https://doi.org/10.1140/epjp/s13360-025-06699-x
Regular Article
On the motion of classical and quantum complex scalar waves in non-homogeneous media
National Institute of Materials Physics, Atomistilor 405A, 077125, Magurele, Romania
Received:
9
May
2025
Accepted:
27
July
2025
Published online:
13
August
2025
The present paper concerns a formalism aimed to describe the behavior and trajectories of complex scalar waves with space-varying amplitude in media characterized by variable refractive index. The classical wave equation is transformed into an eikonal-type equation, without imposing the geometrical optics limit, satisfied by an effective refractive index. The effective index combines the refractive properties of the medium with self-refraction phenomena generated by the space dependence of the wave amplitude. It also induces wave packet dispersion, leading to relations between the group and phase velocities that are particularly relevant in the context of de Broglie waves. The equations of motion are provided, and it is shown that the self-refraction term bends the trajectory of the wave and changes its velocity. The dynamics is dictated by the curvature of the trajectory expressed in terms of the gradient of the effective refractive index. The uncovered findings point toward a possible generalization of geometrical optics with relevant implications for its relation to quantum mechanics, general relativity and transformation optics.
© The Author(s) 2025
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