Examples#
Runnable, self-contained scripts demonstrating every module of
physicskit.semiclassical – WKB/Bohr-Sommerfeld quantization,
semiclassical propagators, the Gutzwiller trace formula, and quantum
scars.
See also the narrative tutorials:
Each script in this gallery is self-contained and can be run directly with
python examples/semiclassical/<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#
wkb – classical momentum, turning points, WKB wavefunctions, and Bohr-Sommerfeld (EBK) quantization, checked against the exact harmonic oscillator spectrum and eigenstates.
propagators – the Van Vleck-Morette single-trajectory propagator (with its Maslov-index caustic count) and the Herman-Kluk multi-trajectory frozen-Gaussian propagator, both checked against exact quantum evolution.
gutzwiller – the exact 1D Gutzwiller trace formula, reconstructing a spectrum from a single classical periodic orbit’s action and period, and the general isolated-orbit stability amplitude.
scarring – quantum scars on the stadium billiard’s bouncing-ball orbit family, and a Husimi phase-space view of a scarred eigenstate.
The Gutzwiller trace formula#
The exact one-dimensional Gutzwiller trace formula, reconstructing a spectrum purely from a single classical periodic orbit’s action and period, and the general isolated-orbit stability amplitude used beyond the exactly-solvable 1D case.
Feynman paths and the classical limit#
Feynman’s sum-over-histories path integral, sampled by Monte Carlo for the exactly-solvable free particle and harmonic oscillator, showing the phasor sum concentrate onto the classical (stationary-action) trajectory as \(\hbar\to 0\).
Semiclassical propagators#
The Van Vleck-Morette single-trajectory propagator, the Maslov index’s topological caustic-counting correction, and the Herman-Kluk multi-trajectory frozen-Gaussian propagator, each checked against exact quantum evolution of the harmonic oscillator.
Herman-Kluk frozen-Gaussian wavepacket propagation
The Maslov index: caustics and Keller’s quantization condition
Quantum scars#
The stadium billiard’s bouncing-ball scarred eigenstate family, the scar-enhancement diagnostic, and a Husimi phase-space view of a scarred state’s wall slice.
WKB and Bohr-Sommerfeld quantization#
The classical momentum and turning points, the WKB wavefunction, and the Bohr-Sommerfeld (EBK) quantization rule, checked against the exact harmonic-oscillator spectrum and eigenstates.
