator that eventually made Planck's quantization rule precise. *References:* M. Planck, Verhandlungen der Deutschen Physikalischen Gesellschaft 2, 237-245 (1900). .. minigallery:: ../../examples/quantum/harmonic_oscillator/plot_ladder_operators.py 1913 -- Bohr's Atomic Model --------------------------- Niels Bohr proposed that an electron orbiting a nucleus is restricted to a discrete set of "stationary states" with quantized angular momentum :math:`L = n\hbar`, in which -- contrary to classical electrodynamics -- it does not radiate, jumping between orbits only by emitting or absorbing a photon of energy :math:`\Delta E`. For hydrogen this postulate, combined with a classical circular-orbit force balance, gives the energy levels .. math:: E_n = -\frac{1}{2n^2}\,\frac{m e^4}{\hbar^2} \quad\text{(Hartree atomic units)}, reproducing the empirical Rydberg formula for hydrogen's spectral lines to remarkable precision. Bohr's model was frankly *ad hoc* -- a classical orbit with a quantization rule bolted on -- but it was the first successful quantum theory of an atom, and every one of its energy levels survives unchanged in the exact 1926 treatment that replaced it. *Connection:* :class:`physicskit.quantum.chapters.hydrogen_am.HydrogenOrbital` reproduces Bohr's :math:`n`-dependence exactly in its ``energy`` property, :math:`E_n = -Z^2/(2n^2)`, even though it is derived from the full Schrodinger equation rather than Bohr's semiclassical orbits. See it in :doc:`/api/gallery/quantum/hydrogen/plot_hydrogen_orbitals`. *References:* N. Bohr, "On the Constitution of Atoms and Molecules," Phil. Mag. Ser. 6, 26, 1-25, 476-502, 857-875 (1913). .. minigallery:: ../../examples/quantum/hydrogen/plot_hydrogen_orbitals.py 1913 -- The Zeeman and Stark Effects ------------------------------------ Pieter Zeeman had already shown, in 1896, that spectral lines split into several components in a magnetic field (Nobel Prize, 1902, shared with Hendrik Lorentz, who supplied the classical explanation). In 1913 Johannes Stark discovered the electrical analogue: hydrogen's Balmer lines split linearly under a strong external electric field. Both effects became essential tools of "old quantum theory" spectroscopy, and both later became textbook applications of perturbation theory -- the Zeeman effect of non-degenerate perturbation theory, and the *linear* Stark effect of hydrogen specifically of the *degenerate* variety, made possible only by hydrogen's accidental level degeneracy in :math:`l`. *Implementation:* :func:`physicskit.quantum.chapters.perturbation.zeeman_splitting` and :func:`~physicskit.quantum.chapters.perturbation.zeeman_spectrum` compute the first-order Zeeman sublevel shifts; :func:`~physicskit.quantum.chapters.perturbation.linear_stark_shift` and :func:`~physicskit.quantum.chapters.perturbation.stark_n2_quartet` reproduce the classic linear Stark quartet of hydrogen's :math:`n=2` shell. *References:* P. Zeeman, Phil. Mag. 43, 226-239 (1897) (originally published in Dutch in 1896); J. Stark, Sitzungsber. Preuss. Akad. Wiss. 1913, 932-946. .. minigallery:: ../../examples/quantum/perturbation/plot_perturbation_and_floquet.py 1915-1916 -- Wilson-Sommerfeld Quantization ------------------------------------------- William Wilson and Arnold Sommerfeld independently generalized Bohr's quantization rule from circular orbits to any periodic classical motion, replacing it with the phase-space integral .. math:: \oint p\,dq = n h, \qquad n = 1, 2, 3, \dots, taken once around a full period. Applied to elliptical Kepler orbits this reproduced hydrogen's Bohr energies unchanged (adding a second quantum number for orbital shape), and Sommerfeld's relativistic refinement of the same rule correctly predicted hydrogen's fine structure -- an early triumph of "old quantum theory." But the rule only quantizes systems whose classical motion is periodic and separable; it could not be extended to the helium atom or to molecular spectra, a failure that motivated the wholesale replacement Heisenberg and Schrodinger supplied over a decade later. *Connection:* :func:`physicskit.semiclassical.core.wkb.bohr_sommerfeld_energies` implements the modern, Maslov-corrected descendant of this same phase-space quantization integral, :math:`\int_{x_1}^{x_2}p\,dx=(n+\tfrac12)\pi\hbar`, turning Wilson and Sommerfeld's postulate into a working numerical bound-state solver for an arbitrary one-dimensional potential. *References:* W. Wilson, Phil. Mag. 29, 795-802 (1915); A. Sommerfeld, Ann. Phys. 356(17), 1-94 (1916). .. minigallery:: ../../examples/semiclassical/wkb/plot_wkb_bohr_sommerfeld.py 1922 -- The Stern-Gerlach Experiment ------------------------------------- Otto Stern proposed, and with Walther Gerlach carried out at the University of Frankfurt, an experiment to test whether the "space quantization" of old quantum theory -- the claim that an atom's magnetic moment can point only along a discrete set of directions relative to an external field, rather than any classical angle -- was physically real rather than a mathematical bookkeeping device. They sent a beam of silver atoms, each with one unpaired valence electron, through a strongly inhomogeneous magnetic field: a magnetic moment free to point at any angle should smear the beam into one broadened band, while a moment restricted to a discrete set of orientations should split it into a corresponding number of separate spots. In the early hours of 8 February 1922 they found the beam split cleanly in two -- not smeared, confirming that spatial quantization is real, though the *number* of spots (two, not the odd count old quantum theory's integer orbital angular momentum would have implied for silver's ground state) could only be explained after Uhlenbeck and Goudsmit posited electron spin three years later, in 1925. .. math:: F_y = \pm\, \mu\,\frac{\partial B_z}{\partial y}, the spin-dependent transverse force -- its sign set by which way the atom's magnetic moment projects along the field gradient -- that pushes the two spin branches apart into separately resolved lobes. Stern received the 1943 Nobel Prize in Physics for the molecular-beam method this experiment inaugurated; Gerlach, who did much of the hands-on experimental work (including the slit refinement that made the clean February 1922 result possible), was never awarded a share, a historical omission usually attributed to wartime politics rather than to any dispute over credit. *Implementation:* :class:`physicskit.quantum.chapters.spin.SternGerlach` implements