Time-dependent Quantum Mechanics Research Articles

Cellular dynamics: A new semiclassical approach to time-dependent quantum mechanics

A new semiclassical approach that constructs the full semiclassical Green’s function propagation of any initial wave function directly from an ensemble of real trajectories, without root searching, is presented. Each trajectory controls a cell of initial conditions in phase space, but the cell area is not constrained by Planck’s constant. The method is shown to be accurate for rather long times in anharmonic oscillators, indicating the semiclassical time-dependent Green’s function is clearly worthy of more study. The evolution of wave functions in anharmonic potentials is examined and a spectrum from the semiclassical correlation function is calculated, comparing with exact fast Fourier transform results.

Application of a coupled-surface time-dependent Hartree grid method to excited state optical spectroscopy

The dynamics of multidimensional wave packet motion on nonadiabatically coupled electronic potential surfaces is explored by numerically exact time-dependent quantum mechanics and by the time-dependent Hartree grid (TDHG) approximation. Excellent agreement is found between the TDHG and the exact evolution of the wave packet; in particular, the approximation yields accurate total cross sections for electronic absorption, excitation profiles for resonance Raman scattering, and partial cross sections for photofragmentation in a two degree of freedom model of direct dissociation (which is qualitatively related to experimentally observed processes in methyl halides and ICN).

Multichannel quantum wave packet propagation in the interaction picture: application to gas-surface scattering

Propagation of quantum wave packet in the interaction picture substantially improves the efficiency of time-dependent quantum mechanical (TDQM) methods for solving various scattering or half scattering problems. This paper generalizes the idea in a straightforward manner to treat coupled channel quantum wave packet propagation when basis set expansions of the total wavefunction in certain degrees of freedom are desired.

Some useful discrete variable representations for problems in time-dependent and time-independent quantum mechanics

We develop a simple, yet powerful approach to constructing discrete variable representations for the solution of quantum mechanical problems by focusing on properties of the underlying basis set. Two examples, one involving fixed-node boundary conditions and the other periodic boundary conditions, are explicitly derived and applied to a problem in spherical polar coordinates with azimuthal symmetry. These two DVRs have equally spaced points and equal weights.

Time dependent quantum mechanics using picosecond time steps: Application to predissociation of HeI2

Time-dependent quantum mechanics has proved to be a useful conceptual and calculational tool for a wide range of femtosecond time scale processes. In this paper we develop a method of extending time-dependent methodology to processes taking many picoseconds, through use of the interaction representation of quantum mechanics. Two implementations are presented, one in the energy frame and one in the coordinate frame; both of these implementations scale as N2 with the number of basis functions, as opposed to N3 for a matrix diagonalization. The interaction representation approach allows the use of time steps much greater than those that can be used in the Schrödinger representation, and is expected to be useful for the broad class of problems where the Hamiltonian can be partitioned into H0+V, where V≪H0. Converged quantum mechanical results are presented for the predissociation of T-shaped HeI2 from highly excited vibrational levels, for times up to 200 ps.

Determinantal solution of the density matrix in quantum transport theory.

A new method previously elaborated on in time-dependent quantum mechanics is recalled and used to establish, on the basis of the unperturbed Hamiltonian, explicit and tractable expressions of the long-term density matrix, well suited to transport problems. A constant collision potential and a constant external field, switched on at t=0, are assumed. As an illustration, the general results are applied to the quantum theory of electrical conductivity in the case of quasifree electrons.

