Ehrenfest Theorem Research Articles

Conditional Values in Quantum Mechanics.

We consider the local value of an operator for a given position or momentum and, more generally on the value of another arbitrary observable. We develop a general approach that is based on breaking up Aψ(x) as Aψ(x)ψ(x)=Aψ(x)ψ(x)R+iAψ(x)ψ(x)I where A is the operator whose local value we seek and ψ(x) is the position wave function. We show that the real part is related to the conditional value for a given position and the imaginary part is related to the standard deviation of the conditional value. We show that the uncertainty of an operator can be expressed in two parts that depend on the real and imaginary parts. In the case of the position representation, the expression for the uncertainty of an operator shows that there are two fundamental contributions, one due to the amplitude of the wave function and the other due to the phase. We obtain the equation of motion for the conditional values, and in particular, we generalize the Ehrenfest theorem by deriving a local version of the theorem. We give a number of examples, including the local value of momentum, kinetic energy, and Hamiltonian. We also discuss other approaches for obtaining a conditional value in quantum mechanics including using quasi-probability distributions and the characteristic function approach, among others.

Quantum and classical properties of the universe according to unimodular quantum cosmology

Abstract We investigate the dynamical behaviour of wavepacket solutions due to unimodular quantum cosmology. We find early and late time quantum eras of the universe.
An Ehrenfest theorem ensures quasi-classical behaviour for the late time universe if the quantum uncertainties are small. The surprising prediction of late time quantum effects for a flat, homogeneous and isotropic universe with a scalar field with an exponential potential is obtained by the analysis of the uncertainty dynamics using the results of the corresponding classical dynamical system. Furthermore quantum correction terms to the Hubble parameter and the matter density are derived. A positive matter expectation value for the de-Sitter universe raises the question if correction terms due to quantum gravity could contribute to solve the dark matter problem.

Orbital angular momentum and rotational properties of off-axis vortex beams in two-dimensional media with anisotropic nonlocal nonlinearity.

By introducing anisotropy into nonlinear propagations, off-axis vortex beams exhibit significantly different characteristics compared to the isotropic case. The orbital angular momentum (OAM) is non-conservative and can periodically change between positive and negative values. Accordingly, the rotation of phase singularity can transit between clockwise and counterclockwise directions. Furthermore, the phase singularity can move to infinity when the OAM approaches zero. By using the Ehrenfest theorem, the motion of the beam center is obtained. Its trajectory can be circular and parabolic or follow other complex shapes, depending closely on the anisotropy of the nonlinearity. The rotational velocity of the beam center can be modulated by the nonlinearity anisotropy and can far exceed the initial value during its propagation. These results may find potential applications in beam shaping and optical manipulation.

Expanding wave packets and the quantum to classical transition

The standard description of the transition from quantum to classical mechanics presented in most text books is the proof that the quantum expectation values of position and momentum obey equations of Newtonian form. This is the Ehrenfest theorem. It is combined with the requirement that wave packets remain localised to describe a single particle moving according to classical mechanics. Hence, the natural spreading of wave packets is viewed as a quantum effect. In contradiction to this view, here it is argued that the spreading, where different momentum components separate, is the signature of the quantum to classical transition. The asymptotic spatial wave function becomes proportional to the initial momentum space wave function, which mirrors exactly the well-known far-field diffraction pattern in optics. Trajectories, defined as the locus of the normals to the expanding wave front, are used to illustrate the transition from quantum to classical motion. Again this is the analogue of the wave to beam transition in optics. It is suggested that this analysis of the quantum to classical transition should be incorporated routinely into introductory quantum mechanics courses.

Simulation of Gaussian wave packets used to illustrate elementary quantum mechanics scenarios

In this paper we numerically solve the time dependent Schrödinger equation for scenarios using wave packets. These examples include the free wave packet, which we use to show the difference between group and phase velocities, the packet in a harmonic oscillator potential with non-trivial initial conditions in one and two dimensions, which is compared with their classical analogs to show how Ehrenfest theorem holds. We also include simulations of the diffraction through the single and double slit potentials, the refraction with a step potential and the dispersion by a central potential. The aim of this paper is to illustrate with simulations, nowadays easy to implement, scenarios that can help explaining the basics of the wave-particle duality.

