Time-independent Hamiltonian Research Articles

The solution of the time-dependent Schrodinger equation by the (t,t') method: application to intense field molecular photodissociation

Using the (t,t') method as introduced by Peskin and Moiseyev (1993) the resonance quasivibrational energies and lifetimes of a time-dependent (not necessarily time-periodic) Hamiltonian are obtained by computational methods originally developed for time-independent Hamiltonians. As an illustrative numerical example the multiphoton resonances of the dissociating H2+ are calculated using three Fourier basis functions. The calculation of resonances at high intensity fields by direct methods (i.e. diagonalization of the Floquet Hamiltonian matrix or a coupled-channel propagative approach) requires, however, the inclusion of 10 or more Floquet channels.

Systematic Treatment of General Time-Dependent Harmonic Oscillator in Classical and Quantum Mechanics

A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding time-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example.

Systematic Treatment of General Time-Dependent Harmonic Oscillator in Classical and Quantum Mechanics

A systematic way of treating a general time-dependent harmonic oscillator in classical and quantum mechanics is given. By a general canonical transformation in classical mechanics, the time-dependent Hamiltonian can be transformed to a time-independent one (the Lewis-Riesenfeld invariant), explicitly separating a total time-derivative term. In quantum mechanics, one can obtain the phase of the wave function straightforwardly from this total-derivative term, together with the corresponding time-independent Hamiltonian. We solve analytically and algebraically the general time-dependent harmonic oscillator driven by a time-dependent inverse cubic force as an example.

Semiclassical quantization of KAM resonances in time-periodic systems

A semiclassical theory for the quasi-energy spectrum of time-periodic systems with accidental classical resonances is presented. The primitive EBK quantum conditions for integrable systems are extended to multiply periodic flux tubes occurring in resonant systems. Replacing classical actions by appropriate differential operators in a classical resonance Hamiltonian yields a uniform quantization of states related to a classical resonance region. The derivation being general for time-periodic systems unfolds the organization of the quasi-energy spectrum, reducing it to the spectrum of a single time-independent Hamiltonian of one degree of freedom with additional rational shifts of h(cross) omega . In a first-order approximation the resonance Hamiltonian is reduced to a pendulum leading to a differential equation of the Mathieu type for the quasi-energies. It is rigorously shown how parameters of the differential equation can be drawn from classical dynamics, using the data of the 'essential' orbits in the resonance zone, i.e. stability coefficients and actions of hyperbolic and elliptic orbits as well as actions of homoclinic orbits. The quasi-energy spectrum of a forced quartic oscillator is studied numerically and evaluated. Semiclassical quasi-energies related to a resonance of period three are computed and compared with exact quantum mechanical eigenvalues.

Geometric phases and Mielnik's evolution loops

The cyclic evolutions and associated geometric phases induced by time-independent Hamiltonians are studied for the case when the evolution operator becomes the identity (those processes are called {\it evolution loops}). We make a detailed treatment of systems having equally-spaced energy levels. Special emphasis is made on the potentials which have the same spectrum as the harmonic oscillator potential (the generalized oscillator potentials) and on their recently found coherent states.

Multiphoton detachment of H-. II. Intensity-dependent photodetachment rates and threshold behavior-complex-scaling generalized pseudospectral method.

We extend our previous perturbative study of the multiphoton detachment of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ [Phys. Rev. A 48, 4654 (1993)] to stronger fields by considering the intensity-dependent photodetachment rates and threshold behavior. An accurate one-electron model potential, which reproduces exactly the known ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ binding energy and the low-energy e-H(1s) elastic-scattering phase shifts, is employed. A computational technique, the complex-scaling generalized pseudospectral method, is developed for accurate and efficient treatment of the time-independent non-Hermitian Floquet Hamiltonian H${\mathrm{^}}_{\mathit{F}}$. The method is simple to implement, does not require the computation of potential matrix elements, and is computationally more efficient than the traditional basis-set-expansion--variational method. We present detailed nonperturbative results of the intensity- and frequency-dependent complex quasienergies (${\mathit{E}}_{\mathit{R}}$,-\ensuremath{\Gamma}/2), the complex eigenvalues of H${\mathrm{^}}_{\mathit{F}}$, providing directly the ac Stark shifts and multiphoton detachment rates of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$. The laser intensity considered ranges from 1 to 40 GW/${\mathrm{cm}}^{2}$ and the laser frequency covers 0.20--0.42 eV (in the c.m. frame). Finally we perform a simulation of intensity-averaged multiphoton detachment rates by considering the experimental conditions of the laser and ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ beams. The results (without any free parameters) are in good agreement with experimental data, both in absolute magnitude and in the threshold behavior.

