The Forgotten Dimension

Ashok Kumar Kapoor, Prasanta Kumar Panigrahi, S. Sree Ranjani

Miloslav Znojil

An unusual type of the exact solvability is reported. It is exemplified by the Coulomb plus harmonic oscillator in D dimensions after a complexification of its Hamiltonian which keeps the energies real. Infinitely many bound states are found in closed form which generalizes the even-parity harmonic-oscillator states at zero charge. Apparently, the model is halfway between exact and quasi-exact.

Frances Y. Kuo, Ian H. Sloan, G.W. Wasilkowski +1 more

José Luis Hevia, Guido Peterssen, Christof Ebert +1 more

Have you ever tried to retrieve that forgotten key code for your suitcase? After one year without traveling, many of us found themselves having forgotten the combination and manually trying all permutations. The same situation, but more complex, would be to systematically try identifying that forgotten access code for an online app that you had not used for a while. Cyberattackers are doing exactly this, of course, at high speed and with increasing computing performance. The recommended security key length is thus getting longer by the year. Yet, the stepwise process to achieve this is tedious or consumes lots of computing power. Now imagine that all of these possible states could be tried in a single step. This would be good for your own number lock, but frightening for our security infrastructure.

Josh Izaac, Jingbo B. Wang

S. Antoci, Dierck Ekkehard Liebscher

Quantum mechanics can be formulated in three ways, as Heisenberg, Schrödinger and Feynman did respectively. For the last way, an unknown (i.e. forgotten) forerunner exists, that we have found in a paper by Gregor Wentzel, published before the famous works by Heisenberg and Schrödinger, and contemporary with the fundamental works of L. de Broglie. In that paper, one can find the basic formulae and their interpretation as they were adopted by Feynman twenty years later. We believe that Wentzel's work was forgotten for several reasons: (I) Schrödinger's equation was much simpler to deal with (Wentzel himself contributed to its development in the same way as L.Brillouin and H.Kramers did). (II) The first application was rejected by Heisenberg and Kramers. (III) The approximation used by Wentzel was too na\"ıve and failed. Nevertheless, the foundation laid by Wentzel was sound, as it has been shown by Feynman's work. Therefore, Wentzel has to be considered as one of the founders of quantum mechanics.