Spread complexity in saddle-dominated scrambling

Recently, the concept of spread complexity, Krylov complexity for states, has been introduced as a measure of the complexity and chaoticity of quantum systems. In this paper, we study the spread complexity of the thermofield double state within integrable systems that exhibit saddle-dominated scrambling. Specifically, we focus on the Lipkin-Meshkov-Glick model and the inverted harmonic oscillator as representative examples of quantum mechanical systems featuring saddle-dominated scrambling. Applying the Lanczos algorithm, our numerical investigation reveals that the spread complexity in these systems exhibits features reminiscent of chaotic systems, displaying a distinctive ramp-peak-slope-plateau pattern. Our results indicate that, although spread complexity serves as a valuable probe, accurately diagnosing true quantum chaos generally necessitates additional physical input. We also explore the relationship between spread complexity, the spectral form factor, and the transition probability within the Krylov space. We provide analytical confirmation of our numerical results, validating the Ehrenfest theorem of complexity and identifying a distinct quadratic behavior in the early-time regime of spread complexity.

Self-adjoint momentum operator for a particle confined in a multi-dimensional cavity

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.

Nonadiabatic Field on Quantum Phase Space: A Century after Ehrenfest.

Nonadiabatic transition dynamics lies at the core of many electron/hole transfer, photoactivated, and vacuum field-coupled processes. About a century after Ehrenfest proposed "Phasenraum" and the Ehrenfest theorem, we report a conceptually novel trajectory-based nonadiabatic dynamics approach, nonadiabatic field (NAF), based on a generalized exact coordinate-momentum phase space formulation of quantum mechanics. It does not employ the conventional Born-Oppenheimer or Ehrenfest trajectory in the nonadiabatic coupling region. Instead, in NAF the equations of motion of the independent trajectory involve a nonadiabatic nuclear force term in addition to an adiabatic nuclear force term of a single electronic state. A few benchmark tests for gas phase and condensed phase systems indicate that NAF offers a practical tool to capture the correct correlation of electronic and nuclear dynamics for processes where the states remain coupled all the time as well as for the asymptotic region where the coupling of electronic states vanishes.

Relativistic potential energy of a non-dissipative classical harmonic oscillator

The well-known relation of Einstein's relativistic energy E=mc2 for a free particle is extended to cover the total relativistic energy of a classical harmonic oscillator (CHO) by calculating the relativistic potential energy. This study is essentially concerned with the relativistic mass m=γm0, where the Lorentz factor γ transforms the state of the second-order differential equation of a CHO from linear into nonlinear. Although the nonlinear solution still remains periodic, the amplitude A(β) and angular frequency ω(β) are determined only by the dimensionless factor β=X˙(0)/c (the ratio of the initial velocity X˙(0) to the light speed c). It is demonstrated that the time period T(β) and Hook force respectively tend to infinity and zero when β→1 so that the state of the bound particle approaches that of a free particle. By contrast, the relativistic behavior of a quantum harmonic oscillator (QHO) in subatomic scales, such as the relativistic ro-vibrational motion of an electron in circular Bohr orbits, is completely different. The relativistic model of Bohr atom is discussed in detail by using the relativistic relations of a CHO according to the Ehrenfest theorem. Finally, the results are confirmed by demonstrating energy conservation.

Universal chaotic dynamics from Krylov space

Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.

Helical beams of electrons in a magnetic field: new analytic solutions of the Schrödinger and Dirac equations

We derive new solutions of the Schrödinger, Klein–Gordon and Dirac equations which describe the motion of particles in a uniform magnetic field. In contrast to the well known stationary solutions, our solutions exhibit the behavior of quantum particles which very closely resembles classical helical trajectories. These solutions also serve as an illustration of the meaning of the Ehrenfest theorem in relativistic quantum mechanics.