The solution of the time dependent Schrödinger equation by the (t,t′) method: The use of global polynomial propagators for time dependent Hamiltonians

Using the (t,t′) method as introduced in Ref. [J. Chem. Phys. 99, 4590 (1993)] computational techniques which originally were developed for time independent Hamiltonians can be used for propagating an initial state for explicitly time dependent Hamiltonians. The present paper presents a time dependent integrator of the Schrödinger equation based on a Chebychev expansion, of the operator Û(x,t′,t0→t), and the Fourier pseudospectral method for calculating spatial derivatives [(∂2/∂x2),(∂/∂t′)]. Illustrative numerical examples for harmonic and Morse oscillators interacting with CW and short pulsed laser fields are given.

Systematic method for deriving effective Hamiltonians.

A systematic procedure for deriving effective Hamiltonians to any order is presented, which is applicable to any time-independent Hamiltonian. The method is based on a continued-fraction approach and avoids the singularities which may occur with perturbation theory. It is illustrated by deriving the effective Hamiltonian for the one-photon, dressed-state laser to second order.

Remark on the propagator of the radial oscillator.

Using a hybrid approach, based on the recursion relations for shape-invariant potentials developed by Das and Huang [Phys. Rev. D 41, 3241 (1990)] and a time-dependent transformation of variables, we derive the propagator for a radial oscillator. Although this is not a new result, we explicitly show that time-dependent transformations are very beneficial even within the context of time-independent Hamiltonians in quantum mechanics.

Exact solution of the generalized time-dependent Jaynes-Cummings Hamiltonian

A time-dependent generalization of the Jaynes-Cummings Hamiltonian is studied using the maximum entropy formalism. The approach, related to a semi-Lie algebra, allows one to find three differentsets of physically relevant operators which describes the dynamics of the system for any temporal dependence. It is shown how the initial conditions of the operators are determined via the maximum entropy principle density operator, where the inclusion of the temperature turns the description of the problem into a thermodynamical one. The generalized time-independent Jaynes-Cummings Hamiltonian is exactly solved as a particular example.

Quantum system driven by periodic external field

A quantum Floquet Theory for a periodically driven system is studied. For this purpose the periodic external fields are changed to have an increasing amplitude with exp(ηt), where η is an infinitestimal positive number to be taken to be zero at the end. By using the expansions in terms of inverse powers of the driving frequency, periodic factors of the time evolution operator are factorized successively. Each step corresponds to a periodically driven system with different strength of external field. This approach produces a time-independent effective Hamiltonian. The effectiveness of the method is examined by applying it to simple models; 1) a forced harmonic oscillator, 2) a particle in the double-well potential, and 3) a hydrogen atom in an electric potential.

How fast can a quantum state change with time?

Lower and upper bounds are derived for the decay and transitions of quantum states, evolving under a time-dependent Hamiltonian, in terms of the energy uncertainty of the initial and final state. The bounds are simultaneously a rigorous version of Fermi's golden rule and of the time-energy uncertainty relation. They are sharp, refer to short times, and are compared with recent long-time results for time-independent Hamiltonians. Illustrations for tunneling systems, laser-driven processes, and neutron interferometry in time-dependent magnetic fields are given.

The pressure dependence of tunnelling of coupled pairs of methyl groups in lithium acetate dihydrate

The pressure dependence of tunnelling of coupled pairs of methyl groups in lithium acetate dihydrate has been investigate using the neutron scattering spectrometer IRSIS. Methyl tunnelling spectra recorded at a temperature of 5 K and pressures up to 9.5 kbar, reveal structure which has been attributed to coupling. The data, which extend an earlier investigation by Heidemann et al. (1987) have been fitted to a time-independent Hamiltonian, and the fit parameters include the potential barriers of the individual methyl groups and the coupling potential between them. The magnitudes of both parameters increase with increasing pressure and their pressure dependence has been analyzed to provide information on the inter-molecular forces.

Quasienergies, Stark Hamiltonians, and growth of energy for driven quantum rings.

We study time-dependent Schrodinger operators in Aharonov-Bohm geometries where the flux threading the hole increases linearly with time. We show that the quasienergy operator has, in these cases, the same spectrum as the time-independent Stark Hamiltonian on the universal covering space. Combining known results on Stark Hamiltonians with a theorem of Bellissard, we prove that the energy of a particle on a finire ring, with smooth background potential, increases without bound as t→∞

Normalization corrections to perturbation theory for the time-independent nonlinear Schrödinger equation.

In higher orders of the Rayleigh-Schr\"odinger-type perturbation theory for nonlinear time-independent Hamiltonians care should be taken about the normalization in the wave-function-dependent perturbation operator. Low-order results, which have been presented earlier [P. R. Surj\'an and J. \'Angy\'an, Phys. Rev. A 28, 45 (1983)], remain valid; nevertheless, small corrections appear for higher orders (n\ensuremath{\ge}3). Explicit expressions are given for the energy correction in the generalized reaction-field theory in terms of nonlocal polarizabilities and reaction-field response functions.

