The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.

arXiv:1903.00583Physical review. EJun 17, 201952 citations

Damped Harmonic Oscillator with Time-Dependent Frictional Coefficient and Time-Dependent Frequency

Eun Ji Jang, Jihun Cha, Young Kyu Lee +1 more

In this paper we extend the so-called dual or mirror image formalism and Caldirola's- Kanai's formalism for damped harmonic oscillator to the case that both frictional coefficient and time-dependent frequency depend on time explicitly. As an solvable example, we consider the case that frictional coefficient $ \ga (t) = \frac{ \ga_0}{1 + q t} , (q > 0 )$ and angular frequency function $ w(t) = \frac{ w_0}{ 1 + q t } $. For this choice, we construct the quantum harmonic Hamiltonian and express it in terms of $su(2)$ algebra generators. Using the exact invariant for the Hamiltonian and its unitary transform, we solve the time-dependent Schro\dinger equation with time-dependent frictional coefficient and time-dependent frequency.In this paper we extend the so-called dual or mirror image formalism and Caldirola's- Kanai's formalism for damped harmonic oscillator to the case that both frictional coefficient and time-dependent frequency depend on time explicitly. As an solvable example, we consider the case that frictional coefficient $ \ga (t) = \frac{ \ga_0}{1 + q t} , (q > 0 )$ and angular frequency function $ w(t) = \frac{ w_0}{ 1 + q t } $. For this choice, we construct the quantum harmonic Hamiltonian and express it in terms of $su(2)$ algebra generators. Using the exact invariant for the Hamiltonian and its unitary transform, we solve the time-dependent Schro\dinger equation with time-dependent frictional coefficient and time-dependent frequency.

Oct 1, 20150 citations

su(2)

algebra generators. Using the exact invariant for the Hamiltonian and its unitary transform, we solve the time-dependent Schro\dinger equation with time-dependent frictional coefficient and time-dependent frequency.