Time-dependent Unitary Transformations Research Articles

Employing an interaction picture to remove artificial correlations in multilayer multiconfiguration time-dependent Hartree simulations

The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method is implemented in the interaction picture to allow a more effective description of correlation effects. It is shown that the artificial correlation present in the original Schrödinger picture can be removed with an appropriate choice of the zeroth-order Hamiltonian. Thereby, operators in the interaction picture are obtained through time-dependent unitary transformations, which have negligible computational cost compared with other parts of the ML-MCTDH algorithm. The efficiency of the method is demonstrated by application to a model of vibrationally coupled charge transport in molecular junctions.

Some Remarks on Quantum Mechanics in a Curved Spacetime, Especially for a Dirac Particle

Some precisions are given about the definition of the Hamiltonian operator H and its transformation properties, for a linear wave equation in a general spacetime. In the presence of time-dependent unitary gauge transformations, H as an operator depends on the gauge choice. The other observables of QM and their rates also become gauge-dependent unless a proper account for the gauge choice is done in their definition. We show the explicit effect of these non-uniqueness issues in the case of the Dirac equation in a general spacetime with the Schwinger gauge. We show also in detail why, the meaning of the energy in QM being inherited from classical Hamiltonian mechanics, the energy operator and its mean values ought to be well defined in a general spacetime.

Extracting complex resonance energies from real calculations

The dynamics of quantum resonances, their energies and their widths, are investigated by means of time-dependent unitary and non-unitary transformations of the Hamiltonian. The aim of the change of representation of the Schrödinger equation is to shape the wave function of the resonances to transform non-square integrable wave functions into localized square-integrable functions. The approach has two advantages: first, it closely relates the energy of the resonances to their dynamics, second, the computational scheme should be compatible with the standard codes of quantum chemistry. The real energies are directly converted into the complex energies of the resonances by analytical continuation. The approach is applied to the discrete Fano model, to a simple model of shape resonance and to the molecular anion H.

Exact time dependence of solutions to the time-dependent Schrödinger equation

Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.

EVOLUTION OF QUANTUM SYSTEMS WITH A SCALING TYPE TIME-DEPENDENT HAMILTONIANS

We introduce a new class of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, the time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the scaling interaction potential. The exact solvability of a "dual" time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence.

Quantum canonical transformations and exact solution of the Schrödinger equation

Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.

Adiabatic product expansion

The time-evolution operator for an explicitly time-dependent Hamiltonian is expressed as the product of a sequence of unitary operators. These are obtained by successive time-dependent unitary transformations of the Hilbert space followed by the adiabatic approximation at each step. The resulting adiabatic product expansion yields a generalization of the quantum adiabatic approximation. Furthermore, it leads to an infinite class of exactly solvable models.

Invariance of the geometrical phase under time-dependent unitary transformations

The effects of time-dependent unitary transformations on the geometrical phase are investigated in the general non adiabatic setting. A detailed study considering both paths of an interference experiment shows that even though the geometry of the fibre bundle: Hilbert space to space of states changes. The measured relative phase, geometrical phase and dynamical phase are all invariant under these transformations.

Broken gauge equivalence of Hamiltonians due to time-evolution and the Berry phase

It is shown that the Berry phase is invariant under both time-dependent unitary and gauge transformations. The initial condition of the quantum system plays a role in uniquely determining the Berry phase. The quantum states transported separately by two Hamiltonians, which are initially related with a static gauge transformation, differ by a Berry phase. The phase in this case cannot be removed by a valid unitary transformation.

Nonuniqueness of the current operators and gauge invariance

For a point-like Dirac particle we describe a class of time-dependent unitary transformations which bring about an explicitly gauge invariant one-body current operator. We do not resort to the 1/ m expansion scheme but retain all orders in 1/ m. We exploit the nonuniqueness (unitary equivalence) inherent to the time-dependent unitary transformation method, and investigate its consequences on the nuclear electromagnetic interactions. We show that the nonuniqueness of the one-body current should influence the exchange current up to the quantity which is expressed in terms of the commutator of the nuclear interaction hamiltonian with the generator of the unitary transformation. We apply the result to the electromagnetic interaction of a nucleon in a relativistic mean field.

Операторные калибровочные преобразования в нерелятивистской квантовой электродинамике Применения к мультипольному Гамильтониану

Operator gauge transformations on the electromagnetic potentials in quantum electrodynamics are obtained by requiring form invariance of the Schrodinger equation of a single particle under arbitrary space- and time-dependent unitary transformations. These operator gauge transformations are found to have the same form as in non-Abelian gauge field theories. By considering the Lorentz force on the particle, we show that the definition of the electromagnetic-field strength operator in terms of the potentials must also have the same form as in non-Abelian gauge field theory. Since the Hamiltonian is gauge dependent, a distinction is made between the Hamiltonian and the gauge-invariant energy operator of the particle. Local conservation of energy for the particle is derived and it is shown that the particle gains (loses) energy at the same rate as expected classically. An operator gauge transformation is made to obtain potentials in the multipolar gauge and the multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian in the multipolar gauge.

On classical hydrodynamics and collective motion as approximation to the nuclear many-body problem

Using time-dependent unitary transformations, one can cast a one-body equation of the time-dependent Hartree-Fock type into a form which is closely related to equations for a classical irrotational fluid. The hydrodynamic equation of state finds its counterpart in a stationary constrained field equation. The hydrodynamic equations in turn can be translated into a classical Hamiltonian formalism with an infinite number of generalised coordinates, which are given as all possible spatial moments of the density. The reduction to a few ones, the “natural collective coordinates” is possible by the choice of appropriate initial conditions. The lowest of the hydrodynamical frequencies can be calculated in closed form by harmonic approximations. For the quadrupole frequency a value ofω=31 A−1/3 MeV/h is obtained. As expected, the value does not agree with the experiment, but rather is in between the characteristic frequency for theβ-vibration and the isoscalar giant quadrupole vibration.

Unitary declension of dynamical symmetries for the time-dependent harmonic oscillator

Elaboration is given to a previous discussion concerning dynamical symmetries of the Lewis–Riesenfeld time-dependent harmonic oscillator. A boson operator formalism is used to define a generator of impicitly time-dependent unitary transformations. Bilinear combinations of the transformed boson operators are shown to span the symplectic algebra Sp(2n,IR) regarded as a larger noninvariance group for the Lewis–Riesenfeld Hamiltonian. The multiplet structure of the embeddings, Sp(2n) ↓SU(n), and SU (n+1) ↓SU(n), is determined for the case n=3, using established branching rules. Boson operator realizations are presented for the multiplet structures in each SU(n) declension of these dynamical groups.

Most-general minimality-preserving Hamiltonian

This paper is concerned with Hamiltonians that preserve the minimum value of the uncertainty product. Using the method of time-dependent unitary transformations, we derive the most general such Hamiltonian.

Quantization of interacting fields

It is shown that the canonical quantization scheme does not lead to a unique formulation of quantum field theory because of the freedom to make time dependent unitary transformations involving a (classical) Boson field, before quantization. As examples, alternative formulations of both non-relativistic and relativistic quantum electrodynamics, in agreement with the correspondence principle, are given. These yield an S-matrix which is free of all divergences, at least to the second order.