Berry phases and Hamiltonian time dependence

Shows that (non-adiabatic) Berry phases in time-independent systems arise for fundamentally different reasons than those for time-dependent systems. For time-independent Hamiltonians the Berry phases are seen to only depend on the chosen cyclic initial state, the Hamiltonian merely providing the appropriate period. The author also discusses two time-dependent examples where the time dependence arises from different sources, namely a two-level atom in an intense laser and a harmonic oscillator with a periodic forcing term.

Time-suspension multiple-pulse sequences: applications to solid-state imaging

In this Communication, we introduce a 48-pulse homonuclear dipolar decoupling cycle which also refocuses chemical shifts and other resonance offsets. When applied by itself to a system of spins then it appears that the spin system does not evolve in the absence of irreversible effects. By carefully combining this multiple-pulse sequence with a time-dependent magnetic field gradient, the gradient-induced evolution can be preserved without destroying the line-narrowing benefits of the cycle. With this method we hope to obtain liquid-like images of solid samples. One approach to solid-state imaging, then, is to combine a multiple-pulse cycle which averages time-independent linear and bilinear Z, Hamiltonians with a temporally and spatially dependent linear Z, Hamiltonian in such a manner that this spatially dependent Hamiltonian does not interfere (I ) with the multiple-pulse averaging of the time-independent Hamiltonians. This approach allows high-resolution solid-state images to be obtained with modest magnetic field gradient strengths thereby avoiding the sensitivity cost associated with large detection bandwidths (2,3). In addition, the resulting images have nearly uniform spatial resolution unlike solid-state images that are acquired with multiple-pulse sequences in a time-independent magnetic field gradient. These “time-suspension” cycles average both linear and bilinear Z, Hamiltonians by toggling Z, and Z,Z: operators through a variety of states the temporal average of which is zero. We use time suspension here to connote setting to zero the time evolution of a propagator rather than the equivalent picture in which the spin-dependent part of the Hamiltonian is set to zero: this phrase is chosen in analogy to “time-reversal” sequences which change the sign of the spin Hamiltonian. Such discussions are only useful when one remembers that irreversible effects are excluded, and spin-lattice relaxation and molecular motions place a limit on the interval over which manipulations of the Hamiltonian may be equated to manipulations of time. Naturally, timesuspension only refers to the sampling point for the cycle, and spin evolution during the cycle is much more complicated. Time-suspension sequences are well known in solid-state imaging: an eight-pulse version was the basis of an early imaging scheme suggested by Mansfield and Grannel (4); Weitekamp et al. have used this approach for radio frequency gradient solid-state imaging (5, 6), and we have employed this approach for solid-state imaging with

Problem of the choice of time in cosmology

A class of gauge conditions is obtained from an isotropic cosmological model with a conformally invariant scalar field, leading to a Hamiltonian dynamics with a time independent Hamiltonian which coincides with that of a harmonic oscillator. The quantum dynamics of the model is considered, and the inevitability of a singularity in its quantum version is shown.

Quadratic Hamiltonians in phase-space quantum mechanics

The dynamical evolution is described within the phase-space formalism by means of the Moyal propagator, which is the symbol of the evolution operator. Quadratic Hamiltonians on the phase space are distinguished in that their Moyal bracket with any function equals their Poisson bracket. It is shown that, for general time-independent quadratic Hamiltonians, the Moyal propagators transform covariantly under linear canonical transformations; they are then derived and classified in a fully explicit manner using the theory of Hamiltonian normal forms. The authors present several tables of propagators. It is proved that these propagators belong to the Moyal algebra of distributions, and that the spectrum of the Hamiltonian may be obtained directly as the support of the Fourier transform of the Moyal propagator with respect to time. From that, the quantum-mechanical problem for these Hamiltonians is, in principle, completely solved. The appropriate path-integral formalism for phase-space quantum mechanics, leading back to the same results, is outlined in an appendix.

A static ensemble approximation for stochastically modulated quantum systems

A short-time approximation for the evolution of quantum systems governed by a Hamiltonian with stochastic time dependence is derived. The evolution operator of the system is replaced by an ensemble of evolution operators with time-independent Hamiltonians, weighted by a distribution function related to the ‘‘line shape function’’ of a randomly modulated harmonic oscillator. The approximation conserves the trace of the density matrix and converges to the exact solution in the case of very slow and in the case of very rapid stochastic modulation. The approximation is applied to hyperfine-induced singlet–triplet transitions of a biradical-like system with fluctuating exchange interaction.