Position Probability

2018 ◽  
pp. 87-98
Author(s):  
Jochen Pade
Keyword(s):  
Position Probability

Derivation of Position-Probability Density for the Transient Nano-Tunnel Problem in SET

2010 ◽  
Author(s):  
Sheam-Chyun Lin ◽  
Hsien-Chang Shih ◽  
Fu-Sheng Chuang ◽  
Ming-Lun Tsai ◽  
Harki Apri Yanto ◽  
...  
Keyword(s):  
Probability Density ◽  
Error Function ◽  
Closed Form Solution ◽  
Form Solution ◽  
Single Electron ◽  
Mathematical Operation ◽  
Similarity Parameter ◽  
Set Up ◽  
Position Probability ◽  
Good Agreement

This theoretical investigation intends to study the nano-tunnel problem of the single electron transistor (SET), which is one of the most important components in the nano-electronics industry. With a combined effort of quantum mechanics and similarity parameter, the partial differential equation of transient position-probability density is attained and can be applied to predict the electron’s position inside the nano tunnel. Also, an appropriate set of the initial and the boundary conditions is set up in accordance to the actual electron behavior for solving this PDE of probability density function. Thereafter, a simple, closed-form solution for the probability density is obtained and expressed in terms of the error function for a new similarity variable η. Note that this analytic similarity solution is easy to perform the calculation and suitable for any further mathematical operation, such as the optimization applications. In addition, it is shown that these predications are reasonable and in good agreement to the physical meanings, which are evaluated from both microscopic and macroscopic viewpoints. In conclusions, this is an innovative approach by using the Schro¨dinger equation directly to solve the nano-tunnel problem. Moreover, with the aids of this analytic position-probability-density solution, it is illustrated that the free single electron in the SET’s tunnel can only appear at some specified regions, which are defined by a dimensionless parameter η within a range of 0 ≤ η ≤ 2. This result can be served as a valuable design reference for setting the practical manufacture requirement.

scholarly journals GENERALIZED CONTINUITY EQUATION AND MODIFIED NORMALIZATION IN PT-SYMMETRIC QUANTUM MECHANICS

2001 ◽  
Vol 16 (31) ◽  
pp. 2047-2057 ◽  
Author(s):  
B. BAGCHI ◽  
C. QUESNE ◽  
M. ZNOJIL
Keyword(s):  
Quantum Mechanics ◽  
Probability Density ◽  
Current Density ◽  
Conservation Law ◽  
Continuity Equation ◽  
Normalization Condition ◽  
Generalized Continuity ◽  
Probability Current ◽  
Symmetric Quantum ◽  
Position Probability

The continuity, equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.

Predicting Equation of Transient Position-Probability Density for Nano-Tunnel Problem in SET

2013 ◽  
Vol 284-287 ◽  
pp. 2570-2574
Author(s):  
Sheam Chyun Lin ◽  
Hsien Chang Shih
Keyword(s):  
Probability Density ◽  
Dimensionless Parameter ◽  
Error Function ◽  
Closed Form Solution ◽  
Form Solution ◽  
Single Electron ◽  
Analytic Investigation ◽  
Similarity Parameter ◽  
Set Up ◽  
Position Probability

This analytic investigation intends to study the nano-tunnel problem of the single electron transistor (SET), which is the most important component in the nano-electronics industry. With a combined effort of quantum mechanics and similarity parameter, the PDE equation of transient position-probability density is attained and can be applied to predict the electron’s position inside the nano tunnel. Also, appropriate initial and the boundary conditions are set up in accordance to the actual electron behavior for solving this PDE of probability density function. Thereafter, a simple, closed-form solution for the probability density is obtained and expressed in terms of the error function for a new similarity variable η. In conclusions, this is an innovative approach by using the Schrödinger equation directly to solve the nano-tunnel problem. Moreover, with the aids of this analytic position-probability-density solution, it is illustrated that the free single electron in the SET’s tunnel can only appear at some specified regions, which are defined by a dimensionless parameter η within a range of 0≤η≤2. This result can be served as a valuable design reference for setting the practical manufacture requirement.

scholarly journals On competitive bidding: Scoring and position probability graphs

2013 ◽  
Vol 31 (3) ◽  
pp. 434-448 ◽  
Author(s):  
Pablo Ballesteros-Pérez ◽  
Ma. Carmen González-Cruz ◽  
Antonio Cañavate-Grimal
Keyword(s):  
Competitive Bidding ◽  
Position Probability

Bi-Probability Fatigue Life Prediction for Bridge Crane Structures

2012 ◽  
Vol 482-484 ◽  
pp. 736-740
Author(s):  
Xiao Mei ◽  
Da Shan Dong ◽  
Yuan Yuan Teng
Keyword(s):  
Fatigue Life ◽  
Life Prediction ◽  
Prediction Method ◽  
Steel Structures ◽  
Fatigue Life Prediction ◽  
Bridge Crane ◽  
Stress Cycle ◽  
Damage Theory ◽  
Rainflow Counting ◽  
Position Probability

Fatigue crack is very dangerous for safely operating of steel structures. To estimate precisely fatigue life of bridge cranes, the randomness of lifted load and trolley’s position should be considered. Therefore, bi-probability fatigue life prediction method, namely load and position probability, is put forward based on the miner linear cumulative damage theory. Stress cycle spectrum is constructed based on real-time monitoring data by rainflow counting method. This method can successfully explain the existence of girder cracks in a typical bridge crane RMG, so it would provide valuable reference for maintenance decision of in-service cranes.

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