From Quantum Probabilities to Quantum Amplitudes

The task of reconstructing the system’s state from the measurements results, known as the Pauli problem, usually requires repetition of two successive steps. Preparation in an initial state to be determined is followed by an accurate measurement of one of the several chosen operators in order to provide the necessary “Pauli data”. We consider a similar yet more general problem of recovering Feynman’s transition (path) amplitudes from the results of at least three consecutive measurements. The three-step histories of a pre- and post-selected quantum system are subjected to a type of interference not available to their two-step counterparts. We show that this interference can be exploited, and if the intermediate measurement is “fuzzy”, the path amplitudes can be successfully recovered. The simplest case of a two-level system is analysed in detail. The “weak measurement” limit and the usefulness of the path amplitudes are also discussed.Financial support of MCIU, through the grant PGC2018-101355-B-100(MCIU/AEI/FEDER, UE) (SMG, MP, DS), of Spanish MINECO, project FIS2016-80681-P (MP), and of the Basque Government Grant No IT986-16 (SMG, MP, DS) is gratefully acknowledged

Speed-up and slow-down of a quantum particle

[EN] We study non-relativistic propagation of Gaussian wave packets in one-dimensional Eckart potential, a barrier, or a well. In the picture used, the transmitted wave packet results from interference between the copies of the freely propagating state with different spatial shifts (delays), x′, induced by the scattering potential. The Uncertainty Principle precludes relating the particle’s final position to the delay experienced in the potential, except in the classical limit. Beyond this limit, even defining an effective range of the delay is shown to be an impracticable task, owing to the oscillatory nature of the corresponding amplitude distribution. Our examples include the classically allowed case, semiclassical tunnelling, delays induced in the presence of a virtual state, and scattering by a low barrier. The properties of the amplitude distribution of the delays, and its pole representation are studied in detail.Financial support through the grants PGC2018-101355-B-100 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”, PID2019-107609GB-I00 by MCIN, and the Basque Government Grant No IT986-16, is acknowledged by MP and DS

Klein paradox for bosons, wave packets and negative tunnelling times

We analyse a little known aspect of the Klein paradox. A Klein-Gordon boson appears to be able to cross a supercritical rectangular barrier without being reflected, while spending there a negative amount of time. The transmission mechanism is demonstrably acausal, yet an attempt to construct the corresponding causal solution of the Klein-Gordon equation fails. We relate the causal solution to a divergent multiple-reflections series, and show that the problem is remedied for a smooth barrier, where pair production at the energy equal to a half of the barrier's height is enhanced yet remains finite.Financial support of MCIU, through the Grant PGC2018-101355-B-100(MCIU/AEI/FEDER,UE) (XGdC, MP, DS), of Spanish MINECO, project FIS2016-80681-P (MP), and of the Basque Government Grant no. IT986-16 (MP, DS) is gratefully acknowledged