We introduce a self-adjoint operator that indicates the direction of time within the framework of standard quantum mechanics. That is, as a function of time its expectation value decreases monotonically for any initial state. This operator can be defined for any system governed by a Hamiltonian with a uniformly finitely degenerate, absolutely continuous and semibounded spectrum. We study some of the operator's properties and illustrate them for a large equivalence class of scattering problems. We also discuss some previous attempts to construct such an operator, and show that the no-go theorems developed in this context are not applicable to our construction.
In nonrelativistic quantum mechanics time enters as a parameter in the Schrödinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable—i.e., a self-adjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admits a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the original quantum mechanical problem to an equivalent irreversible representation. In addition, it is proven that in the irreversible representation there exists a natural time ordering observable splitting the Hilbert space at each t>0 into past and future subspaces.
The equation describing the change of the state of the quantum system with respect to energy is introduced within the framework of the self-adjoint operator of time in nonrelativistic quantum mechanics. In this proposal, the operator of time appears to be the generator of the change of the energy, while the operator of energy that is conjugate to the operator of time generates the time evolution. Two examples, one with discrete time and the other with continuous one, are given and the generalization of Schrödinger equation is proposed.
An alternative quantization rule, in which time becomes a self-adjoint operator and position is a parameter, was proposed by Dias and Parisio [Phys. Rev. A 95, 032133 (2017)]. In this approach, the authors derived a space-time-symmetric (STS) extension of quantum mechanics (QM) where a new quantum state (intrinsic to the particle) $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ is defined at each point in space. The quantum state $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ obeys a space-conditional (SC) Schr\"odinger equation and its projection on $|t\ensuremath{\rangle}, \ensuremath{\langle}t|\ensuremath{\phi}(x)\ensuremath{\rangle}$, represents the probability amplitude of the particle's arrival time at
x. In this work we provide an interpretation of the SC Schr\"odinger equation and the eigenstates of observables in the STS extension. Analogous to the usual QM, we propose that by knowing the initial state $|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, which predicts any measurement on the particle performed by a detector localized at
x0, the SC Schr\"odinger equation provides $|\ensuremath{\phi}(x)\ensuremath{\rangle}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbb{U}}(x,{x}_{0})|\ensuremath{\phi}({x}_{0})\ensuremath{\rangle}$, enabling us to predict measurements when the detector is at $x\ensuremath{\lessgtr}{x}_{0}$. We also verify that for space-dependent potentials, momentum eigenstates in the STS extension $|{P}_{b}(x)\ensuremath{\rangle}$ depend on position just as energy eigenstates in the usual QM depend on time for time-dependent potentials. In this context, whereas a particle in the momentum eigenstate in the standard QM, $|\ensuremath{\psi}(t)\ensuremath{\rangle}={|P\ensuremath{\rangle}|}_{t}$, at time
t, has momentum
P (and indefinite position), the same particle in the state $|\ensuremath{\phi}(x)\ensuremath{\rangle}=|{P}_{b}(x)\ensuremath{\rangle}$ arrives at position
x with momentum
Pb(x) (and indefinite arrival time). By investigating the fact that $|\ensuremath{\psi}(t)\ensuremath{\rangle}$ and $|\ensuremath{\phi}(x)\ensuremath{\rangle}$ describe experimental data of the same observables collected at
t and
x, respectively, we conclude that they provide complementary information about the same particle. Finally, we solve the SC Schr\"odinger equation for an arbitrary space-dependent potential. We apply this solution to a potential barrier and compare it with a generalized Kijowski distribution, showing that they can predict distinct traversal times.
We demonstrate that the time operator that measures the time of arrival of a quantum particle into a chosen state can be defined as a self-adjoint quantum-mechanical operator using periodic boundary conditions and applied to wave functions in energy representation. The time becomes quantized into discrete eigenvalues; and the eigenstates of the time operator, i.e., the stroboscopic wave packets introduced recently [Phys. Rev. Lett. 101, 046402 (2008)], form an orthogonal system of states. The formalism provides simple physical interpretation of the time-measurement process and direct construction of normalized, positive definite probability distribution for the quantized values of the arrival time. The average value of the time is equal to the phase time but in general depends on the choice of zero time eigenstate, whereas the uncertainty of the average is related to the traversal time and is independent of this choice. The general formalism is applied to a particle tunneling through a resonant tunneling barrier in one dimension.
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.