Abstract Although special relativity and quantum mechanics revolutionised physics in the early 20th century, the consequences of combining these two theories are still being explored a hundred years later, usually using the formidable theoretical machinery of quantum field theory. However, a formalism accessible to undergraduates has been recently developed which shows how the centre of mass and internal dynamics of classical and quantum systems is relativistically coupled with interesting consequences. Here we explore some of the implications of this coupling, first classically, where we find that the dynamics of the system is time dilated when moving relative to another inertial frame. We then apply the dynamics to a quantum 2-level atom bound in a one-dimensional infinite potential well, and show that the coupling produces collapses and revivals in quantum interference. This example provides an illustration of how the combination of special relativity and quantum mechanics can be studied in situations familiar to most undergraduates.
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit
ℏ→0. The particles' motions are completely determined by the initial conditions. In this theory, the wave function
ψ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation,
ρ=∣ψ∣2. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.
General structural features of dynamical theories can be exhibited in relations between classical and quantum mechanics; the essential structure is a Lie algebra of basic dynamical-variable fanctions providing transformationgroup infinitesimal generators. It is shown that the particular representation chosen is not important for the dynamical structure analysis by considering possibilities of transcribing classical and quantum mechanics each into the natural representation of the other. This leads to an outlining of the formal theory of generalized dynamics by constructing a class of Lie algebras including those of classical and quantum mechanics as special cases. (L.N.N.)