Breakthroughs in Semiclassical Physics ========================================= .. include:: /_generated/nav/semiclassical.rst .. epigraph:: "The most important application of the quantum theory... is that in which one lets Planck's constant go to zero." -- paraphrasing the spirit of the WKB program Semiclassical mechanics asks what quantum mechanics looks like in the limit :math:`\hbar\to0` without simply discarding quantum effects -- instead, it reorganizes them entirely around classical trajectories, actions, and periodic orbits. This chronology traces that program behind :mod:`physicskit.semiclassical`, from the 1920s WKB approximation to Heller's 1984 discovery that individual chaotic eigenstates still remember the classical orbits a naive random-matrix picture says they should have forgotten, with a pointer to the corresponding implementation in this package at each stop. .. contents:: Timeline :local: :depth: 1 1926 -- The WKB Approximation ------------------------------------ Gregor Wentzel, Hendrik Kramers, and Leon Brillouin (building on earlier work by Harold Jeffreys) independently found the leading term of an asymptotic expansion of the Schrodinger equation in powers of :math:`\hbar`: away from classical turning points, the wavefunction is well approximated by :math:`\psi(x)\sim p(x)^{-1/2}\exp(\pm i\int p\,dx/\hbar)`, an oscillation whose local wavelength and amplitude both track the classical momentum :math:`p(x)=\sqrt{2m(E-V(x))}`. Matching this oscillatory solution through the breakdown region at each turning point (via the Airy function) costs a phase of :math:`\pi/4`, and demanding the wavefunction close consistently around a full oscillation gives the Bohr-Sommerfeld quantization rule that had, until then, been an ad hoc postulate of the old quantum theory rather than a consequence of wave mechanics. *Implementation:* :func:`physicskit.semiclassical.core.wkb.wkb_wavefunction` builds exactly this approximate wavefunction; :func:`~physicskit.semiclassical.core.wkb.bohr_sommerfeld_energies` implements the resulting quantization condition, reproducing the exact harmonic-oscillator spectrum to machine precision. *References:* G. Wentzel, "Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik," Z. Phys. **38**, 518-529 (1926); H. A. Kramers, "Wellenmechanik und halbzahlige Quantisierung," Z. Phys. **39**, 828-840 (1926); L. Brillouin, "La mécanique ondulatoire de Schrödinger: une méthode générale de resolution par approximations successives," C. R. Acad. Sci. **183**, 24-26 (1926); H. Jeffreys, "On Certain Approximate Solutions of Linear Differential Equations of the Second Order," Proc. London Math. Soc. **23**, 428-436 (1925). .. minigallery:: ../../examples/semiclassical/wkb/plot_wkb_bohr_sommerfeld.py 1928 -- Van Vleck's Semiclassical Propagator ------------------------------------------------ John Van Vleck showed that the WKB idea extends from stationary states to the full quantum propagator: the leading :math:`\hbar\to0` approximation to :math:`K(x_f,t;x_i,0)` is built entirely from a single classical trajectory connecting :math:`x_i` to :math:`x_f` in time :math:`t`, weighted by a prefactor set by that trajectory's stability -- how sensitively the endpoint depends on the launch momentum. Cecile Morette's 1951 path-integral derivation later gave the same object a cleaner name, the Van Vleck-Morette determinant, and a systematic route to higher dimensions. *Implementation:* :func:`physicskit.semiclassical.core.propagators.propagate_trajectory_monodromy_action` integrates the classical trajectory together with its monodromy matrix; :func:`~physicskit.semiclassical.core.propagators.van_vleck_propagator_1d` assembles the propagator from it, matching the exact quantum propagator of the free particle and the harmonic oscillator to better than :math:`10^{-5}`. :func:`~physicskit.semiclassical.visualizers.propagators.plot_classical_trajectory_on_wigner` overlays that same classical :math:`(q,p)` trajectory directly on the exact quantum Wigner distribution, showing the classical orbit tracing the ridge the quantum phase-space density concentrates on. *References:* J. H. Van Vleck, "The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics," Proc. Natl. Acad. Sci. **14**, 178-188 (1928); C. Morette, "On the Definition and Approximation of Feynman's Path Integrals," Phys. Rev. **81**, 848-852 (1951). .. minigallery:: ../../examples/semiclassical/propagators/plot_semiclassical_propagators.py 1958 -- Keller's Quantization Condition and Maslov's Topological Index ------------------------------------------------------------------------ Joseph Keller reformulated Bohr-Sommerfeld quantization for classically integrable systems with more than one degree of freedom, showing that the naive phase-space integral needs a correction term set by the number of caustics the trajectory crosses -- turning points in 1D, or focal points more generally -- each contributing a further :math:`-\pi/2` to the phase. This "corrected Bohr-Sommerfeld" or Einstein-Brillouin-Keller (EBK) condition already contains, for the periodic-orbit case, the phase-counting rule that Viktor Maslov would later place on rigorous, coordinate-independent footing: in his 1965 monograph (English translation 1972), Maslov showed this phase count is a topological invariant of the trajectory -- the Maslov index -- rather than an artifact of the coordinates used to compute it, valid far beyond the periodic-orbit setting Keller started from. *Implementation:* :func:`physicskit.semiclassical.core.propagators.count_caustics` counts sign changes of the monodromy matrix's :math:`\partial q_t/\partial p_0` element along a trajectory to extract exactly this index. *References:* J. B. Keller, "Corrected Bohr-Sommerfeld Quantum Conditions for Nonseparable Systems," Ann. Phys. **4**, 180-188 (1958); V. P. Maslov, *Théorie