The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.

Physical Review AJun 1, 199012 citations

The Heisenberg Equation of Motion

D. Dubbers, H.‐J. Stöckmann

Graduate texts in physicsJan 1, 20133 citations

Integration of the Heisenberg equations for inverse power-law potentials

M. Hron, Mohsen Razavy

The direct method of integration of the operator Heisenberg equations of motion is extended to the solution of quantum tunneling when the central potential is a sum of inverse powers of the radial distance. By obtaining the equation of motion for the Weyl-ordered basis set { ${\mathit{S}}_{\mathit{m},\mathit{n}}$ (t)} formed from r(t) and ${\mathit{p}}_{\mathit{r}}$ (t), one can express the time evolution of any member of the set as an infinite sum involving the operators { ${\mathit{S}}_{\mathit{m},\mathit{n}}$ (0)}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. \textcopyright{} 1996 The American Physical Society.The direct method of integration of the operator Heisenberg equations of motion is extended to the solution of quantum tunneling when the central potential is a sum of inverse powers of the radial distance. By obtaining the equation of motion for the Weyl-ordered basis set {${\mathit{S}}_{\mathit{m},\mathit{n}}$(t)} formed from r(t) and ${\mathit{p}}_{\mathit{r}}$(t), one can express the time evolution of any member of the set as an infinite sum involving the operators {${\mathit{S}}_{\mathit{m},\mathit{n}}$(0)}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. \textcopyright{} 1996 The American Physical Society.

Physical Review ANov 1, 19964 citations

Dynamics in the state space and Heisenberg equations

В. М. Максимов

Theoretical and Mathematical PhysicsMar 1, 19760 citations

{\mathit{S}}_{\mathit{m},\mathit{n}}

(t)} formed from r(t) and

{\mathit{p}}_{\mathit{r}}

(t), one can express the time evolution of any member of the set as an infinite sum involving the operators {

{\mathit{S}}_{\mathit{m},\mathit{n}}

(0)}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. \textcopyright{} 1996 The American Physical Society.