I. quantization as group extension

Quantization is described as group extension of the linear space F of classical objects considered as an Abelian group by the (Abelian) group of complex numbers. The factor of the extension is studied in some detail and its general form is derived under the assumption of analyticity. A certain class of representations of the extended group is described in terms of functions over F ∗ (the dual to F ) with multiplication determined by a factor. The antisymmetric part of this factor determines the commutation relations. The method is extended to the case when F consists of generators of a Grassmann algebra. The procedure is illustrated on examples in which F consists of functions over an index set J . Four cases are treated: one in which the index set J is discrete and finite (time independent quantum mechanics), another in which J = R × {1, 2,…, n} (time dependent quantum mechanics), a third one with J = M H (quantum field theory of fields over Minkowski space) and finally the fourth case with J = M H × S 2 (quantum field theory of fields over Minkowski space extended in a relativistic way by a two-dimensional space of internal degrees of freedom). Further restrictions upon the factors follow if one requires invariance with respect to the group of motions of the considered space.

Generalized Gaussian wave packet dynamics, Schrödinger equation, and stationary phase approximation

We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.

Exact time-dependent quantum mechanical dissociation dynamics of I2He: Comparison of exact time-dependent quantum calculation with the quantum time-dependent self-consistent field (TDSCF) approximation

The vibrational predissociation dynamics of a collinear model of the I2(v)He cluster is studied by numerically exact time-dependent quantum mechanics, and by the time-dependent self-consistent field (TDSCF) approximation. The time evolution for the initial excitation levels v=5, 11, 22 is explored. Excellent agreement is found between the TDSCF and the exact evolution of the wave packet; in particular the approximation reproduces well the dephasing events in the dynamics, and the measurable predissociation lifetimes. The results are very encouraging as to the applicability of quantum TDSCF as a quantitative tool in the study of van der Waals predissociation dynamics.

A Gaussian wave packet method for studying time dependent quantum mechanics in a curve crossing system: Low energy motion, tunneling, and thermal dissipation

We explore numerically the behavior of a method of describing the time dependent quantum mechanics of a curve crossing system. The two nuclear wave functions corresponding to the two electronic states are each described by a Gaussian wave packet. The packet describing the incident state mimics the initial wave function, and the other packet is created by the time dependent Schrödinger equation. They are both propagated by using a variational method. The packets interact and we do not assume that they have a small width. Exploratory calculations are made for curve crossing dynamics at low kinetic energy above the barrier of the lowest adiabatic state, for tunneling, for multiple crossings, and for a curve crossing system which is strongly coupled to a harmonic bath whose motion is described by a mean trajectory classical Langevin method.

Time-dependent invariants and quantum mechanics in two dimensions

A class of time-dependent problems in two space dimensions possessing time-dependent invariants, bilinear in momenta, is considered. Explicit expression of the potentials and the corresponding invariants are derived. Quantum mechanics is introduced in these time-dependent problems directly through a Feynman propagator defined as a path integral involving the classical action. The propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators. Equivalence of the present theory to that of Lewis and Riesenfeld [H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)] is discussed.

An alternative to the magnus expansion in time-dependent perturbation theory

An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.

Determinantal solution of density matrix equations in time-dependent quantum mechanics. II. Harmonic perturbation

The determinantal formalism previously built up in the evolution operator problem is extended to the density matrix equation in the case, here, of a Heaviside step external perturbation, and in the presence of internal collision potential. Reduced determinantal forms, matched to those first derived for the evolution operator, are obtained and compared with both the iterative expansion and the initial Fredholm-Laplace solution. They not only exhibit the multiple transition structure of the physical response, along with the associated transition width upon expansion, but also the trace conservation requirement is shown to be satisfied at any order and all times. An illustration is given in a simplified physical system.

Transition from Adiabatic to Sudden Excitation of Core Electrons

A model based on time-dependent quantum mechanics is presented for calculating the crossover from the adiabatic to the sudden regime in core ionization. With reasonable parameters the predictions of this model are in agreement with newly reported experimental results on shakeup as well as with older data on shakeoff.

Determinantal solution of density matrix equations in time-dependent quantum mechanics. I. Constant perturbation

The determinantal formalism previously built up in the evolution operator problem is extended to the density matrix equation in the case, here, of a Heaviside step external perturbation, and in the presence of internal collision potential. Reduced determinantal forms, matched to those first derived for the evolution operator, are obtained and compared with both the iterative expansion and the initial Fredholm-Laplace solution. They not only exhibit the multiple transition structure of the physical response, along with the associated transition width upon expansion, but also the trace conservation requirement is shown to be satisfied at any order and all times. An illustration is given in a simplified physical system.

