The dynamics of a particle in an expanding cavity is investigated in the Klein-Gordon framework in a regime in which the single-particle picture remains valid. The cavity expansion represents a time-dependent boundary condition for the relativistic wave function. We show that this expansion induces a non-local effect on the current density throughout the cavity. Our results indicate that a relativistic treatment still contains apparently spurious effects traditionally associated with the unbounded velocities inherent in non-relativistic solutions obtained from the Schrödinger equation. Possible reasons for this behaviour are discussed.
The non-local boundary condition introduced by Bogolyubov et al. is useful for eliminating the difficulty of negative probability concerning the indefinite metric in Hilbert space. Their theory is developed in the S-matrix formalism and does not give a causal description of the physical state. A causal description is given here in the form of an ordinary Schrödinger equation. The Hamiltonian is not hermitian in general, though it gives the unitary S-matrix. Such a characteristic situation is discussed by using a simple model.
We consider exact time-dependent analytic solutions to the Schr\"odinger equation for tunneling in one dimension with cutoff wave initial conditions at
t=0. We obtain that as soon as $t\ensuremath{\ne}0$ the transmitted probability density at any arbitrary distance rises instantaneously with time in a linear manner. Using a simple model we find that the above nonlocal effect of the time-dependent solution is suppressed by consideration of low-energy relativistic effects. Hence at a distance
x0 from the potential the probability density rises after a time
t0=x0/c restoring Einstein causality. This implies that the tunneling time of a particle can never be zero.