the standard semiclassical two-branch treatment: each spin projection is modeled as an independent Gaussian wavepacket subject to exactly this constant transverse force, so its :meth:`~physicskit.quantum.chapters.spin.SternGerlach.joint_density` (and its time-stacked :meth:`~physicskit.quantum.chapters.spin.SternGerlach.joint_density_stack`) show the two lobes separating in real time as the beam propagates through the apparatus, animated with :func:`~physicskit.quantum.visualizers.wavefunctions.animate_density_2d`. See it in :doc:`/api/gallery/quantum/measurement/plot_stern_gerlach`. *References:* W. Gerlach and O. Stern, Z. Phys. 9, 349-352 (1922); precursor O. Stern, Z. Phys. 7, 249-253 (1921). .. minigallery:: ../../examples/quantum/measurement/plot_stern_gerlach.py 1924 -- de Broglie's Matter Waves --------------------------------- In his doctoral thesis, Louis de Broglie proposed that the wave-particle duality Einstein had already established for light should run in reverse: every material particle of momentum :math:`p` has an associated wavelength .. math:: \lambda = \frac{h}{p}. The hypothesis was speculative -- de Broglie had no direct evidence for it -- until Davisson and Germer observed electron diffraction from a nickel crystal in 1927, confirming it experimentally. De Broglie's wave is the first appearance of the traveling-wave factor :math:`e^{ikx}` that appears in every wavepacket and plane-wave state used throughout quantum mechanics. *Implementation:* :func:`physicskit.quantum.chapters.wave_packets.free_gaussian_wavepacket` builds a localized wavepacket carrying exactly this de Broglie phase, :math:`e^{ik_0 x}`, modulating a Gaussian envelope centered on momentum :math:`p_0 = \hbar k_0`. The same packet's subsequent free evolution -- :class:`~physicskit.quantum.chapters.wave_packets.GaussianDispersion`, the exact analytic solution of the free-particle Schrodinger equation for a Gaussian initial condition -- is animated frame by frame with its :meth:`~physicskit.quantum.chapters.wave_packets.GaussianDispersion.trajectory` method and :func:`~physicskit.quantum.visualizers.wavefunctions.animate_density`, showing the de Broglie phase riding along a center that advances ballistically at :math:`v=\hbar k_0/m` while the envelope spreads. The animated dispersion is shown in :doc:`/api/gallery/quantum/wave_packets/plot_wave_packet_dynamics`. :class:`~physicskit.quantum.visualizers.phase_space.WignerVisualizer` renders the same packet's Wigner phase-space distribution directly, a single blob centered on :math:`(x_0, p_0)` that is position space and momentum space at once -- the modern phase-space picture of exactly the wave-particle duality de Broglie proposed. *References:* L. de Broglie, Ann. Phys. (Paris) 10e série, 3 (1925) (thesis defended 1924); experimental confirmation C. Davisson and L. Germer, Phys. Rev. 30, 705-740 (1927). .. minigallery:: ../../examples/quantum/wave_packets/plot_de_broglie_wavepacket.py 1925-1927 -- Pauli's Exclusion Principle and Spin Matrices ------------------------------------------------------------ Wolfgang Pauli proposed, in 1925, that no two electrons in an atom can share the same complete set of quantum numbers -- the exclusion principle that finally explained the shell structure of the periodic table old quantum theory could only fit by hand. Two years later he made the electron's spin degree of freedom, already inferred by Uhlenbeck and Goudsmit from the Stern-Gerlach result, mathematically explicit, representing a spin-1/2 particle's two-valued internal state with a set of three :math:`2\times2` matrices, .. math:: \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, satisfying :math:`\sigma_i\sigma_j = \delta_{ij}I + i\epsilon_{ijk}\sigma_k`. The exclusion principle is a statement about many-electron wavefunctions (their required antisymmetry under particle exchange, a connection Pauli himself only made fully explicit with the 1940 spin-statistics theorem); the Pauli matrices, by contrast, are the working algebra of any single two-level quantum system, and appear throughout this package wherever a spin-1/2 or qubit degree of freedom is modeled. *Implementation:* :data:`physicskit.quantum.core.operators.sigma_x`, :data:`~physicskit.quantum.core.operators.sigma_y`, and :data:`~physicskit.quantum.core.operators.sigma_z` are exactly these three matrices, and :func:`~physicskit.quantum.core.operators.spin_operator` generalizes them to arbitrary spin quantum number :math:`s`. They are the computational basis for two-level physics elsewhere in this chronology: :class:`physicskit.quantum.chapters.spin.RabiProblem` builds its RWA Hamiltonian and propagator directly from ``sigma_z`` and ``sigma_x``, and :class:`physicskit.quantum.chapters.entanglement.BellCorrelations` uses the same two matrices to build its spin-projection measurement operators. The exclusion principle itself -- multi-electron antisymmetrization -- has no direct counterpart in this package, which works throughout in the single- and two-qubit/single-particle regime. *References:* W. Pauli, "Uber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren," Z. Phys. 31, 765-783 (1925) (exclusion principle); W. Pauli, "Zur Quantenmechanik des magnetischen Elektrons," Z. Phys. 43, 601-623 (1927) (spin matrices). .. minigallery:: ../../examples/quantum/entanglement/plot_bloch_sphere_spin_dynamics.py 1926 -- Schrodinger's Wave Equation and the Hydrogen Atom --------------------------------------------------------- Erwin Schrodinger, seeking a wave equation whose stationary states would reproduce Bohr's energy levels, published the time-independent equation .. math:: -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi in a rapid sequence of four papers in 1926, and immediately solved it exactly for the Coulomb potential -- recovering Bohr's hydrogen spectrum :math:`E_n = -Z^2/(2n^2)` not as a postulate but as an eigenvalue, together with the full three-dimensional orbital structure Bohr's model could never supply. Within months Schrodinger also showed his wave mechanics to be mathematically equivalent to Heisenberg's matrix mechanics, unifying the two independently-discovered quantum theories. *Implementation:* :class:`physicskit.quantum.core.eigensolvers.NumerovSolver` solves exactly this time-independent equation numerically for an arbitrary 1D potential; :func:`physicskit.quantum.chapters.hydrogen_am.radial_wavefunction` and :func:`~physicskit.quantum.chapters.hydrogen_am.spherical_harmonic` give