des Perturbations et Méthodes Asymptotiques* (Dunod, Paris, 1972; Russian original, Moscow State University Press, 1965); V. P. Maslov and M. V. Fedoriuk, *Semi-Classical Approximation in Quantum Mechanics* (Reidel, Dordrecht, 1981). .. minigallery:: ../../examples/semiclassical/propagators/plot_maslov_index_caustics.py 1971 -- The Gutzwiller Trace Formula ------------------------------------------ Martin Gutzwiller derived a semiclassical formula for the quantum density of states of a general -- in particular, classically chaotic -- system directly from its classical periodic orbits, with no reference to any quantum wavefunction at all: each isolated periodic orbit contributes an oscillatory term to :math:`g(E)` at a frequency set by its action and an amplitude set by its linear instability. For chaotic systems, whose periodic orbits proliferate exponentially with period, the resulting sum is only conditionally convergent -- famously "the most divergent series you'll ever want to use" -- yet reproduces real quantum spectra with striking accuracy when truncated sensibly. *Implementation:* :func:`physicskit.semiclassical.core.gutzwiller.gutzwiller_density_of_states` implements the exact bound-1D specialization of the trace formula (via Poisson summation of the EBK spectrum), reconstructing the harmonic oscillator's Bohr-Sommerfeld levels from its classical action and period alone; :func:`~physicskit.semiclassical.core.gutzwiller.gutzwiller_amplitude_from_monodromy` gives the general isolated-unstable-orbit stability amplitude the full multi-dimensional formula uses. *References:* M. C. Gutzwiller, "Periodic Orbits and Classical Quantization Conditions," J. Math. Phys. **12**, 343-358 (1971). .. minigallery:: ../../examples/semiclassical/gutzwiller/plot_gutzwiller_trace_formula.py 1976 -- The Berry-Tabor Formula for Integrable Systems ------------------------------------------------------------ Michael Berry and Michael Tabor showed that the Gutzwiller trace formula's assumption of isolated periodic orbits fails for integrable systems -- where, since every energy has a whole torus of orbits rather than one isolated orbit, the trace formula must be replaced by a different (still exact, for linear tori) sum organized around rational winding numbers. The distinction between the Gutzwiller (chaotic, isolated orbits) and Berry-Tabor (integrable, orbit families) regimes remains the basic dichotomy of semiclassical spectral theory. *Connection:* the one-dimensional trace formula in this package is the :math:`f=1` special case where the distinction is moot (a single degree of freedom has no room for chaos), which is exactly why :func:`~physicskit.semiclassical.core.gutzwiller.gutzwiller_density_of_states` can be *exact* rather than merely leading-order in :math:`\hbar`. *References:* M. V. Berry and M. Tabor, "Closed Orbits and the Regular Bound Spectrum," Proc. R. Soc. Lond. A **349**, 101-123 (1976). .. minigallery:: ../../examples/semiclassical/gutzwiller/plot_gutzwiller_trace_formula.py 1984 -- Herman and Kluk's Frozen Gaussians ------------------------------------------------ Michael Herman and Edward Kluk proposed replacing the single classical trajectory of Van Vleck theory with an entire phase-space family: launch one fixed-width ("frozen") Gaussian wavepacket from every point of a phase-space grid, propagate each one along its own classical trajectory, and sum the results with a monodromy-built prefactor and classical action phase. Unlike a single semiclassical trajectory, this multi-trajectory ("initial value representation") sum needs no troublesome root-finding to connect fixed endpoints, generalizes cleanly to many degrees of freedom, and -- for any potential at most quadratic in position -- is exact. *Implementation:* :func:`physicskit.semiclassical.core.propagators.herman_kluk_propagate_wavepacket` sums exactly such a grid of Numba-compiled classical trajectories, built from :func:`~physicskit.semiclassical.core.propagators.frozen_gaussian_1d` and :func:`~physicskit.semiclassical.core.propagators.herman_kluk_prefactor`. *References:* M. F. Herman and E. Kluk, "A Semiclassical Justification for the Use of Non-Spreading Wavepackets in Dynamics Calculations," Chem. Phys. **91**, 27-34 (1984). .. minigallery:: ../../examples/semiclassical/propagators/plot_herman_kluk_wavepacket.py 1984 -- Heller's Discovery of Quantum Scars ------------------------------------------------ Eric Heller, numerically diagonalizing the Bunimovich stadium billiard -- a paradigm chaotic system -- found that a small fraction of its eigenstates were *not* the featureless, ergodically-spread blobs that random-matrix universality had led theorists to expect: instead they showed a clear, robust enhancement of probability density along the path of one particular unstable classical periodic orbit. These "scars" are not a violation of quantum ergodicity (which holds for *almost every* eigenstate as :math:`\hbar\to0`) but a finite-:math:`\hbar` imprint of the same periodic orbits that weight the Gutzwiller trace formula's sum -- a direct, visible link between individual eigenstates and classical chaos. *Implementation:* :func:`physicskit.semiclassical.systems.scarring.bouncing_ball_energies` and :func:`~physicskit.semiclassical.systems.scarring.bouncing_ball_orbit_points` concern the stadium billiard's most famous scarred family; :func:`~physicskit.semiclassical.systems.scarring.scar_enhancement` quantifies the effect directly in position space, and :func:`~physicskit.semiclassical.systems.scarring.husimi_projection_1d` gives the phase-space (Husimi) view Heller himself used to first see it. *References:* E. J. Heller, "Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits," Phys. Rev. Lett. **53**, 1515-1518 (1984). .. minigallery:: ../../examples/semiclassical/scarring/plot_quantum_scars.py