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Physical Review A

Theory of quantum friction and the deep inelastic process

Somnath Pal

Published Apr 1, 1989• 7 citations
Publisher

Abstract

The damped harmonic oscillator is used to represent the damping processes in nature. Therefore it is necessary to find the Hamiltonian for the damped oscillator. The Hamiltonian is developed for the harmonic oscillator with frictional force proportional to velocity. The solution of the Schr\"odinger equation gives nonstationary states. The quantum friction coefficient is derived to be zero for massless particles. It is thought that velocity-dependent forces exist near the nucleus and therefore, the scattering amplitude for the deep inelastic process is derived.

DOI
10.1103/physreva.39.3825

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The damped harmonic oscillator is used to represent the damping processes in nature. Therefore it is necessary to find the Hamiltonian for the damped oscillator. The Hamiltonian is developed for the harmonic oscillator with frictional force proportional to velocity. The solution of the Schr"odinger equation gives nonstationary states. The quantum friction coefficient is derived to be zero for massless particles. It is thought that velocity-dependent forces exist near the nucleus and therefore, the scattering amplitude for the deep inelastic process is derived.
The Hamiltonian quantum theory of the damped harmonic oscillator is investigated by using an explicitly time-dependent Hamiltonian including a commutative stochastic force. The propagator is computed and shown to describe an inhomogeneous quantum dynamical semigroup evolution on the Banach cone of density matrices. The moments tend to the thermal equilibrium values independently of the initial conditions, when time tends to infinity and some rescaling procedure is performed. With respect to the rescaling neither the Heisenberg nor the Schrodinger picture exists. Furthermore the energy dissipates below the zero-point energy in violation of the Heisenberg uncertainty relation.
An explicitly time independent Lagrangian functional of a three-dimensional damped harmonic oscillator has been proposed. I derive results for the motion of the three-dimensional damped harmonic oscillator with a pure imaginary three dimensional vector and oscillator’s position-dependent friction coefficient. The Hamiltonians corresponding to the Lagrangian is also explicitly time independent. The choice of functional form of the friction coefficient on the oscillator position determines and plays a vital role in the form of the equation of motion classically and quantum mechanically. One choice of the form of the friction coefficient I made lead to breaking the symmetry of the isotropy of oscillations in the three dimensional space.
Considering the Lagrangian proposed by Havas, that describes the classical damped motion of a particle, new momentum and position are defined in order to write a Hamiltonian that is subsequently quantized and expressed in terms of non-Hermitian operators. Using the c-number formalism proposed by Lax and Yuen, we associate to the quantum Liouville equation a Fokker–Planck one in terms of c-numbers. From the properties of this equation we obtain the mean values of the position, momentum, and energy of a brownian particle and we also verify the uncertainty principle. We observe that when the system is considered under the Markov hypothesis, the stochastic force is intimately related to the uncertainty principle and to the zero point energy.
A quantum-mechanical model of a damped harmonic oscillator (both with time-independent and time-dependent parameters) is studied in the framework of the linear Schr\"odinger equation with a Hermitian nonstationary Hamiltonian. Integrals of the motion of this equation and their eigenstates, including coherent states, are constructed. The influence of an external harmonic force to the time evolution of various average values calculated over coherent states is considered, including the resonance case. The specific symmetry of the Hamiltonian leading to the new concept of loss-energy states is discussed.
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)su(2)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.