the exact analytic 3D hydrogen solution :math:`\psi_{nlm}=R_{nl}(r)Y_l^m(\theta,\phi)`. A coherent superposition of two such eigenstates is no longer stationary: :func:`~physicskit.quantum.chapters.hydrogen_am.orbital_superposition_psi` and :func:`~physicskit.quantum.chapters.hydrogen_am.orbital_superposition_density` evolve :math:`\psi = c_a\psi_a e^{-iE_a t/\hbar}+c_b\psi_b e^{-iE_b t/\hbar}` and show its density beating at the Bohr transition frequency :math:`\omega_{ab}=(E_a-E_b)/\hbar`, rendered as a Plotly play-button isosurface animation by :func:`~physicskit.quantum.visualizers.orbitals.animate_orbital_beating`. The exact hydrogen solution, including the beating animation, is shown in :doc:`/api/gallery/quantum/hydrogen/plot_hydrogen_orbitals`. *References:* four "Mitteilungen" papers, E. Schrodinger, Ann. Phys. 384(4), 361-376; 384(6), 489-527; 385(13), 437-490; 386(18), 109-139 (1926); equivalence proof, Ann. Phys. 384(8), 734-756 (1926). .. minigallery:: ../../examples/quantum/potentials/plot_bound_states_numerov.py 1926 -- Born's Probabilistic Interpretation ------------------------------------------- While working out scattering theory, Max Born proposed that Schrodinger's wavefunction :math:`\psi` has no direct physical reality itself; rather, :math:`\lvert\psi(x)\rvert^2` is a probability density for finding the particle at :math:`x` upon measurement, .. math:: P(x)\,dx = \lvert\psi(x)\rvert^2\, dx. This "Born rule" was the interpretive break that made quantum mechanics irreducibly statistical -- not merely as a matter of incomplete knowledge, as in classical statistical mechanics, but as a fundamental feature of nature -- a conclusion Einstein famously never accepted ("God does not play dice"). Born received the Nobel Prize for this insight in 1954. *Implementation:* :func:`physicskit.quantum.utils.measure.simulate_position_measurement` draws simulated measurement outcomes directly from the Born rule, :math:`x_i \sim \lvert\psi(x)\rvert^2`; :func:`~physicskit.quantum.utils.measure.position_expectation` computes the resulting expectation value :math:`\langle x\rangle`. *References:* M. Born, Z. Phys. 37, 863-867 (1926), corrected and extended in Z. Phys. 38, 803-827 (1926). .. minigallery:: ../../examples/quantum/measurement/plot_born_rule_measurement.py 1926 -- The WKB Approximation ----------------------------- Building on de Broglie's matter wave, Gregor Wentzel, Hendrik Kramers, and Leon Brillouin -- following Harold Jeffreys' earlier, independent mathematical treatment of the same asymptotic expansion -- showed in 1926 how to recover Schrodinger's exact wave mechanics as a controlled semiclassical limit. Writing the wavefunction as an amplitude times a rapidly oscillating phase, .. math:: \psi(x) \sim \frac{1}{\sqrt{p(x)}}\, \exp\!\left(\pm\frac{i}{\hbar}\int p(x')\,dx'\right), \qquad p(x) = \sqrt{2m\bigl(E-V(x)\bigr)}, and expanding order by order in :math:`\hbar`, the WKB approximation reproduces the classical limit exactly where it should: the amplitude :math:`p(x)^{-1/2}` is precisely the classical probability of finding a particle where it moves slowest, and matching the oscillatory solution through each turning point (via the Airy-function connection formulas) costs a phase of :math:`\pi/4`, refining Wilson-Sommerfeld's integer quantization into the half-integer Einstein-Brillouin-Keller (EBK) rule. *Implementation:* :func:`physicskit.semiclassical.core.wkb.classical_momentum` and :func:`~physicskit.semiclassical.core.wkb.turning_points` build the semiclassical momentum :math:`p(x)` and locate its turning points; :func:`~physicskit.semiclassical.core.wkb.wkb_action` and :func:`~physicskit.semiclassical.core.wkb.wkb_wavefunction` assemble the resulting approximate wavefunction, reproducing the node-counting theorem and the exact harmonic-oscillator spectrum to machine precision. *References:* G. Wentzel, Z. Phys. 38, 518-529 (1926); H. Kramers, Z. Phys. 39, 828-840 (1926); L. Brillouin, C. R. Acad. Sci. 183, 24-26 (1926); precursor H. Jeffreys, Proc. London Math. Soc. 23, 428-436 (1925). .. minigallery:: ../../examples/semiclassical/wkb/plot_wkb_bohr_sommerfeld.py 1927 -- Heisenberg's Uncertainty Principle ------------------------------------------ Werner Heisenberg had already given quantum mechanics its first formulation in 1925, replacing classical trajectories with infinite arrays ("matrices") of transition amplitudes -- matrix mechanics, in which position and momentum are non-commuting operators satisfying :math:`[\hat x,\hat p] = i\hbar`. Heisenberg's initial paper treated only one degree of freedom and left the general algebra incomplete; its full development into a systematic many-body matrix mechanics is jointly credited to Heisenberg, Max Born, and Pascual Jordan, whose "Drei-Manner- Arbeit" ("three-man paper") supplied the rigorous multi-dimensional formulation, including the canonical commutation relation in the form above. In 1927 Heisenberg drew out that algebra's starkest physical consequence: position and momentum cannot both be known to arbitrary precision, .. math:: \Delta x \, \Delta p \ge \frac{\hbar}{2}. This is not a statement about measurement clumsiness but a structural property of any state -- a direct consequence of :math:`x` and :math:`p` failing to commute, and the first hint that quantum mechanics would demand a wholesale revision of what "knowing a system's state" even means. *Implementation:* :func:`physicskit.quantum.utils.measure.uncertainty` computes :math:`\Delta x`, :math:`\Delta p`, and their product directly from a wavefunction, checking the Heisenberg bound; :func:`physicskit.quantum.core.operators.commutator` together with :func:`~physicskit.quantum.core.operators.position_operator` and :func:`~physicskit.quantum.core.operators.momentum_operator` verify the underlying canonical commutation relation :math:`[\hat x,\hat p]=i\hbar` in a truncated oscillator basis. *References:* W. Heisenberg, Z. Phys. 33, 879-893 (1925); M. Born and P. Jordan, Z. Phys. 34, 858-888 (1925); M. Born, W. Heisenberg, and P. Jordan, Z. Phys. 35, 557-615 (1926); the uncertainty principle itself, W. Heisenberg, Z. Phys. 43, 172-198 (1927). .. minigallery:: ../../examples/quantum/measurement/plot_heisenberg_uncertainty.py 1927 -- Ehrenfest's Theorem ---------------------------- Paul Ehrenfest asked how classical mechanics could ever emerge from a theory built entirely on wavefunctions and operators, and found that it already had: the expectation values