Ehrenfest Principle Research Articles

Ehrenfest principle and unitary dynamics of quantum-classical systems with general potential interaction

Representation of classical dynamics by unitary transformations has been used to develop a unified description of hybrid classical-quantum systems with a particular type of interaction, and to formulate abstract systems interpolating between classical and quantum ones. We solved the problem of a unitary description of two interpolating systems with general potential interaction. The general solution is used to show that with arbitrary potential interaction between the two interpolating systems the evolution of the so-called unobservable variables is decoupled from that of the observable ones if and only if the interpolation parameters in the two interpolating systems are equal.

A resonance shift prediction based on the Boltzmann–Ehrenfest principle for cylindrical cavities with a rigid sphere

An investigation on the resonance frequency shift for a plane-wave mode in a cylindrical cavity produced by a rigid sphere is reported in this paper. This change of the resonance frequency has been previously considered as a cause of oscillational instabilities in single-mode acoustic levitation devices. It is shown that the use of the Boltzmann-Ehrenfest principle of adiabatic invariance allows the derivation of an expression for the resonance frequency shift in a simpler and more direct way than a method based on a Green's function reported in literature. The position of the sphere can be any point along the axis of the cavity. Obtained predictions of the resonance frequency shift with the deduced equation agree quite well with numerical simulations based on the boundary element method. The results are also confirmed by experiments. The equation derived from the Boltzmann-Ehrenfest principle appears to be more general, and for large spheres, it gives a better approximation than the equation previously reported.

Cubic response functions in time-dependent density functional theory

We present density-functional theory for time-dependent response functions up to and including cubic response. The working expressions are derived from an explicit exponential parametrization of the density operator and the Ehrenfest principle, alternatively, the quasienergy ansatz. While the theory retains the adiabatic approximation, implying that the time-dependency of the functional is obtained only implicitly-through the time dependence of the density itself rather than through the form of the exchange-correlation functionals-it generalizes previous time-dependent implementations in that arbitrary functionals can be chosen for the perturbed densities (energy derivatives or response functions). In particular, general density functionals beyond the local density approximation can be applied, such as hybrid functionals with exchange correlation at the generalized-gradient approximation level and fractional exact Hartree-Fock exchange. With our implementation the response of the density can always be obtained using the stated density functional, or optionally different functionals can be applied for the unperturbed and perturbed densities, even different functionals for different response order. As illustration we explore the use of various combinations of functionals for applications of nonlinear optical hyperpolarizabilities of a few centrosymmetric systems; molecular nitrogen, benzene, and the C(60) fullerene. Considering that vibrational, solvent, and local field factors effects are left out, we find in general that very good experimental agreement can be obtained for the second dynamic hyperpolarizability of these systems. It is shown that a treatment of the response of the density beyond the local density approximation gives a significant effect. The use of different functional combinations are motivated and discussed, and it is concluded that the choice of higher order kernels can be of similar importance as the choice of the potential which governs the Kohn-Sham orbitals.

Density-functional theory of linear and nonlinear time-dependent molecular properties

We present density-functional theory for linear and nonlinear response functions using an explicit exponential parametrization of the density operator. The response functions are derived using two alternative variation principles, namely, the Ehrenfest principle and the quasienergy principle, giving different but numerically equivalent formulas. We present, for the first time, calculations of dynamical hyperpolarizabilities for hybrid functionals including exchange-correlation functionals at the general gradient-approximation level and fractional exact Hartree–Fock exchange. Sample calculations are presented of the first hyperpolarizability of the para-nitroaniline molecule and of a porphyrin derived push–pull molecule, showing good agreement with available experimental data.

De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory

A quantization of field theory based on the De Donder-Weyl (DW) covariant Hamiltonian formulation is discussed. A hypercomplex extension of quantum mechanics, in which the space-time Clifford algebra replaces that of the complex numbers, appears as a result of quantization of Poisson brackets on differential forms which were put forward for the DW theory earlier. The proposed covariant hypercomplex Schrödinger equation is shown to lead in the classical limit to the DW Hamilton-Jacobi equation and to obey the Ehrenfest principle in the sense that the DW canonical field equations are satisfied for expectation values of properly chosen operators.

