Extracting trajectory equations of classical periodic orbits from the quantum eigenmodes in two-dimensional...

Area of Science:

• Quantum mechanics
• Classical mechanics
• Wave phenomena

Background:

• Periodic orbits in classical billiards are fundamental but complex.
• Quantum stationary coherent states offer a novel perspective on these orbits.

Purpose of the Study:

• To systematically extract classical periodic orbit equations from quantum stationary coherent states.
• To establish the relationship between quantum state properties and classical orbit initial conditions.
• To explore the connection between multi-particle quantum states and multiple periodic orbits.

Main Methods:

• Derivation of trajectory equations from quantum stationary coherent states.
• Analytical derivation of the relationship between phase factors and initial positions.
• Analysis of stationary coherent states with noncoprime parametric numbers.

Main Results:

• Successful extraction of classical periodic orbit equations for triangular and circular billiards.
• Analytical derivation of the phase factor-initial position relationship.
• Demonstration that noncoprime parametric numbers in coherent states correspond to multiple periodic orbits.
• Verification of the link between stationary coherent states and experimental resonant modes.

Conclusions:

• Quantum stationary coherent states provide a direct link to classical periodic orbits.
• The study bridges quantum and classical descriptions in billiard systems.
• The findings support the importance of classical features in experimental wave systems and suggest methods for manipulating mesoscopic wave functions.