In this paper, particle tunneling through a 1-dimensional double square potential barrier is considered under fluctuations of an observer’s frame of reference (OFR). The concept of duration density—the proportion of the duration of a measurement that a particle has a particular energy—is adopted to formulate the average transmission probability of a particle for two types of fluctuations in the OFR. For single-period square-wave fluctuations, the average transmission probability is found to oscillate with a period of π or 2π as a function of the energy of the particle; the period of the oscillations rapidly switches from one value to another as a result of extremely small changes in the wavenumber configuration. For single-period sawtooth-wave fluctuations, the period of the oscillations in the transmission probability gradually increases from π to 2π as the wavenumber configuration changes. It is concluded, therefore that the average transmission probability has the potential to provide information about the nature of the fluctuations in an OFR.
In this paper, particle tunneling through a 1-dimensional double square potential barrier is considered under fluctuations of an observer’s frame of reference (OFR). The concept of duration density—the proportion of the duration of a measurement that a particle has a particular energy—is adopted to formulate the average transmission probability of a particle for two types of fluctuations in the OFR. For single-period square-wave fluctuations, the average transmission probability is found to oscillate with a period of π or 2π as a function of the energy of the particle; the period of the oscillations rapidly switches from one value to another as a result of extremely small changes in the wavenumber configuration. For single-period sawtooth-wave fluctuations, the period of the oscillations in the transmission probability gradually increases from π to 2π as the wavenumber configuration changes. It is concluded, therefore that the average transmission probability has the potential to provide information about the nature of the fluctuations in an OFR.
This study reports tunneling through a one-dimensional (1D) square potential barrier (SPB) under fluctuations in an observer’s frame of reference (OFR). To date, tunneling through an SPB has been studied under the assumption that the OFR remains constant throughout the tunneling measurements; therefore, the change of the tunneling probability when the OFR is assumed to fluctuate remains unanswered. In this paper, a 1D SPB is considered under fluctuations of an OFR. The average transmission probability of a particle through an SBP for two types of OFR fluctuations (periodic-square-wave and periodic-sawtooth-wave fluctuations) is formulated in time representations. Under these types of fluctuations, the average transmission probability gradually increases with a particle’s energy, which is saturated to the transmission probability in the case of the stationary OFR at a much greater energy than the amplitude of the fluctuations. The average transmission probability is much higher at the amplitude of the fluctuations in the case of periodic-square-wave fluctuations. Therefore, the average transmission probability with a particle’s energy has the potential to reveal the distribution of OFR fluctuations.
The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.
The average current through a single-electron transistor (SET) under fluctuations of an observer’s frame of reference (OFR) is reported. To date, the average current through a SET has been studied under the assumption that an OFR remains constant throughout the performance of measurements of the current; thus, it remains an unsolved problem as to what is measured of the current when the OFR is assumed to fluctuate. In this paper, a SET comprising a source, drain, and single channel is considered, where an OFR is assumed to be matched to the electrochemical potential energy of the drain of the SET. The average current through the SET for two types of OFR fluctuation is formulated: periodic-square-wave fluctuation and periodic-sawtooth-wave fluctuation, in time representations. Under these types of fluctuation, the average current exhibits a zero-bias Coulomb peak—the amplitude of which gradually increases with the amplitude of the fluctuation type divided by temperature. The amplitude of the zero-bias Coulomb peak is greater in the case of periodic-square-wave fluctuations. Therefore, the amplitude of the zero-bias Coulomb peak together with a varying of both the energy of the channel and the temperature has the potential to reveal the distribution of fluctuations of an OFR.
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 or [ , where and 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.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.