Global uniform semiclassical approximation for Clebsch-Gordan coefficients.

Area of Science:

• Quantum mechanics
• Angular momentum theory
• Mathematical physics

Background:

• Clebsch-Gordan coefficients are essential for coupling angular momenta in quantum mechanics.
• Existing semiclassical approximations often rely on Gaussian coherent states, which may limit accuracy.

Purpose of the Study:

• To develop novel semiclassical integral representations for Clebsch-Gordan coefficients.
• To introduce non-Gaussian kernels adapted for angular momentum variables.
• To assess the accuracy of these new methods, particularly for classically forbidden regimes.

Main Methods:

• Formulating semiclassical integral representations analogous to propagator initial value expressions.
• Developing two types (L and R) of non-Gaussian kernels tailored for angular momentum variables.
• Comparing the accuracy of the new methods against uniform Airy approximations.

Main Results:

• The proposed non-Gaussian kernels achieve accuracy comparable to uniform Airy approximations.
• The semiclassical treatments accurately predict results for both classically allowed and strongly forbidden quantum numbers.
• The methods utilize real-valued angle variables and classical transformations.

Conclusions:

• The novel semiclassical integral representations provide an accurate and versatile tool for calculating Clebsch-Gordan coefficients.
• These methods extend the applicability of semiclassical approximations to challenging quantum number regimes.
• The use of non-Gaussian kernels represents a significant advancement in semiclassical treatments of angular momentum coupling.