Numerical validation of Ehrenfest theorem in a Bohmian perspective for non-conservative systems

Numerical validation of Ehrenfest theorem in a Bohmian perspective for non-conservative systems

Newton’s equations from quantum mechanics for a macroscopic body in the vacuum

Newton’s force law [Formula: see text] is derived from the Schrödinger equation for isolated macroscopic bodies, composite states of e.g. [Formula: see text] atoms and molecules, at finite body temperatures. We first review three aspects of quantum mechanics (QM) in this context: (i) Heisenberg’s uncertainty relations for their center of mass (CM), (ii) the diffusion of the CM wave packet, and (iii) a finite body temperature which implies a metastable (mixed-) state of the body: photon emissions and self-decoherence. They explain the origin of the classical trajectory for a macroscopic body. The ratio between the range [Formula: see text] over which the quantum fluctuations of its CM are effective, and the body’s (linear) size [Formula: see text], [Formula: see text] or [Formula: see text], tells whether the body’s CM behaves classically or quantum mechanically, respectively. In the first case, Newton’s force law for its CM follows from the Ehrenfest theorem. We illustrate this for weak gravitational forces, a harmonic-oscillator potential, and for constant external electromagnetic fields slowly varying in space. The derivation of the canonical Hamilton equations for many-body systems is also discussed. Effects due to the body’s finite size such as the gravitational tidal forces appear in perturbation theory. Our work is consistent with the well-known idea that the emergence of classical physics in QM is due to the environment-induced decoherence, but complements and completes it, by clarifying the conditions under which Newton’s equations follow from QM, and by deriving them explicitly.

Ring Polymer Molecular Dynamics with Electronic Transitions.

Full quantum dynamics of molecules and materials is of fundamental importance, which requires a faithful description of simultaneous quantum motions of the electron and nuclei. A new scheme is developed for nonadiabatic simulations of coupled electron-nuclear quantum dynamics with electronic transitions based on the Ehrenfest theorem and ring polymer molecular dynamics. Built upon the isomorphic ring polymer Hamiltonian, time-dependent multistate electronic Schrödinger equations are solved self-consistently with approximate equation of motions for nuclei. Each bead bears a distinct electronic configuration and thus moves on a specific effective potential. This independent-bead approach provides an accurate description of the real-time electronic population and quantum nuclear trajectory, maintaining a good agreement with the exact quantum solution. Implementation of first-principles calculations enables us to simulate photoinduced proton transfer in H_{2}O-H_{2}O^{+} where we find a good agreement with experiment.

Cosmic-Time Quantum Mechanics and the Passage-of-Time Problem

A new dynamical paradigm merging quantum dynamics with cosmology is discussed. We distinguish between a universe and its background space-time. The universe here is the subset of space-time defined by Ψτ(x)≠0, where Ψτ(x) is a solution of a Schrödinger equation, x is a point in n-dimensional Minkowski space, and τ≥0 is a dimensionless ‘cosmic-time’ evolution parameter. We derive the form of the Schrödinger equation and show that an empty universe is described by a Ψτ(x) that propagates towards the future inside some future-cone V+. The resulting dynamical semigroup is unitary, i.e., ∫V+d4x|Ψτ(x)|2=1 for τ≥0. The initial condition Ψ0(x) is not localized at x=0. Rather, it satisfies the boundary condition Ψ0(x)=0 for x∉V+. For n=1+3 the support of Ψτ(x) is bounded from the past by the ‘gap hyperboloid’ ℓ2τ=c2t2−x2, where ℓ is a fundamental length. Consequently, the points located between the hyperboloid and the light cone c2t2−x2=0 satisfy Ψτ(x)=0, and thus do not belong to the universe. As τ grows, the gap between the support of Ψτ(x) and the light cone increases. The past thus literally disappears. Unitarity of the dynamical semigroup implies that the universe becomes localized in a finite-thickness future-neighbourhood of ℓ2τ=c2t2−x2, simultaneously spreading along the hyperboloid. Effectively, for large τ the subset occupied by the universe resembles a part of the gap hyperboloid itself, but its thickness Δτ is non-zero for finite τ. Finite Δτ implies that the three-dimensional volume of the universe is finite as well. An approximate radius of the universe, rτ, grows with τ due to Δτrτ3=Δ0r03 and Δτ→0. The propagation of Ψτ(x) through space-time matches an intuitive picture of the passage of time. What we regard as the Minkowski-space classical time can be identified with ctτ=∫d4xx0|Ψτ(x)|2, so tτ grows with τ as a consequence of the Ehrenfest theorem, and its present uncertainty can be identified with the Planck time. Assuming that at present values of τ (corresponding to 13–14 billion years) Δτ and rτ are of the order of the Planck length and the Hubble radius, we estimate that the analogous thickness Δ0 of the support of Ψ0(x) is of the order of 1 AU, and r03∼(ctH)3×10−44. The estimates imply that the initial volume of the universe was finite and its uncertainty in time was several minutes. Next, we generalize the formalism in a way that incorporates interactions with matter. We are guided by the correspondence principle with quantum mechanics, which should be asymptotically reconstructed for the present values of τ. We argue that Hamiltonians corresponding to the present values of τ approximately describe quantum mechanics in a conformally Minkowskian space-time. The conformal factor is directly related to |Ψτ(x)|2. As a by-product of the construction, we arrive at a new formulation of conformal invariance of m≠0 fields.

