A previous calculation (see J.Phys. A:Math. Gen., vol.8, p.533 (1975)) of delay time in electromagnetic barrier penetration is improved using a simple device for elimination of non-causal effects.
The time delay connected with the transmission of an electromagnetic wave across a barrier is investigated. The stationary phase method implies that the effective group velocity of the transmitted wave, inside the barrier, can exceed the velocity of light in vacuo. Consequently the transmission of a signal with a sharp front is studied to determine the upper bound for the signal velocity. The approximation employed does not yield a conclusive answer in general, while for the special case of critical incidence the proof is given that the effective signal velocity does not exceed the velocity of light in vacuo.
The total delay time for a particle tunnelling through a non-rectangular potential barrier is calculated as an explicit function of the potential energy U(x), on a rigorous mathematical basis. The resulting expression is valid for sufficiently smooth potentials. It represents a physically well defined quantity that is measured in the experiment directly. In particular, an explicit expression for the tunnelling time related to the potential is obtained. It is valid if the particle's wavelength, taken far away from the barrier, is small compared with the barrier's effective length l, irrespective of the value of the barrier penetration integral. Exact sufficient conditions for the validity of Connor's parabolic connection formulae are established.
We present a substantially improved version of the method proposed by Sawicki et al (2012 Phys. Rev. A 86 040304(R) , 2014 Rev. Math. Phys. 26 1450004 ) for finding critical points of the linear entropy for an L -qubit system. The new approach is based on the correspondence between the momentum maps for abelian and non-abelian groups, as described in Kirwan (1984 Cohomology of Quotients in Symplectic and Algebraic Geometry ( Mathematical Notes vol 31) (Princeton: Princeton Univ. Press)). The proposed method can be implemented numerically and is much easier than the previous method.