of position and momentum obey Newton's equations of motion exactly, for *any* potential, not merely a harmonic one, .. math:: \frac{d\langle x\rangle}{dt} = \frac{\langle p\rangle}{m}, \qquad \frac{d\langle p\rangle}{dt} = -\left\langle\frac{dV}{dx}\right\rangle. This is a precise structural fact about the Schrodinger equation, not an approximation valid only for slowly varying potentials -- the subtlety is that the second equation involves :math:`\langle dV/dx\rangle`, the average of the force over the whole spread-out wavefunction, which coincides with the classical force evaluated at the mean position, :math:`-dV/dx(\langle x\rangle)`, only when :math:`V` is quadratic. For any other potential a wavepacket's centroid drifts away from the corresponding classical trajectory as it spreads, broadens, or splits -- the precise sense in which quantum and classical dynamics agree exactly in expectation value, yet disagree about everything a single classical trajectory would predict. *Implementation:* :class:`physicskit.quantum.core.solvers.SplitOperatorSolver1D` propagates an arbitrary wavepacket through an arbitrary potential, and :class:`physicskit.quantum.utils.measure.ExpectationMonitor` records :math:`\langle x\rangle(t)` and :math:`\langle p\rangle(t)` from each snapshot; comparing their numerical time derivatives against :math:`\langle p\rangle/m` and :math:`-\langle dV/dx\rangle`, computed independently via :func:`~physicskit.quantum.utils.measure.expectation_value`, verifies both Ehrenfest identities directly from a real, propagated wavefunction in a genuinely anharmonic (quartic) well; the same propagated frames also feed :class:`~physicskit.quantum.visualizers.phase_space.WignerVisualizer`, showing the state's phase-space distribution shear away from a rigid rotation as the anharmonic potential acts, with the :math:`(\langle x\rangle,\langle p\rangle)` trajectory traced on top of it. *References:* P. Ehrenfest, Z. Phys. 45, 455-457 (1927). .. minigallery:: ../../examples/quantum/wave_packets/plot_ehrenfest_theorem.py 1927 -- Dirac's Time-Dependent Perturbation Theory -------------------------------------------------- Paul Dirac, developing the quantum theory of radiation, worked out how a weak, time-dependent perturbation drives transitions between the unperturbed stationary states of a system -- the foundation of time-dependent perturbation theory. Two decades later Enrico Fermi gave the leading-order transition-rate result a name that stuck: the "golden rule," so called in his course-derived textbook *Nuclear Physics*, compiled from his University of Chicago lectures and published in 1950 -- not, as sometimes stated, unpublished lecture notes. The same machinery describes a system driven periodically in time, where transitions become resonant multiphoton processes whenever an integer number of drive quanta :math:`\hbar\omega` bridges an energy gap. *Implementation:* :class:`physicskit.quantum.chapters.perturbation.FloquetDrivenBox` propagates a periodically driven infinite well with the FFT split-operator method, and its :meth:`~physicskit.quantum.chapters.perturbation.FloquetDrivenBox.transition_probability` method exhibits exactly these resonant multiphoton transitions between box eigenstates. *References:* P. A. M. Dirac, Proc. R. Soc. A 114(767), 243-265 (1927); "golden rule" naming, E. Fermi, Nuclear Physics (University of Chicago Press, 1950), a textbook compiled from his lecture course. .. minigallery:: ../../examples/quantum/perturbation/plot_floquet_driven_box.py 1928 -- Gamow, Gurney and Condon: Quantum Tunneling --------------------------------------------------- George Gamow, and independently Ronald Gurney and Edward Condon, explained why radioactive nuclei undergo alpha decay at all: classically, the alpha particle sits in a potential well surrounded by a Coulomb barrier far higher than its kinetic energy, and should never escape. Quantum mechanically, the wavefunction does not vanish inside a classically forbidden region -- it decays exponentially but remains nonzero -- giving a small but nonzero probability of finding the particle on the far side of the barrier. This single idea, quantum tunneling, explained the enormous range of observed alpha-decay half-lives (many orders of magnitude, for barriers differing only modestly in height) and remains the archetypal example of a genuinely quantum phenomenon with no classical counterpart. *Implementation:* :meth:`physicskit.quantum.chapters.potentials.FiniteSquareWell.scattering` gives the exact transmission probability through a barrier, continuing correctly into the sub-barrier tunneling regime via complex arithmetic; :meth:`physicskit.quantum.chapters.potentials.DoubleWellSimulator.tunneling_oscillation` and :meth:`~physicskit.quantum.chapters.potentials.DoubleWellSimulator.left_well_probability` reproduce the related phenomenon of coherent tunneling oscillation between the two wells of a symmetric double well. Beyond the stationary transmission coefficient, :meth:`~physicskit.quantum.chapters.potentials.FiniteSquareWell.wavepacket_scattering` propagates an actual moving Gaussian wavepacket through the same barrier (or, with a positive well depth instead, the related resonant-scattering case) with the FFT split-operator method, splitting it into a reflected and a transmitted piece in real time; and :meth:`~physicskit.quantum.chapters.potentials.DoubleWellSimulator.tunneling_wavefunction` gives the complex two-state wavefunction underlying the density-only ``tunneling_oscillation`` above. Both are rendered frame by frame with :func:`~physicskit.quantum.visualizers.wavefunctions.animate_density`. See the barrier-tunneling animation in :doc:`/api/gallery/quantum/potentials/plot_barrier_tunneling`, the animated well-scattering case in :doc:`/api/gallery/quantum/potentials/plot_ramsauer_townsend_resonance`, and the double-well oscillation, including its animated complex-valued version, in :doc:`/api/gallery/quantum/potentials/plot_double_well_tunneling`. *References:* G. Gamow, Z. Phys. 51, 204-212 (1928); R. Gurney and E. Condon, Nature 122, 439 (1928), and Phys. Rev. 33, 127-140 (1929). .. minigallery:: ../../examples/quantum/potentials/plot_barrier_tunneling.py ../../examples/quantum/potentials/plot_ramsauer_townsend_resonance.py ../../examples/quantum/potentials/plot_double_well_tunneling.py 1928 -- The Van Vleck-Morette Semiclassical Propagator -------------------------------------------------------- The same semiclassical logic behind