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Sistemes quàntics en una dimensió: hidrogen unidimensional

Vidal Arias, Alfredo José

Published Jun 22, 2015• 0 citations
OpenAlex

Abstract

Out of all atoms, hydrogen possesses the smallest mass, which results in very strong quantum effects at low temperatures. The hydrogen isotopes have different masses (hydrogen H, deuterium D and tritium T) and follow as significantly different behavior. The role of quantum fluctuations is further increased in low-dimensional geometry. A precise description of the equation of state and correlation functions of one-dimensional hydrogen is of a large fundamental and experimental importance.

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Out of all atoms, hydrogen possesses the smallest mass, which results in very strong quantum effects at low temperatures. The hydrogen isotopes have different masses (hydrogen H, deuterium D and tritium T) and follow as significantly different behavior. The role of quantum fluctuations is further increased in low-dimensional geometry. A precise description of the equation of state and correlation functions of one-dimensional hydrogen is of a large fundamental and experimental importance.
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge -e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the centre of mass coordinate and the relative coordinates. The existence of degenerate energy eigenvalues implies that linear combination of the corresponding eigenfunctions are solutions of the wave equation with the same energy. The isotopes of atomic hydrogen, deuterium and tritium also provide examples of hydrogenic systems. Since the mass of deuterium is md˜2mp and that of tritium is mt˜3mp, the reduced masses of them will be slightly different from that of hydrogen. In addition to deuterium and tritium, there exists other hydrogenic systems in which the role of proton is played by another positive particle.
La ecuación de onda parcial-diferencial temporalmente dependiente o independiente de Schroedinger para el átomo de hidrógeno se puede solucionar en coordenadas espaciales en cuatro sistemas distintos. Sin embargo, las propiedades de dichas soluciones e incluso los números cuánticos en conjunto que caracterizan esas funciones de onda o amplitud son muy propias a cada sistema. Por lo anterior, se discuten las relaciones entre las funciones de onda del átomo de hidrógeno, la mecánica ondulatoria y la estructura molecular. Por otro lado, se argumenta que la mecánica cuántica es en gran medida irrelevante en la educación química.
We review the recent activity in the theoretical description of spin-polarized atomic hydrogen and its isotopes at very low temperatures. Spin-polarized hydrogen is the only system in nature that remains stable in the gas phase even in the zero temperature limit due to its small mass and weak interatomic interaction. Hydrogen and its heavier isotope tritium are bosons, the heavier mass of tritium producing a self-bound (liquid) system at zero temperature. The other isotope, deuterium, is a fermion with nuclear spin one making possible the study of three different quantum systems depending on the population of the three degenerate spin states. From the theoretical point of view, spin-polarized hydrogen is specially appealing because its interatomic potential is very accurately known making possible its precise quantum many-body study. The experimental study of atomic hydrogen has been very difficult due to its high recombination rate, but it finally led to its Bose-Einstein condensate state in 1998. Degeneracy has also been observed in thin films of hydrogen adsorbed on the 4He surface allowing for the possibility of observing the Berezinskii–Kosterlitz–Thouless superfluid transition.
The ground-state properties of one-dimensional electron-spin-polarized hydrogen 1 H, deuterium 2 H, and tritium 3 H are obtained by means of quantum Monte Carlo methods. The equations of state of the three isotopes are calculated for a wide range of linear densities. The pair correlation function and the static structure factor are obtained and interpreted within the framework of the Luttinger liquid theory. We report the density dependence of the Luttinger parameter and use it to identify different physical regimes: Bogoliubov Bose gas, super-Tonks–Girardeau gas, and quasi-crystal regimes for bosons; repulsive, attractive Fermi gas, and quasi-crystal regimes for fermions. We find that the tritium isotope is the one with the richest behavior. Our results show unambiguously the relevant role of the isotope mass in the properties of this quantum system.