Quantum-Classical Hybrid Systems and Ehrenfest's Theorem.

The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum-classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum-classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest's theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum-classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum-classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum-classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum-classical case. We expect that these theoretical results will be useful both to analyze quantum-classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations.

Bouncing wave packets, Ehrenfest theorem, and uncertainty relation based upon a new concept for the momentum of a particle in a box

For a particle in a box, the operator −i∂x is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg–Robertson–Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.

Time-dependent multiconfiguration self-consistent-field and time-dependent optimized coupled-cluster methods for intense laser-driven multielectron dynamics

We reviewed the time-dependent multiconfiguration self-consistent-field (TD-MCSCF) method and the time-dependent optimized coupled-cluster (TD-OCC) method for first-principles simulations of high-field phenomena such as tunneling ionization and high-order harmonic generation in atoms and molecules irradiated by a strong laser field. These methods provide a flexible and systematically improvable description of the multielectron dynamics by expressing the all-electron wavefunction by configuration interaction expansion or coupled-cluster expansion, using time-dependent one-electron orbital functions. The time-dependent variational principle plays a key role in deriving these methods while satisfying gauge invariance and Ehrenfest theorem. The real-time/real-space implementation with an absorbing boundary condition enables the simulation of high-field processes involving multiple excitation and ionization. We present a detailed, comprehensive discussion of such features of the TD-MCSCF and TD-OCC methods.

Pauli’s Electron in Ehrenfest and Bohm Theories, a Comparative Study

Electrons moving at slow speeds much lower than the speed of light are described by a wave function which is a solution of Pauli's equation. This is a low-velocity limit of the relativistic Dirac equation. Here we compare two approaches, one of which is the more conservative Copenhagen's interpretation denying a trajectory of the electron but allowing a trajectory to the electron expectation value through the Ehrenfest theorem. The said expectation value is of course calculated using a solution of Pauli's equation. A less orthodox approach is championed by Bohm, and attributes a velocity field to the electron also derived from the Pauli wave function. It is thus interesting to compare the trajectory followed by the electron according to Bohm and its expectation value according to Ehrenfest. Both similarities and differences will be considered.

THz Response of Charge Carriers in Nanoparticles: Microscopic Master Equations Reveal an Unexplored Equilibration Current and Nonlinear Mobility Regimes

Herein, the THz mobility of charge carriers in low‐dimensional semiconductors based on a density matrix approach involving master equations for population and polarization dynamics is modeled. Pulsed THz fields induce intraband transitions between quantized subband states, creating polarization and subsequent charge transport that governs the electron mobility. It is shown that an equilibration current emerges—a purely quantum mechanical contribution understood via the Ehrenfest theorem in 1D—reshaping the low‐frequency mobility. Apart from thermal population, the results further demonstrate that the frequency‐dependent mobility becomes THz field strength and spectrum, as well as pulse width and chirp dependent, already at moderate THz probe fields of 1 kV cm−1, e.g., for 1D CdSe or GaAs nanostructures. The parametric nature of the underlying master differential equations for polarization and population, driving the intraband conductivity, results in a nonlinear, third‐order mobility and susceptibility, causing a nontrivial field dependence as well as power broadening even at moderate field strength. The obtained results are in good agreement with experiments. The observed high nonlinearities strongly impact the design and interpretation of THz charge carrier mobility experiments and further allow applications like coherent control, frequency mixing, or synthesis through a field‐controlled nonlinearity or high harmonics generation, especially interesting for future 6G telecommunication.