WKB applies equally to time evolution rather than stationary states: in 1928 John Van Vleck (extended into a full path-integral formulation by Cecile DeWitt-Morette in 1951) showed that the quantum propagator :math:`\langle x\rvert e^{-i\hat Ht/\hbar}\lvert x_0\rangle` is dominated, to leading order in :math:`\hbar`, by a single classical trajectory connecting :math:`x_0` to :math:`x` in time :math:`t`, .. math:: K(x,t;x_0,0) \approx \sqrt{\frac{1}{2\pi i\hbar}\left\lvert\frac{\partial p_0}{\partial x}\right\rvert}\; e^{iS(x,t;x_0,0)/\hbar - i\sigma\pi/2}, where :math:`S` is the classical action along that trajectory and :math:`\sigma` (the Maslov index) counts the caustics -- focal points where nearby trajectories cross -- it passes through. This single-trajectory picture of propagation, two decades before Richard Feynman's 1948 path-integral formulation summed over *every* trajectory rather than just the classical one, is the direct ancestor of every semiclassical propagation method used in modern quantum chaos and chemical physics. *Implementation:* :func:`physicskit.semiclassical.core.propagators.propagate_trajectory_monodromy_action` integrates a classical trajectory together with its monodromy matrix and action; :func:`~physicskit.semiclassical.core.propagators.van_vleck_prefactor`, :func:`~physicskit.semiclassical.core.propagators.count_caustics`, and :func:`~physicskit.semiclassical.core.propagators.van_vleck_propagator_1d` assemble these into exactly the propagator amplitude above, Maslov phase included. *References:* J. Van Vleck, PNAS 14(2), 178-188 (1928); C. Morette, Phys. Rev. 81, 848-852 (1951); R. P. Feynman, "Space-Time Approach to Non-Relativistic Quantum Mechanics," Rev. Mod. Phys. 20, 367-387 (1948). .. minigallery:: ../../examples/semiclassical/propagators/plot_semiclassical_propagators.py 1930 -- Dirac's Operator Method and the Harmonic Oscillator ----------------------------------------------------------- In *The Principles of Quantum Mechanics* (1930), Dirac popularized an algebraic shortcut for the quantum harmonic oscillator that sidesteps solving the Schrodinger equation as a differential equation entirely: factor the Hamiltonian into raising and lowering ("ladder") operators :math:`\hat a^\dagger, \hat a` satisfying :math:`[\hat a,\hat a^\dagger]=1`, build the entire spectrum by repeated raising from a ground state annihilated by :math:`\hat a`, and read off :math:`E_n=\hbar\omega(n+\tfrac12)` directly from the operator algebra. The technique, generalized to Glauber's 1963 coherent states :math:`\lvert\alpha\rangle` -- eigenstates of :math:`\hat a` that behave as classically as a quantum state can -- became the standard language of quantum optics. *Implementation:* :func:`physicskit.quantum.core.operators.annihilation_operator`, :func:`~physicskit.quantum.core.operators.creation_operator`, and :func:`~physicskit.quantum.core.operators.number_operator` build exactly this ladder-operator algebra in a truncated Fock basis; :meth:`physicskit.quantum.chapters.harmonic_spin.HarmonicOscillator.coherent_wavefunction` and :meth:`~physicskit.quantum.chapters.harmonic_spin.HarmonicOscillator.squeezed_vacuum_wavefunction` build Glauber coherent states and their squeezed generalizations from this same Fock-state expansion. The coherent/squeezed states are shown in :doc:`/api/gallery/quantum/harmonic_oscillator/plot_harmonic_oscillator_suite`. *References:* P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, 1930); coherent states, R. Glauber, Phys. Rev. 131, 2766-2788 (1963). .. minigallery:: ../../examples/quantum/harmonic_oscillator/plot_ladder_operators.py 1935 -- The EPR Paradox ----------------------- Einstein, Podolsky, and Rosen argued, in a paper meant to demonstrate quantum mechanics' incompleteness, that if two particles are prepared in a correlated ("entangled") state and then separated by an arbitrary distance, measuring one instantaneously determines the corresponding outcome for the other. Since no real influence should propagate faster than light, EPR concluded that the particles must have possessed definite values all along, carried by some "hidden variable" quantum mechanics simply fails to describe -- entanglement, to them, was evidence the theory was unfinished rather than a genuinely new feature of nature. It would take Bell twenty-nine years to show that this intuition, however reasonable, is testable -- and wrong. *Implementation:* :class:`physicskit.quantum.chapters.entanglement.IsingEntangler` builds exactly the kind of correlated two-particle state EPR had in mind, but shows how it actually arises rather than simply positing it: two qubits, each individually unbiased and initially unentangled (:math:`\lvert+\rangle\otimes\lvert+\rangle`), accumulate genuine entanglement purely through an Ising coupling :math:`\hat H=\hbar J\,\sigma_z^{(1)}\otimes\sigma_z^{(2)}`, with its :meth:`~physicskit.quantum.chapters.entanglement.IsingEntangler.concurrence` and :meth:`~physicskit.quantum.chapters.entanglement.IsingEntangler.purity` methods tracking the buildup from a product state (concurrence 0) to a maximally entangled, Bell-equivalent state (concurrence 1) at :math:`Jt=\pi/4`, animated with :func:`~physicskit.quantum.visualizers.entanglement.animate_entanglement_growth`. See it in :doc:`/api/gallery/quantum/entanglement/plot_ising_entanglement_growth`. *References:* A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777-780 (1935). .. minigallery:: ../../examples/quantum/entanglement/plot_ising_entanglement_growth.py 1935 -- Schrodinger Names "Entanglement" ----------------------------------------- Responding directly to Einstein, Podolsky, and Rosen within months of their paper, Erwin Schrodinger wrote a long analysis of what their correlated state actually implies, and in it coined the word that has described the phenomenon ever since: *Verschrankung*, "entanglement." Schrodinger argued that entanglement -- not any single particle's uncertainty -- is the truly characteristic trait of quantum mechanics, the feature that most sharply separates it from any classical theory, and in the same paper introduced his now-famous cat, whose life hangs on the entangled fate of a radioactive atom, to dramatize how strange it is that this feature seems to survive all the way up to macroscopic scale. The package's own entanglement measures and Bell-state machinery are built on precisely the concept Schrodinger named here, three decades before Bell showed it was testable and half