Shannon entropy of asymmetric rectangular multiple well with unequal width barrier

Abstract

We present the Shannon entropy of the Schrödinger equation with an asymmetric rectangular multiple well, in which the potential well width is equal but the barrier width is in an arithmetic sequence. When the confined potential is small, we notice that the wave function of the particle is distributed inside the entire interval, but its distribution is mainly confined inside the wells as the confined potential increases. For the higher excited state, we find that the magnitude inside the barrier is always bigger than that inside the well. However, in the double-well case the magnitude is the biggest in the leftmost second barrier for Vconf=1000 meV, but it is in the rightmost barrier in the triple well case. For large confined potential, the momentum entropy density (MED) will be concentrated to the center of the total rectangular multiple well with the increasing confined potential. Likewise, we see that the momentum entropy densities of the ground, first and second excited states are overlapped in the triple well case, but they are all overlapped in the four well case. Particularly, in the double well case, the position entropy densities of ground state and 2nd excited state as well as the 1st and 3rd excited states occupy each well. In the triple well case, however, the position entropy densities of ground state and 2nd excited state occupy each one of three wells, but the 1st and 3rd excited states occupy one of four wells. The position entropy density (PED) of each state in four well case is distributed in each well of them as the confined potential is very large. When the number of wells increases, there exist some additional oscillations in the PED. For the higher excited state, the oscillator frequency of the PED increases inside the well but decreases inside the barrier as the confined potential depth increases. For the lower states, as the potential well depth increases the MED extends outward from the original central distribution, but the main distribution is concentrated to the center of the total system for the higher excited state. We also note that both the position entropy Sx and the momentum entropy Sp do not decrease or increase as the parameter Vconf increases, but their sum always satisfies the BBM inequality.