Time-dependent Force Constant Research Articles

Phase space path integral approach to harmonic oscillator with a time-dependent force constant

The quantum statistical mechanical propagator for a harmonic oscillator with a time-dependent force constant, mω2(t), has been investigated in the past and was found to have only a formal solution in terms of the solutions of certain ordinary differential equations. Such path integrals are frequently encountered in semiclassical path integral evaluations and having exact analytical expressions for such path integrals is of great interest. In a previous work, we had obtained the exact propagator for motion in an arbitrary time-dependent harmonic potential in the overdamped limit of friction using phase space path integrals in the context of Lévy flights - a result that can be easily extended to Brownian motion. In this paper, we make a connection between the overdamped Brownian motion and the imaginary time propagator of quantum mechanics and thereby get yet another way to evaluate the latter exactly. We find that explicit analytic solution for the quantum statistical mechanical propagator can be written when the time-dependent force constant has the form ω2(t)=λ2(t)−dλ(t)dt, where λ(t) is any arbitrary function of t and use it to evaluate path integrals which have not been evaluated previously. We also employ this method to arrive at a formal solution of the propagator for both Lévy flights and Brownian subjected to a time-dependent harmonic potential in the underdamped limit of friction.

Work fluctuations in a time-dependent harmonic potential: Rigorous results beyond the overdamped limit

We investigate the stochastic motion of a Brownian particle in the harmonic potential with a time-dependent force constant. It may describe the motion of a colloidal particle in an optical trap where the potential well is formed by a time-dependent field. We use the path integral formalism to solve the Langevin equation and the associated Fokker-Planck (Kramers) equation. Rigorous relations are derived to generate the probability density function for the time-dependent nonequilibrium work production beyond the overdamped limit. We find that the work distribution exhibits an exponential tail with a power-law prefactor, accompanied by an interesting oscillatory feature (multiple pseudo-locking-unlocking transitions) due to the inertial effect. Some exactly solvable cases are discussed in the overdamped limit.

Work distribution in a time-dependent logarithmic–harmonic potential: exact results and asymptotic analysis

We investigate the distribution of work performed on a Brownian particle in a time-dependent asymmetric potential well. The potential has a harmonic component with a time-dependent force constant and a time-independent logarithmic barrier at the origin. For an arbitrary driving protocol, the problem of solving the Fokker–Planck equation for the joint probability density of work and particle position is reduced to the solution of the Riccati differential equation. For a particular choice of the driving protocol, an exact solution of the Riccati equation is presented. An asymptotic analysis of the resulting expression yields the tail behavior of the work distribution for small and large work values. In the limit of a vanishing logarithmic barrier, the work distribution for the breathing parabola model is obtained.

Several Theorems in Time-Dependent Density Functional Theory

The time dependence of the exchange-correlation energy in density functional theory is given in terms of the exchange-correlation potential. The virial theorem for the exchange-correlation potential is shown to hold for time-dependent electronic systems and is illustrated by an exactly solved model: Hooke’s atom with a time-dependent force constant. A relation between the coupling constant and functionals evaluated on scaled densities is derived. [S0031-9007(98)08169-1]

The dynamics of a harmonic oscillator with time-dependent force constant and perturbed by weak quartic anharmonicity

A harmonic oscillator with time-dependent force constant when perturbed by a weak static quartic anharmonicity λ x 4, λ small, is shown to generate new features in the dynamics. For weak coupling, both time-dependent perturbation theoretical analysis and near-exact numerical calculations predict that the excitation probability to an accessible excited state is a parabolic function of λ for a short time when all other parameters of the system are fixed. This is approximately independent of the form of the time-dependence of the harmonic force constant. Bound state populations are shown to oscillate in time when anharmonic coupling is present, due to mutual interference of the relaxing force constant and anharmonicity.

Time-dependent raising and lowering operators for the ageing, forced and damped oscillator in quantum mechanics

Raising and lowering operators are constructed for the harmonic oscillator with a time-dependent force constant and for the damped and linearly forced oscillator. The demand that the total derivative [ a ( x, p, t), H ] + i ħ ∂/∂ t a ( x, p, t) of the time-dependent operator a ( x, p, t) and its Hermitian adjoint a †( x, p, t) be zero makes these the lowering and raising operators for a set of eigenvectors to the dynamic harmonic oscillators. The lowest eigenvector can therefore be constructed solving a first-order differential equation in ∂/∂ x, whereafter a complete set can be generated by the application of a †( x, p, t). The functional time dependence of a ( t) and a †( t) is simply given as the solution to Newton's second law of motion.

The general quantum-mechanical solution to the harmonic oscillator with a time-dependent force constant

The solution to Newton's second law for a harmonic oscillator with a time-dependent force constant allows the solutions to the corresponding time-dependent Schrödinger equation to be written down by analogy.

Wigner phase space method: Exact solution of a harmonic oscillator with time dependent force constant

The Wigner phase space method, suggested by Heller, is applied to the problem of a harmonic oscillator with a time dependent force constant, which may be thought of as a model for collisions involving harmonic oscillators. The method leads to an exact solution of the problem.

String-plucking model for vibrational excitation of molecules

A ``string-plucking'' model has been developed to describe the translational to vibrational energy transfer between an atom and a harmonic oscillator in terms of the time-dependent force constant and equilibrium distance of the molecule. The model is compared with exact classical trajectory calculations of collinear and ``breathing sphere'' collisions involving repulsive exponential or Lennard-Jones interaction potentials. In the adiabatic limit for simple collisions (i.e., in the absence of multiple collisions produced by a light middle atom or a deep potential well), the energy transfer predicted by the model is in good agreement with the exact trajectory results. In the case of the Lennard-Jones potential, both model and exact calculations predict that the effect of decreasing the initial kinetics energy is to increase the fractional energy transfer up to 100%. For breathing sphere collisions with a Lennard-Jones potential, the model correctly predicts a sharp decline in energy transfer at a critical impact parameter which is found in the exact trajectory calculations to equal the rainbow impact parameter. For simple collisions with either potential both model and exact calculations show that the effect of increasing the vibrational energy is equivalent to diminishing the initial radial kinetic energy by an equal amount. The model is also compared with the Landau-Teller, Nikitin, and Attermeyer-Marcus theories, and their regions of validity are explored.

On the Exponential Form of Time-Displacement Operators in Quantum Mechanics

We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrödinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time-displacement operator in every order of approximation. We study the usefulness of the first- and second-order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two-state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time-dependent force constant.