a century before it acquired experimental and technological weight of its own. *Connection:* :class:`physicskit.quantum.chapters.entanglement.IsingEntangler` and :func:`physicskit.quantum.chapters.entanglement.bell_state`, together with the concurrence and purity measures used throughout the entanglement entries in this chronology, quantify exactly the non-classical correlation -- entanglement -- that Schrodinger's paper first named and argued was quantum mechanics' defining feature. *References:* E. Schrodinger, "Die gegenwartige Situation in der Quantenmechanik," Naturwissenschaften 23, 807-812, 823-828, 844-849 (1935); English translation, "The Present Situation in Quantum Mechanics," Proc. Am. Phil. Soc. 124, 323-338 (1980). .. minigallery:: ../../examples/quantum/entanglement/plot_ising_entanglement_growth.py 1937-1946 -- Rabi's Magnetic Resonance Method and the Bloch Sphere -------------------------------------------------------------------- In 1937, at Columbia University, Isidor Rabi devised the molecular-beam magnetic-resonance method: a beam of molecules passes through a static magnetic field and a region of oscillating field tuned near the natural (Larmor) precession frequency of the nuclear magnetic moment, and that moment's projection flips with near-unit probability exactly when the drive is resonant -- the phenomenon now called a "Rabi oscillation." The theoretical paper was completed in February and published in April 1937; the first experimental resonance curve, obtained with a lithium chloride beam, was submitted to *Physical Review* in January 1938. The method gave, for the first time, a direct and extraordinarily precise measurement of nuclear magnetic moments, and Rabi received the 1944 Nobel Prize in Physics for it. Nine years later, at Stanford, Felix Bloch (and independently, at Harvard, Edward Purcell) extended the same resonance idea from a molecular beam in vacuum to nuclear spins in bulk matter, detecting the induced radiofrequency signal directly rather than a deflected beam -- "nuclear induction," published in 1946. In that same paper Bloch introduced the geometric device that now carries his name: representing bulk nuclear magnetization as a classical vector precessing (under a static field) or nutating (under a resonant drive) on the surface of a unit sphere, exactly as a classical magnetic moment would. Bloch's own sphere was a picture of a real classical vector -- an ensemble average over many nuclear spins -- rather than the quantum state of one two-level system; the now-standard generalization of the same geometric picture to the pure quantum state of *any* single two-level system (a single spin, or, in modern usage, a single qubit) is usually credited to a 1957 paper by Richard Feynman, Frank Vernon Jr., and Robert Hellwarth, which showed that any two-state Schrodinger evolution can be mapped onto exactly Bloch's precession/nutation geometry. Bloch and Purcell shared the 1952 Nobel Prize in Physics for the original discovery, the direct ancestor of nuclear magnetic resonance (NMR) spectroscopy, MRI, and -- via the Feynman-Vernon-Hellwarth generalization -- the standard way of visualizing a single qubit's state in modern quantum computing. .. math:: U(t) = \cos\!\left(\frac{\Omega_R t}{2}\right) I - i\sin\!\left(\frac{\Omega_R t}{2}\right) \frac{\Delta\,\sigma_z + \Omega\,\sigma_x}{\Omega_R}, \qquad \Omega_R = \sqrt{\Delta^2+\Omega^2}, the closed-form rotating-wave-approximation propagator for a two-level system driven near resonance -- the "Rabi formula" -- with :math:`\Delta` the drive detuning and :math:`\Omega` the drive strength. *Implementation:* :class:`physicskit.quantum.chapters.spin.RabiProblem` implements exactly this closed-form propagator, and its :meth:`~physicskit.quantum.chapters.spin.RabiProblem.excited_state_population` method reproduces the Rabi formula's population-flopping curve; :func:`~physicskit.quantum.visualizers.bloch_sphere.state_to_bloch_trajectory` and :func:`~physicskit.quantum.visualizers.bloch_sphere.animate_bloch_sphere` trace the resulting state as a vector precessing/nutating on Bloch's own sphere, animated frame by frame with a matplotlib ``FuncAnimation``. See the Rabi-driven Bloch-sphere animation in :doc:`/api/gallery/quantum/entanglement/plot_bloch_sphere_spin_dynamics`. *References:* I. Rabi, Phys. Rev. 51, 652-654 (1937); I. Rabi et al., Phys. Rev. 53, 318 (1938); F. Bloch, Phys. Rev. 70, 460-474 (1946); E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys. Rev. 69, 37-38 (1946); generalization of the Bloch sphere to a single two-state quantum system, R. P. Feynman, F. L. Vernon Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49-52 (1957). .. minigallery:: ../../examples/quantum/entanglement/plot_bloch_sphere_spin_dynamics.py 1959 -- The Aharonov-Bohm Effect -------------------------------- Yakir Aharonov and David Bohm predicted, in 1959, a startling consequence of quantum mechanics with no classical analogue at all: a charged particle can be measurably affected by an electromagnetic potential even in a region where every field it could classically feel -- :math:`\mathbf B` and :math:`\mathbf E` -- vanishes identically. Threading a magnetic flux :math:`\Phi` through the hole of a conducting ring, without ever letting the electron enter the field region, shifts every energy level of the ring, .. math:: E_n(\Phi) = \frac{\hbar^2}{2mR^2}\left(n - \frac{\Phi}{\Phi_0}\right)^2, \qquad \Phi_0 = \frac{2\pi\hbar}{q}, purely through the vector potential's Aharonov-Bohm phase :math:`\Delta\phi = 2\pi\Phi/\Phi_0` picked up by the electron's wavefunction as it circles the ring. Confirmed experimentally by Akira Tonomura's group using electron holography in 1986, the effect showed that the electromagnetic potentials -- not just the fields derived from them -- are physically real, and it seeded the entire field of mesoscopic persistent-current physics. The same phase effect had in fact been derived a decade earlier, in 1949, by Werner Ehrenberg and Raymond Siday, who noted it explicitly in a paper on electron optics that went largely unnoticed at the time; in recognition of that priority the effect is sometimes called the "Ehrenberg-Siday-Aharonov-Bohm effect." *Implementation:* :class:`physicskit.quantum.chapters.entanglement.AharonovBohmRing` reproduces exactly this flux-dependent spectrum in its ``energy`` and ``spectrum`` methods, its ``aharonov_bohm_phase`` method gives :math:`\Delta\phi`, and its ``persistent_current`` method