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Schrödinger's Wave Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
In quantum mechanics, the square of the wave function is the probability of finding the particle at a given position, r, defined as: dP(r)=ψ*(r)ψ(r)d3r
The Schrodinger Wave Equation
Published in Vinod Kumar Khanna, Introductory Nanoelectronics, 2020
The equation is presented in two forms: the time-independent Schrodinger equation (TISE) and time-dependent Schrodinger equation (TDSE), although the two forms are not separate and the time-independent form is derivable from the time-dependent form (Schiff 1968, Schleich et al 2013). The TISE describes the allowed energies of a particle while the TDSE describes how the wave function of the particle evolves with respect to time. The wave function is a complex-valued function that describes the wave characteristics associated with a particle.
Overview of Bohmian Mechanics
Published in Xavier Oriols, Jordi Mompart, Applied Bohmian Mechanics, 2019
The typical orthodox prediction of some experimental property of the quantum system is described through the use of a proper operator Ĝ whose eigenvalues give the possible outcomes of the measurement. When we measure a particular eigenvalue, the initial wave function is transformed into an eigenfunction of the operator. This is the so-called von Neumann (or projective) measurement. Thus, the time evolution of the wave function of a quantum system is governed by two (quite) different laws: The first dynamical evolution is given by the Schrödinger equation. This dynamical law is deterministic in the sense that the final wave function of the quantum system is perfectly determined when we know the initial wave function and the Hamiltonian of the quantum system.The second dynamical law is called the collapse of the wave function. The collapse is a process that occurs when the wave function interacts with a measuring apparatus. The initial wave function before the measurement is substituted by one of the eigenstates of the particular operator Ĝ. Contrarily to the dynamical law given by the Schrödinger equation, the collapse is not deterministic, since the final wave function is randomly selected among the operator’s eigenstates.
Eigenvalues and thermal properties of the A1Σ u + state of sodium dimers
Published in Molecular Physics, 2022
Ridha Horchani, Nidhal Sulaiman, Safa Al Shafii
The main reason for solving the Schrödinger equation is to obtain the energies and the corresponding wave function of the potential energy model for the considered system. Various authors have reported elaborately on this subject matter. The time-independent Schrödinger equation is given by where is the reduced mass of a diatomic molecule, and , where is the Planck constant. We express the wave function as , is the spherical harmonic function. Therefore, by substituting Equation (1) into Equation (5) we obtain the differential equation describing the radial motion of the diatomic molecule: where is the rovibrational energy of the level (, ) where and are the vibrational and rotational quantum numbers, respectively.
Conditional stability for a Cauchy problem for the ultrahyperbolic Schrödinger equation
Published in Applicable Analysis, 2022
İsmet Gölgeleyen, Özlem Kaytmaz
The classical Schrödinger equation is the fundamental equation of quantum theory and describes the evolution of the wave function of a charged particle under the influence of electrical potential. It has important applications in atomic and molecular physics [1–3]. In this paper, we consider an ultrahyperbolic Schrödinger equation of the form in a domain , where is a bounded domain with smooth boundary , for L>0 and ,
Optimizing Berth-quay Crane Allocation considering Economic Factors Using Chaotic Quantum SSA
Published in Applied Artificial Intelligence, 2022
Xia Cao, Zhong-Yi Yang, Wei-Chiang Hong, Rui-Zhe Xu, Yu-Tian Wang
In quantum space, the velocity and position of a particle cannot be determined simultaneously. Consequently, the wave function is used to describe the state of the particle, and the probability density function of the particle at a point in space is obtained by solving the Schrödinger equation. Then, the position of the particle is obtained by Monte Carlo simulation according to Eq. (27) as follows,