Time-dependent theory of electronic-to-vibrational energy transfer. Application to metastable argon atoms with nitrogen molecules

Abstract The product vibrational distribution is calculated for the process Ar( 3 P) + N 2 (X) → Ar + Bi* 2 (C 3 H u ). The evolution of the molecular vibration is treated using simple time-dependent quantum mechanics. The theory fits the oscillations in the vibrational distribution observed recently by Cutshall and Muschlitz.

Stochastic theory of vibrational energy transfer in collinear atom-diatom collisions: the role of non-markovian effects

The role of non-markovian effects in the stochastic treatment of vibrational-translational energy transfer in collinear atom-diatom collisions is examined. A comparison of transition probabilities using both markovian and non-markovian types of master equations, as well as exact time dependent quantum mechanics, is made for various values of the system parameters m and α. We find that for certain ranges of the system parameters, the deviations between markovian and non-markovian theories are substantial. Only in the perturbation theory limit and in the limit of low m/α2 values and high enough initial translational energies such that an impulsive approximation for translational motion is accurate are the markovian and non-markovian results similar. An analysis of the collision dynamics indicates that the markovian and non-markovian probabilities agree with each other and with the exact probabilities when action-angle correlations are weak while none of these theories agree (except by accident) when such ef...

The Green’s function method of sensitivity analysis in quantum dynamics

The paper considers the sensitivity of quantum dynamics systems with respect to parameters or features in the underlying Hamiltonians. The analysis allows for the calculation of the gradient of the wave function with respect to any well defined system parameters in the Hamiltonian. Differential sensitivity equations are derived from the Schrödinger equation and their solution is expressed in terms of a Green’s function. For scattering problems the necessary asymptotic Green’s function can be exclusively expressed in terms of the available solution to the original Schrödinger equation. Also considered are problems from time-dependent quantum mechanics. In this latter situation sensitivity theory allows for the calculation of gradients of the time evolution operator with respect to system parameters. The scattering formalism is illustrated with a study of the anisotropy coefficient sensitivity in the collision of an atom and a ridgid rotor. The time-dependent theory was applied to a collision-free molecular beam translating through a region of space containing electric and magnetic fields. This case examined the sensitivity of the molecular transition probabilities to field parameters. These brief examples serve to illustrate the potential usefulness of sensitivity analysis in quantum dynamics.

Path integrals and optical potentials for elastic and inelastic scattering

When relative degrees of freedom are taken to traverse a fixed trajectory R(τ), the properties of Feynman path integrals are used to derive a series expansion of the logarithm of the internal state elastic scattering amplitude in terms of the internal relative coupling potential. Likewise transition amplitudes for inelastic scattering are expressed in terms of an exponential of an infinite series of 2 x 2 matrices. These results represent a reordering of the usual perturbation series of time dependent quantum mechanics to obtain a series expansion in the exponential as well as truncation from the space of all channels to that of only one or two (or more, if necessary). When the relative degrees of freedom are to be treated quantum mechanically, R(τ) is now the integration variable in a Feynman path integral. The above noted exponentiated series then correspond to the exact optical potentials for elastic and inelastic scattering within the Feynman path integral formulation of quantum mechanics.

Quantum-Mechanical Systems in Monochromatic and Nearly Monochromatic Fields

It is shown how Rabi's treatment of the response of a two-level quantum-mechanical system to a monochromatic external driving field may be rigorously placed within the framework of the time-dependent quantum mechanics of a general system by using unitary transformations to separate out slowly varying contributions in closed form. The method is extended to include a certain class of nonmonochromatic fields and the concept of a linearly time-dependent (induced) transition probability is discussed.