computes the equilibrium ring current the flux dependence implies. *References:* Y. Aharonov and D. Bohm, Phys. Rev. 115, 485-491 (1959); experimental confirmation, A. Tonomura et al., Phys. Rev. Lett. 56, 792-795 (1986); priority, W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. B 62, 8-21 (1949). .. minigallery:: ../../examples/quantum/entanglement/plot_aharonov_bohm_ring.py 1961 -- Jonsson's Electron Double-Slit Experiment ------------------------------------------------- Richard Feynman would later call the electron double-slit experiment "a phenomenon which is impossible ... to explain in any classical way, and which has in it the heart of quantum mechanics." Claus Jonsson had already performed it: using electron-biprism interferometry through up to five parallel micro-slits machined in copper foil, he observed exactly the interference fringes de Broglie's matter waves and Born's probabilistic interpretation predicted, extending Young's 1801 two-slit demonstration for light to individual massive particles for the first time. In 2002, readers of *Physics World* voted the electron double-slit experiment "the most beautiful experiment in physics." .. math:: I(y) = \left\lvert \psi_1(y) + \psi_2(y) \right\rvert^2, the same two-amplitude interference law as classical wave optics, now describing the probability density of finding a single electron -- fired one at a time -- at position :math:`y` on the downstream screen. *Implementation:* :class:`physicskit.quantum.chapters.wave_packets.TwinSlit` builds this two-source interference pattern from coherent Gaussian slit sources propagated to a screen; its ``amplitude`` and ``intensity`` methods reproduce the fringe pattern :math:`\lvert\psi_1+\psi_2\rvert^2` directly. See it in :doc:`/api/gallery/quantum/wave_packets/plot_wave_packet_dynamics`. Rather than that far-field (Fraunhofer) shortcut, :func:`~physicskit.quantum.chapters.wave_packets.double_slit_potential` and :func:`~physicskit.quantum.chapters.wave_packets.propagate_double_slit` build the same two-gap wall as a genuine 2D potential and propagate a Gaussian wavepacket through it with :class:`~physicskit.quantum.core.solvers.SplitOperatorSolver2D`, so the fringe pattern builds up frame by frame from the full time-dependent Schrodinger equation rather than a closed-form formula, animated with :func:`~physicskit.quantum.visualizers.wavefunctions.animate_density_2d`. See it in :doc:`/api/gallery/quantum/measurement/plot_double_slit_propagation`. *References:* C. Jonsson, Z. Phys. 161, 454-474 (1961); English translation, Am. J. Phys. 42, 4-11 (1974). .. minigallery:: ../../examples/quantum/wave_packets/plot_wave_packet_dynamics.py ../../examples/quantum/measurement/plot_double_slit_propagation.py 1964 -- Bell's Theorem and the CHSH Inequality ---------------------------------------------- John Bell showed that *any* theory of the EPR type -- one where measurement outcomes are fixed in advance by local "hidden variables," unknown to quantum mechanics but real nonetheless -- must satisfy an inequality that quantum mechanics itself predicts will be violated. Clauser, Horne, Shimony, and Holt reformulated Bell's inequality in 1969 into the experimentally practical CHSH form, .. math:: S = E(a,b) - E(a,b') + E(a',b) + E(a',b'), \qquad \lvert S\rvert \le 2 \ \text{(local hidden variables)}, which the singlet state violates up to the Tsirelson bound :math:`\lvert S\rvert = 2\sqrt2 \approx 2.828`, established by Boris Tsirelson in 1980 as quantum mechanics' own ceiling. Experiments -- from Aspect's in the early 1980s to the loophole-free tests of 2015 -- have consistently confirmed the quantum prediction, closing the question EPR opened: entanglement is real, and no local hidden-variable theory can reproduce it. *Implementation:* :func:`physicskit.quantum.chapters.entanglement.bell_state` builds the singlet used throughout; :class:`~physicskit.quantum.chapters.entanglement.BellCorrelations` computes the spin :meth:`~physicskit.quantum.chapters.entanglement.BellCorrelations.correlation`, and its :meth:`~physicskit.quantum.chapters.entanglement.BellCorrelations.chsh_S` and :meth:`~physicskit.quantum.chapters.entanglement.BellCorrelations.chsh_optimal` methods reproduce exactly this CHSH violation up to the Tsirelson bound. *References:* J. Bell, Physics 1(3), 195-200 (1964); J. Clauser, M. Horne, A. Shimony, and R. Holt, Phys. Rev. Lett. 23, 880-884 (1969); B. Tsirelson, Lett. Math. Phys. 4, 93-100 (1980); A. Aspect et al., Phys. Rev. Lett. 49, 1804-1807 (1982); loophole-free tests, B. Hensen et al., Nature 526, 682-686 (2015). .. minigallery:: ../../examples/quantum/entanglement/plot_entanglement_and_topology.py 1971 -- Gutzwiller's Trace Formula ---------------------------------- Martin Gutzwiller asked, in a series of papers culminating in 1971, how a quantum energy spectrum could be reconstructed purely from the *classical* mechanics of a system -- including chaotic systems, with no exact quantum numbers at all. His answer, the trace formula, expresses the quantum density of states as a sum over every classical periodic orbit, weighted by that orbit's action, period, and (in higher dimensions) stability: .. math:: g(E) \approx \bar g(E) + \frac{1}{\pi\hbar}\sum_{\text{p.o.}} \sum_{r=1}^{\infty} \frac{T_p}{\sqrt{\lvert\det(M_p^r - I)\rvert}} \cos\!\left(\frac{r S_p(E)}{\hbar} - r\,\sigma_p\frac{\pi}{2}\right), where the outer sum runs over each primitive periodic orbit :math:`p` (action :math:`S_p`, period :math:`T_p`, monodromy matrix :math:`M_p`, Maslov index :math:`\sigma_p`) and the inner sum over its repetitions :math:`r`. For an integrable one-dimensional system this identity is exact rather than approximate, and it founded the field of quantum chaos: the first rigorous bridge between a system's classical dynamics -- regular or chaotic -- and the fine structure of its quantum spectrum. *Implementation:* :func:`physicskit.semiclassical.core.gutzwiller.classical_period` supplies :math:`T(E)=dS/dE` for a 1D bound orbit, and :func:`~physicskit.semiclassical.core.gutzwiller.gutzwiller_density_of_states` sums the resulting exact 1D trace formula, reconstructing the Bohr-Sommerfeld spectrum's delta-function peaks purely from repetitions of a single classical orbit; :func:`~physicskit.semiclassical.core.gutzwiller.gutzwiller_amplitude_from_monodromy` gives the general stability amplitude :math:`1/\sqrt{\lvert 2-\operatorname{tr}M\rvert}` needed beyond the exactly-solvable 1D case. *References:* M. Gutzwiller, J. Math. Phys. 