Semiclassical propagator to evaluate off-diagonal matrix elements of the evolution operator between quantum states.

Area of Science:

• Quantum mechanics
• Classical mechanics
• Mathematical physics

Background:

• Evaluating off-diagonal matrix elements of the evolution operator is crucial in quantum mechanics.
• Existing methods like the Van Vleck propagator can be computationally intensive.
• Understanding quantum dynamics near unstable periodic orbits is a significant challenge.

Purpose of the Study:

• To present a novel semiclassical expression for evaluating off-diagonal matrix elements of the evolution operator.
• To simplify the calculation of quantum dynamics for systems near unstable periodic orbits.
• To validate the new expression using a relevant physical system.

Main Methods:

• Derivation of a semiclassical expression based on heteroclinic orbits and canonical invariants.
• The expression is valid up to the Heisenberg time.
• Utilizing canonical invariants specific to heteroclinic orbits for term computation.

Main Results:

• A powerful and computationally simpler semiclassical expression is derived.
• The expression involves a sum over heteroclinic orbits, with each term computed by canonical invariants.
• Successful verification of the formula in the hyperbola billiard system.

Conclusions:

• The new semiclassical expression offers a significant advantage in evaluating quantum evolution matrix elements.
• The method provides a more tractable approach for studying quantum systems near unstable periodic orbits.
• The successful application in the hyperbola billiard demonstrates the formula's practical utility and accuracy.