Strong Time Operator in Quantum Mechanics

Asao Arai

Some aspects of strong time operators in an abstract algebraic quantum mechanics (including quantum statistical mechanics) and in quantum field theory (QFT) are described.As for the facts presented in QFT, they have essentially been established in a previous paper (A.Arai, Rev. Math.Phys.17(2005), 10711109). §1. Introduction: Strong Time Operators and Their FundamentalProperties This paper is concerned with time operators in algebraic quantum mechanics (including quantum statistical mechanics) and quantum field theory.The concept of time operators itself is a general one and there are some types on them [1, 2, 5, 9].In the present paper we concentrate our attention on strong time operators as defined below and their realizations in algebraic quantum mechanics and quantum field theory.Let \ ma t h f r a k { H} be a complex Hilbert space.We denote the inner product and the norm of \ ma t h f r a k { H} by \l a ng l e \c dot , \ r a n g l e _ { \ m a t h f r a k { H } } (anti-linear in the first variable) and \ V e r t \ c d o t \ V e r t _ { \ ma t h f r a k { H} } respectively.If there is no danger of confusion, then the subscript \ ma t h f r a k { H} in \l a ng l e \c do t , \ r a n g l e _ { \ m a t h f r a k { H } } and \ operator A on a Hilbert space, \mat hrm{D}(A) (resp. Ran (A)) denotes the domain (resp. range) of A . IfA is closable, then we denote its closure by \ov e r l i ne { A} .Let H be a self-adjoint operator on \ ma t h f r a k { H} and T be a symmetric operator on \ ma t h f r a k { H} .If there exists a bounded self-adjoint operator C\neq 0 on \ ma t h f r a k { H} such that \mathrm{D}(C)=\mathf rak{H}, (1.1) e^{-i tH}\mathrm{D}(T)\subset \mathrm{D}(T) , \foral l t\i n \mathbb{R},

Jan Nesemann

In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.

Dmitri M Gitman, Igor Viktorovich Tyutin, Boris Leonidovich Voronov

Quantization of physical systems requires a correct definition of quantum-mechanical observables, such as the Hamiltonian, momentum, etc., as self-adjoint operators in appropriate Hilbert spaces and t

Slobodan Prvanović, D. Arsenović

Within the framework of self-adjoint operator of time in non-relativistic quantum mechanics some properties of solutions of Schroedinger equation, related to Hilbert space formalism, are investigated for two types of time dependent Hamiltonian.

Jerzy Kijowski

Contrary to claims contained in papers by Grot, Rovelli, and Tate [Phys. Rev. A 54, 4676 1996)] and Delgado and Muga [Phys. Rev. A 56, 3425 (1997)], the ``time operator,'' which I have constructed [Rep. Math. Phys. 6, 361 (1974)] in an axiomatic way, is a self-adjoint operator existing in a usual Hilbert space of (nonrelativistic or relativistic) quantum mechanics.

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.