Study On The Tunneling And Energy Splitting Of The Double-well Molecules With Dc Electric Field

Double-well models have been widely used in many physical and chemical problems. Using the physical process or mechanism such as tunneling effect etc., the characteristic of some matter and material have been studied and developed. Such as the potential energy surfaces of NH3,CH4and HF, and its spectrum analysis in external field, the interactions of the particle and the laser field, quantum dots and quantum wire in the double-well, and the Bose-Einstain condensate process using the localization effect in the double-well. The most widely used and primary problem of all is to obtain the eigensolution, so a lot of paper have discussed about it. Gaining the accurate eigenvalues and eigenvectors and the character of them become the common object to all of us. In addition, the study on the quantum effect of particles in the double-well with external field is the other hot topic at present, like tunneling, localization and population. In a word, the quantum behavior of particle in double-well will be changed in external field, and in this paper we developed the study in detail on the tunneling, localization, energy splitting with dc electric field and eigensolutions in all cases. The main conclusions are:1.Eigensolutions:(l)There are approximate doublet states and delocalized symmetric and antisymmetric wavefunctions for tunneling in symmetric double-well models. Because the tunneling dynamics of the shallow double-well is stronger than of the deep, the wavefunctions of it are more continuous. For the deep double-well, with weak tunneling dynamics at the bottom the energy splitting is very small and the wavefunctions are almostly confined in the two wells, but near the top of the barrier the energy splitting is large with relatively strong tunneling. In each pair of the doublets one with the lower energy is symmetric, the other is asymmetric.(2)There are localized wavefunctions in the asymmetric double-well. When the asymmetric term is small, the doublet structure still exists with larger energy splitting obviously than that of symmetric well. The states with lower energy of each pairs are localized in the deeper single well, while the others are localized in the other single well. As the asymmetric term is large, one of the single wells rises evidently, the othercomes down seriously, and the double-well model loses symmetry completely, then the shallow double-well approaches to the single well. For deep double-well, the first several states are all localized in the deeper single well, and when the state energy reaches the bottom of the other well, subsequent levels appear to be localized in alternate wells up to top of the barrier. Here the doublet structure disappears completely and the distribution of the states in the deeper single well is similar to the one in the harmonic potential well. 2.Effect of dc electric field:When a dc electric field applied, the double-well becomes asymmetric. According to the analytical solutions of the Shrodinger equation the dynamic process of the particle localization is obtained. Namely with the intensity of the dc electric field changing from small to relatively large, the process is the dynamic process of the tunneling decreasing and disappearing and the wavefunction from delocalization to localization. The characteristic of the localization process depends upon the intensity of dc electric field nonlinearly and the influence effected by dc electric field to the particles' quantum behavior in the double-well is from weak to strong. Moreover, it is pointed out that the nonlinear influence of the external field on the localization results in the nonlinear dynamic process of the energy splitting in dc electric field. By analysis with the energy splitting it is demonstrated that the energy splitting is affected both by the tunneling and dc electric field in this process, so the nonlinear characteristic appears when dc electric field is small comparatively. When the dc electric field is large, the tunneling disappears, the wavefunction is localized, and the Stark effect will...