The evolution operator in quantum mechanics and its applications

Daniel H. T. Franco, L. S. Lima

Amit Kumar Choudhary, S Musa, S. M Nengem +1 more

ABSTRACT: The main aim of this article is to present the time-evolution as the unitary and self-adjoint operators on Hilbert spaces and to describe its application in the development of quantum mechanics. The initial value problems associated with the quantum mechanical Schrodinger equation, ( ) = ( ) in the Hilbert space is solved by the use of timeevolution. The importance of time-evolution is also seen as the operation of turning machine in quantum mechanics, which is regarded as the time-evolution of the machine control state. Time-evolution of a quantum mechanical system is observed to be unitary and self-adjoint operators of Hilbert space, since it corresponds to an observable, specifically energy position and momentum. It was observed that time-evolution of a quantum mechanical system is generated by a self-adjoint operator, called Hamiltonian, V K H ˆ ˆ ˆ   , expressed by the Schrodinger equation, above.

Michele Cini

Per‐Olov Löwdin

X. D. Dongfang

Quantum mechanics based on Planck hypothesis and statistical interpretation of wave function has achieved great success in describing the discrete law of micro motion. However, the idea of quantum mechanics has not been successfully used to describe the discrete law of macro motion, and the causality implied in the Planck hypothesis and the application scope of the basic principles of quantum mechanics have not been clarified. In this paper, we first introduce the angular motion law and its application, which seems to be of no special significance as a supplement to the perfect classical mechanics, but plays an irreplaceable role in testing whether the core mathematical procedure of quantum mechanics of operator evolution wave equation satisfies the unitary principle. Then, the operator evolution wave equations corresponding to the angular motion law are discussed, and the necessity of generalized optimization of differential equations are illustrated by the form of ordinary differential equations. Finally, the real wave equation which is superior to the Schr\{o}dinger equation in physical meaning but not necessarily the ultimate answer is briefly introduced. The implicit conclusion is that Hamiltonian can not be the only inevitable choice of constructing wave equation in quantum mechanics, and there is no causal relationship between operator evolution wave equation and quantized energy in bound state system, which indicates that whether the essence of quantum mechanics can be completely revealed is the key to unify macro and micro quantized theory.

Stephen R. Addison, John Edward Gray

In this paper, the concept of a post-selection operator current is introduced that is applicable to remote sensing physics as well as quantum mechanics. The post-selection operator (or cross-correlation for classical applications) current density (PSOCD) is defined in this paper, which also allows an operator current density to be. Both polarization operators and three dimensional operators have explicit Lie algebras, which impose lead to an algebra for the current which we discuss. When there is an evolution equation for signal such as the ambiguity function, the time evolution of a post-selected operator can be characterized. The underlying geometry that is implicit with this evolution is also applicable to other evolutionary/transport equations such as the Fokker-Planck equation, Brownian motion, and other equations that are used to model noise.