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Time-Dependent Perturbation Theory
Published in Shabnam Siddiqui, Quantum Mechanics, 2018
In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V(r, t) = V(r). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach (Figure 7.1).
Non-Relativistic Schrödinger Equation
Published in Zbigniew Ficek, Quantum Physics for Beginners, 2017
Thus, when the Hamiltonian of a particle is independent of time, the probability of finding the particle in an arbitrary point is independent of time. Such a state (wave function) is called a stationary state of the particle.
Mixed equilibrium and social joining strategies in Markovian queues with Bernoulli-schedule-controlled vacation and vacation interruption
Published in Quality Technology & Quantitative Management, 2019
In the almost observable M / M / 1 queue with Bernoulli-schedule-controlled vacation and vacations interruption where customers join the system in accordance with a threshold strategy “join if and balk otherwise", the stationary state distribution .
Analytical investigation on the free vibration behavior of rotating FGM truncated conical shells reinforced by orthogonal eccentric stiffeners
Published in Mechanics of Advanced Materials and Structures, 2018
Dao Van Dung, Le Thi Ngoc Anh, Le Kha Hoa
Figures 5 and 6 also show the critical frequency ωcr increase with the increase of Ω. In addition, when the conical shell is in a stationary state (for Ω = 0), a standing wave occurs. However, forward and backward waves will appear, when the conical shell starts to rotate.
Local frame transformation theory for two classes of diatomic molecules
Published in Molecular Physics, 2019
Within this fixed nuclei approximation, the wave function for the electron possesses separable solutions in the prolate spheroidal coordinate system. The coordinates depend on the internuclear separation R and the distances and of the electron from each core. With the two nuclei positioned along the z-axis, the coordinates are given by: Expressed in these coordinates, the stationary state wave function of the electron is where is a normalization constant. The wave functions in ξ and η are solutions of the following differential equations, where and α is related to the third constant of the motion for the two-center Coulomb system. In Equations (6) and (7), the factors and were separated out to remove all first derivatives. If one does not do that, then the third constant of the motion appears explicitly in the differential equations, whose solutions we denote and . Note that we define these solutions to be energy-independent to leading order near the nuclei, in the spirit of Refs. [1] and [4]. Depending on and , the effective potential in Equation (7) can exhibit a potential barrier, whereas the potential in Equation (6) does not include a barrier. If , the potential barrier in the η coordinate has height − 1.