Ehrenfest's principle and the problem of time in quantum gravity

We elaborate on a proposal made by Greensite and others to solve the problem of time in quantum gravity. The proposal states that a viable concept of time and a sensible inner product can be found from the demand for the Ehrenfest equations to hold in quantum gravity. We derive and discuss in detail exact consistency conditions from both Ehrenfest equations as well as from the semiclassical approximation. We also discuss consistency conditions arising from the full field theory. We find that only a very restricted class of solutions to the Wheeler-DeWitt equation fulfills all consistency conditions. We conclude that therefore this proposal must either be abandoned as a means to solve the problem of time or, alternatively, be used as an additional boundary condition to select physical solutions from the Wheeler-DeWitt equation.

Ehrenfest's principle in quantum gravity

The Ehrenfest principle δt〈q〉 = 〈i[H, q]〉 is proposed as 〈part of 〉 a definition of the time variable in canonical quantum gravity. This principle selects a time direction in superspace, and provides a conserved, positive definite probability measure. An exact solution of the Ehrenfest condition is obtained, which leads to constant-time surfaces in superspace generated by the operator d/dτ = Δϑ·Δ, where Δ is the gradient operator in superspace, and ϑ is the phase of the Wheeler-DeWitt wavefunction ψ; the constant-time surfaces are determined by this solution up to a choice of initial t = 0 surface. This result holds throughout superspace, including classically forbidden regions and in the neighborhood of caustics; it also leads to ordinary quantum field theory and classical gravity in regions of superspace where the phase satisfies |δ,ϑ| ≫ |δt In(ψ≠ψ)| and (δtϑ)2 ≫ |δ2tϑ|.

Acoustic levitation and the Boltzmann–Ehrenfest principle

The Boltzmann–Ehrenfest principle of adiabatic invariance relates the acoustic potential acting on a sample positioned in a single-mode cavity to the shift in resonant frequency caused by the presence of this sample. This general and simple relation applies to samples and cavities of arbitrary shape, dimension, and compressibility. Positioning forces and torques can, therefore, be determined from straightforward measurements of frequency shifts. Applications to the Rayleigh disk phenomenon and levitated cylinders are presented.

Use of Ehrenfest's principle for inelastic collisions

By combining Ehrenfest's principle with optimised wavefunctions, closed equations of motion are obtained for mean values of observable quantities. Quantum-mechanical trajectories for average position and momentum of a particle are studied together with their corresponding fluctuations. The method is applied to the Landau-Teller model for the collinear atom-diatomic-molecule collision. Energy transfer and vibrational transition probabilities are extracted from the collision-induced trajectories. A comparison with coupled-channel results demonstrates the high accuracy of the proposed method.

Time operator in quantum mechanics

Within aspace-time description of nonrelativistic quantum objects in terms of wave packets, one may simply consider (for every fixed spatial point\(\bar x\): see eq. (5)) the «wave-packets»\(F(t,\bar x) = \int d Ef(E,\bar x)\), that we shall assume to have as weight functions the vectors of the functional spaceP defined as follows. The spaceP is the space of continuousL2-functions i) defined over the (total) energy interval 0<E<∞, ii) with square-integrable first derivatives and iii) for which a Hermitian energy operator exists. Such a spaceP isdense in the Hilbert space ofL2-functions. It is then shown that a «good» time operator exists,\(\hat t = - (i/2)(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {\partial } /\partial E)\), that acts ontoP and i) is «symmetric» (but not self-adjoint), ii) is canonically conjugate to the (total) energy, and iii) satisfies the Ehrenfest principle and Galilei invariance. The old, known objection by Pauli is recognized to point out merely that our operator\(\hat t\) cannot be hypermaximal, as was clarified by von Neumann. But even nonhypermaximal operators may be given a physical meaning and may represent observables in quantum mechanics. As already emphasized by previous authors, confining one's attention only to self-adjoint operators in quantum mechanics is too restrictive a postulate. Notwithstanding that\(\hat t\) has no true eigenfunctions, nevertheless we succeed in calculating theaverage values of our time operator over our «wave packets» (and over the physical states corresponding to them). The case of wave packets moving freely is first considered. Secondly, the nonfree cases ofscattering by a potential are investigated.

Probability of violation of the Ehrenfest principle in fast passage

Probability of violation of the Ehrenfest principle in fast passage