The effect of stochastic barrier fluctuation on semiclassical transmission probability and Shannon entropy of a symmetric double well potential

Abstract

Stochastic fluctuation of barrier height and width of a symmetric double well plays a very significant role in the corresponding dynamics by increasing the semiclassical transmission probability and Shannon entropy of the system. The population of the system has been observed to be spread into several under barrier states starting from the W L or W R [W L;R 5 1 ffiffi 2 p W 1 6W 2 ð Þ , where W 1 and W 2 are the wave functions describing the two lowest degenerate states] in presence of the stochastic fluctuation. This distribution over several states is manifested by steady increase in Shannon entropy. However, any arbitrary value of the stochastic fluctuation cannot increase the populations of the upper energy states and consequently no gain in the net value of Shannon entropy results. There is an optimum frequency for which the Shannon entropy passes through a maximum, which is also found out in this work. We have also calculated the semiclassical WKB like transmission probability as a function of time and it is clear that the random fluctuation of barrier causes the transmission probability to increase to a significant amount. As the total energy of the system remains below the potential barrier, this transmission probability is equivalent to tunneling probability. It has been clearly shown that if the fluctuation is made to be periodic (without changing the frequency and magnitude of the fluctuation) it cannot effect any significant change in the overall dynamics. K E Y W O R D S barrier fluctuation, Hellmann-Feynman energy, Shannon entropy, TDFGH, transmission probability 1 | I N T R O D U C T I O N Quantum mechanical tunneling through a symmetric double well is an well studied area both in context of theoretical and experimentalresearch. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] Quantum mechanical tunneling plays crucial roles in low energy reactions in condensed phase where the potential energy surface is represented by a symmetric or asymmetric double well. [21][22][23][24][25][26][27][28] The effect of interaction of the particle trapped in a bistable potential with the environment can be mimicked by fluctuating the barrier height and width. As the interaction with the environment is mainly aperiodic and stochastic, one can model the system potential as a symmetric double well whose barrier parameters are getting continuously perturbed by a nonperiodic Gaussian like perturbation. This acts as a very useful model in describing H-atom tunneling in a number of chemical and biological systems. [6,7] There are many chemical systems which involve reaction at the surface which can be seen as a particle tunneling through a potential energy surface. [21][22][23] The potential energy surface obviously depends on the interactions among surface atoms or molecules with the moving gas molecules which are getting absorbed. So the potential energy surface has an explicit time dependence and this effect can be dressed into the fluctuating barrier model. Also many isomerization and tautomeriz...

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