This thesis presents a new method to calculate thermal rate constants for arbitrary one dimensional scattering potentials in the presence of many quasi-bound states. This novel methodology can be proficiently applied to the case of multiple-barrier passages where quasi-bound states are present. After showing that thermal rate constants can be calculated from asymptotic conditions, the Schrödinger equation has been solved as an ordinary differential equation, with the energy as a fixed parameter, by choosing suitable asymptotic boundaries conditions. The method we propose is time-independent and it provides a significant advantage over any available time-dependent method, since time-dependent methods are not adequate for the calculation of rate constants in the presence of long-lived resonance states. The error respect to the exact expression was typically less than 1%, even at extremely low temperatures. Possible multidimensional implementations of the method are under way. Three main applications of our method have been considered: i) Separation of Helium isotopes by resonant tunneling in a double layer Polyphenylene system (2D-PP). Due to the presence of resonant states given by the double barrier potential, the 2D-PP filter was able to select between He3 from He4, even at relatively high temperatures. ii) Extensive studies of the effects of resonant tunneling on the thermal rate constants for double barrier potentials. We numerically observed two important phenomena: the "oscillation" of the thermal rate constant as a function of the distance between the two barriers, and the "Inverse Kinetic Isotope Effect" where the heavier isotope has a larger thermal rate constant with respect to the lighter isotope. iii) Realization of a quantum protocol for the calculation of the thermal rate constant on a quantum computer. In particular, we take advantage of our time-independent method to devise a quantum algorithm with an exponential speed-up with respect to any equivalent classical algorithm.

Deep Nuclear Resonant Tunneling Thermal Rate Constant Calculations / S. Mandra' ; tutor: S. Caracciolo ; co-tutor: M. Ceotto ; coordinatore: M. Bersanelli. - : . UNIVERSITA' DEGLI STUDI DI MILANO, 2013 Mar 19. ((23. ciclo, Anno Accademico 2009/2010. [10.13130/mandra-salvatore_phd2013-03-19].

### Deep Nuclear Resonant Tunneling Thermal Rate Constant Calculations

#### Abstract

This thesis presents a new method to calculate thermal rate constants for arbitrary one dimensional scattering potentials in the presence of many quasi-bound states. This novel methodology can be proficiently applied to the case of multiple-barrier passages where quasi-bound states are present. After showing that thermal rate constants can be calculated from asymptotic conditions, the Schrödinger equation has been solved as an ordinary differential equation, with the energy as a fixed parameter, by choosing suitable asymptotic boundaries conditions. The method we propose is time-independent and it provides a significant advantage over any available time-dependent method, since time-dependent methods are not adequate for the calculation of rate constants in the presence of long-lived resonance states. The error respect to the exact expression was typically less than 1%, even at extremely low temperatures. Possible multidimensional implementations of the method are under way. Three main applications of our method have been considered: i) Separation of Helium isotopes by resonant tunneling in a double layer Polyphenylene system (2D-PP). Due to the presence of resonant states given by the double barrier potential, the 2D-PP filter was able to select between He3 from He4, even at relatively high temperatures. ii) Extensive studies of the effects of resonant tunneling on the thermal rate constants for double barrier potentials. We numerically observed two important phenomena: the "oscillation" of the thermal rate constant as a function of the distance between the two barriers, and the "Inverse Kinetic Isotope Effect" where the heavier isotope has a larger thermal rate constant with respect to the lighter isotope. iii) Realization of a quantum protocol for the calculation of the thermal rate constant on a quantum computer. In particular, we take advantage of our time-independent method to devise a quantum algorithm with an exponential speed-up with respect to any equivalent classical algorithm.
##### Scheda breve Scheda completa Scheda completa (DC)
CARACCIOLO, SERGIO
BERSANELLI, MARCO RINALDO FEDELE
Resonant Tunneling ; Thermal Rate Constant ; Resonant Rate Constant ; Transmission Probability ; Isotope Effects ; Isotope Separation and Enrichment ; Quantum Computation
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
Deep Nuclear Resonant Tunneling Thermal Rate Constant Calculations / S. Mandra' ; tutor: S. Caracciolo ; co-tutor: M. Ceotto ; coordinatore: M. Bersanelli. - : . UNIVERSITA' DEGLI STUDI DI MILANO, 2013 Mar 19. ((23. ciclo, Anno Accademico 2009/2010. [10.13130/mandra-salvatore_phd2013-03-19].
Doctoral Thesis
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/2434/218725`