12, 343-358 (1971). .. minigallery:: ../../examples/semiclassical/gutzwiller/plot_gutzwiller_trace_formula.py 1982 -- The No-Cloning Theorem ------------------------------- William Wootters and Wojciech Zurek, and independently Dennis Dieks, proved a negative result as consequential as any of Bell's: no unitary process can take an arbitrary, unknown quantum state and produce two independent copies of it, :math:`\lvert\psi\rangle\lvert e\rangle \mapsto \lvert\psi\rangle\lvert\psi\rangle`, for *every* :math:`\lvert\psi\rangle` simultaneously. The proof is a direct consequence of the linearity that makes quantum mechanics quantum mechanics: a fixed unitary that faithfully clones two particular non-orthogonal states :math:`\lvert\psi\rangle` and :math:`\lvert\phi\rangle` would, by linearity, act on their superposition in a way that is *not* a faithful copy of that superposition -- a contradiction unless the two states were orthogonal (or identical) to begin with. The no-cloning theorem is the reason quantum cryptography can detect eavesdropping (an eavesdropper cannot covertly copy a qubit to inspect it later), and the reason quantum error correction has to encode information redundantly across *entangled* qubits rather than by simply duplicating a single qubit's state. *Connection:* the package has no dedicated cloning-machine example, but the two-qubit unitary machinery the theorem constrains is exactly the machinery built elsewhere in this chronology: :func:`physicskit.quantum.chapters.entanglement.bell_state` and :class:`~physicskit.quantum.chapters.entanglement.IsingEntangler` both construct genuinely entangled two-qubit states from unitary, linear time evolution, the same linearity whose consequences the Wootters-Zurek-Dieks proof exploits; no combination of a fixed unitary and an ancilla built from these same primitives could instead be made to duplicate an arbitrary input qubit. *References:* W. K. Wootters and W. H. Zurek, Nature 299, 802-803 (1982); D. Dieks, Phys. Lett. A 92, 271-272 (1982). .. minigallery:: ../../examples/quantum/entanglement/plot_no_cloning_theorem.py 1984 -- Heller's Quantum Scars and the Herman-Kluk Propagator --------------------------------------------------------------- Eric Heller discovered, in 1984, a phenomenon that contradicted the prevailing expectation for quantum systems whose classical dynamics is chaotic: rather than spreading uniformly over all of accessible phase space (as random-matrix "quantum ergodicity" predicted they should), many eigenstates of chaotic billiards show statistically significant density enhancement concentrated along a handful of short, unstable periodic orbits -- "scars." That same year, Michael Herman and Edward Kluk jointly published, as co-authors of a single paper, a multi-trajectory semiclassical propagator that made computing such wavepacket dynamics in chaotic and anharmonic systems tractable, replacing the single (and often singular, at a caustic) Van Vleck trajectory with a smooth sum over a whole family of frozen Gaussian wavepackets, .. math:: \psi(x,t) \approx \int\!\frac{dq_0\,dp_0}{2\pi\hbar}\, C_t(q_0,p_0)\, e^{iS(t)/\hbar}\, \langle g_{q_0,p_0}\vert\psi_0\rangle\, g_{q_t,p_t}(x), an integral over classical initial conditions that, unlike the single Van Vleck trajectory, never itself passes through a caustic singularity. *Implementation:* :func:`physicskit.semiclassical.systems.scarring.bouncing_ball_energies` and :func:`~physicskit.semiclassical.systems.scarring.bouncing_ball_orbit_points` locate the stadium billiard's shortest unstable periodic orbit family, and :func:`~physicskit.semiclassical.systems.scarring.scar_enhancement` quantifies exactly Heller's density-enhancement signature, :math:`\eta = \langle\lvert\psi\rvert^2\rangle_{\text{tube}}/\langle\lvert\psi\rvert^2\rangle_{\text{billiard}}`; :func:`~physicskit.semiclassical.core.propagators.herman_kluk_propagate_wavepacket`, built on :func:`~physicskit.semiclassical.core.propagators.frozen_gaussian_1d` and :func:`~physicskit.semiclassical.core.propagators.herman_kluk_prefactor`, implements the Herman-Kluk multi-trajectory propagator above. *References:* E. Heller, Phys. Rev. Lett. 53, 1515-1518 (1984); M. Herman and E. Kluk, Chem. Phys. 91, 27-34 (1984) (a single joint paper, not two independent discoveries). .. minigallery:: ../../examples/semiclassical/scarring/plot_quantum_scars.py ../../examples/semiclassical/propagators/plot_herman_kluk_wavepacket.py 1986-1990 -- Wave-Packet Revivals ---------------------------------- A wavepacket built from many eigenstates of an *anharmonic* potential -- where, unlike the harmonic oscillator, the energy levels are not evenly spaced -- spreads and appears to lose all resemblance to its initial shape within a few classical periods. J. Parker and C. R. Stroud predicted theoretically in 1986, and J. A. Yeazell, M. Mallalieu, and C. R. Stroud confirmed experimentally in Rydberg wave packets in 1990, what the quadratic term in the level spacing actually implies: the packet is not gone, only dephased, and it exactly reconstructs itself at a calculable revival time, with smaller-scale "fractional revival" clones (named and analyzed by Ilya Averbukh and Nathan Perelman in 1989) appearing at rational fractions of that time along the way. For the textbook case of a particle in an infinite square well, where :math:`E_n \propto n^2` exactly, .. math:: t_\text{rev} = \frac{4mL^2}{\pi\hbar}, and a mirror-image replica of the initial packet appears already at :math:`t_\text{rev}/2`. *Implementation:* :class:`physicskit.quantum.chapters.wave_packets.QuantumRevival` expands an arbitrary localized initial state (via ``gaussian_initial_state`` and ``eigenbasis_coefficients``) in the infinite-well eigenbasis and evolves it with ``wavefunction``; its ``revival_time`` property gives :math:`t_\text{rev}` above, and ``fidelity_to_initial`` tracks the wavepacket's overlap with :math:`\psi(x,0)` collapsing and then sharply recovering at that time. See it in :doc:`/api/gallery/quantum/wave_packets/plot_wave_packet_dynamics`. *References:* J. Parker and C. Stroud, Phys. Rev. Lett. 56, 716-719 (1986); J. A. Yeazell, M. Mallalieu, and C. R. Stroud Jr., Phys. Rev. Lett. 64, 2007-2010 (1990); fractional revivals, I. Sh. Averbukh and N. F. Perelman, Phys. Lett. A 139, 449-453 (1989). .. minigallery:: ../../examples/quantum/wave_packets/plot_wave_packet_dynamics.py See Also -------- - :doc:`/api/quantum` - :doc:`/api/gallery/quantum/index`