scholarly journals HEISENBERG OPERATORS, INVARIANT DOMAINS AND HEISENBERG EQUATIONS OF MOTION

2007 ◽  
Vol 19 (10) ◽  
pp. 1045-1069 ◽  
Author(s):  
ASAO ARAI
Keyword(s):  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Adjoint Operator ◽  
Dirac Operators ◽  
Heisenberg Equation ◽  
Abstract Operator ◽  
Heisenberg Equations ◽  
Heisenberg Operators ◽  
Heisenberg Equations Of Motion ◽  
Invariant Domains

An abstract operator theory is developed on operators of the form AH(t) := eitHAe-itH, t ∈ ℝ, with H a self-adjoint operator and A a linear operator on a Hilbert space (in the context of quantum mechanics, AH(t) is called the Heisenberg operator of A with respect to H). The following aspects are discussed: (i) integral equations for AH(t) for a general class of A; (ii) a sufficient condition for D(A), the domain of A, to be left invariant by e-itH for all t ∈ ℝ; (iii) a mathematically rigorous formulation of the Heisenberg equation of motion in quantum mechanics and the uniqueness of its solutions; (iv) invariant domains in the case where H is an abstract version of Schrödinger and Dirac operators; (v) applications to Schrödinger operators with matrix-valued potentials and Dirac operators.

The quantum mechanics of non-holonomic systems

1951 ◽  
Vol 205 (1083) ◽  
pp. 583-595 ◽  
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Classical Dynamics ◽  
Holonomic System ◽  
Holonomic Systems ◽  
Classical Analogue ◽  
Heisenberg Equations ◽  
Subsequent Motion ◽  
Heisenberg Equations Of Motion

Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q (0) r , then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q (0) r .

scholarly journals Exact density matrix elements for a driven dissipative system described by a quadratic Hamiltonian

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Sh. Saedi ◽  
F. Kheirandish
Keyword(s):  
Density Matrix ◽  
Equations Of Motion ◽  
Dissipative System ◽  
Characteristic Functions ◽  
Matrix Elements ◽  
Exact Density ◽  
Quadratic Hamiltonian ◽  
Heisenberg Equations ◽  
Reduced Density ◽  
Heisenberg Equations Of Motion

AbstractFor a prototype quadratic Hamiltonian describing a driven, dissipative system, exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations of motion and operator calculus. The special and limiting cases are discussed.

Heisenberg equations of motion in the Caldirola-Montaldi procedure

1982 ◽  
Vol 67 (2) ◽  
pp. 161-172 ◽  
Author(s):  
A. Jannussis ◽  
A. Leodaris ◽  
P. Filippakis ◽  
Th. Filippakis ◽  
V. Zisis
Keyword(s):  
Equations Of Motion ◽  
Heisenberg Equations ◽  
Heisenberg Equations Of Motion

QUANTUM TREATMENT OF A TIME DEPENDENT SINGLE-TRAPPED ION INTERACTING WITH A BIMODAL CAVITY FIELD

2003 ◽  
Vol 17 (31n32) ◽  
pp. 5925-5941 ◽  
Author(s):  
MAHMOUD ABDEL-ATY ◽  
A.-S. F. OBADA ◽  
M. SEBAWE ABDALLA
Keyword(s):  
Cross Correlation ◽  
Analytic Solution ◽  
Coupling Parameter ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Field Interaction ◽  
Trapped Ion ◽  
Heisenberg Equations ◽  
Quantum Treatment ◽  
Heisenberg Equations Of Motion

In the present communication we consider a time dependent ion-field interaction. Here we discuss the interaction between a single trapped ion and two fields taking into account the coupling parameter to be time dependent and allowing for amplitude modulation of the laser field radiating the trapped ion. At exact resonances the analytic solution for the Heisenberg equations of motion is obtained. We examine the effect of the velocity and the acceleration on the Rabi oscillations by studying the second order correlation function. The phenomenon of squeezing for single and two fields cases is considered. The cross correlation between the fields is discussed.

BOSE OPERATOR HAMILTONIAN MODEL FOR ROTATING ELECTRIC DIPOLE AND GENERALIZED EHRENFEST'S THEOREM

2002 ◽  
Vol 17 (01) ◽  
pp. 45-50 ◽  
Author(s):  
HONGYI FAN
Keyword(s):  
Angular Momentum ◽  
Equations Of Motion ◽  
Classical Equation ◽  
Zero Point ◽  
Rotating System ◽  
Bose Operator ◽  
Hamiltonian Model ◽  
Heisenberg Equations ◽  
Ehrenfest Theorem ◽  
Heisenberg Equations Of Motion

We present a Bose operator Hamiltonian model for totally quantum mechanically describing energy and eigenfunctions of a rotating dipole in the presence of electric field. The corresponding Heisenberg equations of motion are derived which leads us to the generalized Ehrenfest theorem for rotating system. Remarkably, a zero-point angular momentum ℏ/2 appears in the generalized Ehrenfest theorem corresponding to the classical equation Iω=L.

Sign in / Sign up

Export Citation Format

Share Document