Quantum Equations Research Articles

Dynamics of a moving quantum charged particle

Quantum equations describing a moving charged quantum particle are derived. The electric nature is described by the current and charge densities, J→ and ρ; and the quantum nature by the particle mass, wavefunctions, ψ→ and ψ0. The vector wavefunction, ψ→ is found to act along the current direction. The energy of the matter fields concomitant with a particle is found to flow along the current direction. A fluid with a stationary mass density inherently exhibits a Thomas precession.

A new insight to the scaling-law fluid associated with the Mandelbrot scaling law

This paper addresses a non-traditional approach for the scaling-law fluid-flows described by fractal scaling-law vector calculus associated with the Mandelbrot scaling law. Their quantum equations were proposed to control the fluid-flows associated with the Mandelbrot scaling law. This gives a new insight into the descriptions for the scaling-law behaviors of the fluid-flows in the Mandelbrot scaling-law phenomena.

Decoherence as an Inherent Characteristic of Quantum Mechanics

In this study, we show that it is possible to explain the quantum measurement process within the framework of quantum mechanics without any additional postulates. We do not delve into a deep discussion regarding what the measurement problem actually is, and only examine the problems that seem to exist between classical and quantum physics. Relations between quantum and classical equations of motion are briefly reviewed to show that the transition from a superposition of quantum states to an eigenstate, namely, decoherence, is necessary to ensure that the expectation values in quantum mechanics obey the classical equations of motion. Several Bell-type inequalities and the Kochen-Specker theorem are also reviewed to clarify the concepts of nonseparability and counterfactual definiteness in quantum mechanics. The main objective of this study is to show that decoherence is an inherent characteristic of quantum states caused by the quantum uncertainty relation. We conclude that the quantum measurement process can indeed be explained within the framework of pure quantum mechanics. We also show that our conclusion is consistent with the counterfactual indefiniteness of quantum mechanics.

Consequences of minimal length discretization on line element, metric tensor, and geodesic equation

When minimal length uncertainty emerging from a generalized uncertainty principle (GUP) is thoughtfully implemented, it is of great interest to consider its impacts on gravitational Einstein field equations (gEFEs) and to try to assess consequential modifications in metric manifesting properties of quantum geometry due to quantum gravity. GUP takes into account the gravitational impacts on the noncommutation relations of length (distance) and momentum operators or time and energy operators and so on. On the other hand, gEFE relates classical geometry or general relativity gravity to the energy–momentum tensors, that is, proposing quantum equations of state. Despite the technical difficulties, we intend to insert GUP into the metric tensor so that the line element and the geodesic equation in flat and curved space are accordingly modified. The latter apparently encompasses acceleration, jerk, and snap (jounce) of a particle in the quasi‐quantized gravitational field. Finite higher orders of acceleration apparently manifest phenomena such as accelerating expansion and transitions between different radii of curvature and so on.

Quantum Maxwell's Equations Made Simple: Employing Scalar and Vector Potential Formulation

We present a succinct way to quantize Maxwell's equations. We begin by discussing the random nature of quantum observables. Then we present the quantum Maxwell's equations and give their physical meanings. Due to the mathematical homomorphism between the classical and quantum cases [1], the derivation of quantum Maxwell's equations can be simplified. First, one needs only to verify that the classical equations of motion are derivable from a Hamiltonian by energy-conservation arguments. Then the quantum equations of motion follow due to homomorphism, which applies to sum-separable Hamiltonians. Hence, we first show the derivation of classical Maxwell's equations using Hamiltonian theory. Then the derivation of the quantum Maxwell's equations follows in an analogous fashion. Finally, we apply this quantization procedure to the dispersive medium case. (This article is written for the classical electromagnetic community assuming little knowledge of quantum theory, an introduction of which can be found in [2, Lecs. 38 and 39].)

Quantum particles and the ergosphere of the Kerr metric

The existence of the ergosphere of the Kerr metric does not manifest itself in quantum equations for particles of different spins.To justify the Penrose process with energy extraction from the ergosphere, it is necessary to substantiate and prove its existence within the framework of the consistent quantum theory.

Adiabatic Trajectory Approximation within the Framework of Mixed Quantum/Classical Theory.

A hierarchy of approximate methods is proposed for solving the equations of motion within a framework of the mixed quantum/classical theory (MQCT) of inelastic molecular collisions. Of particular interest is a limiting case: the method in which the classical-like equations of motion for the translational degrees of freedom (scattering) are decoupled from the quantum-like equations for time evolution of the internal molecular states (rotational and vibrational). In practice, trajectories are pre-computed during the first step of calculations with driving forces determined solely by the potential energy surface of the entrance channel, which is an adiabatic trajectory approximation. Quantum state-to-state transition probabilities are computed in the second step of calculations with an expanded basis and very efficient step-size adjustment. Application of this method to H2O + H2 rotationally inelastic scattering shows a significant computational speedup by 2 orders of magnitude. The results of this approximate propagation scheme are still rather accurate, as demonstrated by benchmarking against more rigorous calculations in which the quantum and classical equations of motion are held coupled and against the full-quantum coupled-channel calculations from the literature. It is concluded that the AT-MQCT method (the adiabatic trajectory version of MQCT) represents a promising tool for the computational treatment of molecular collisions and energy exchange.

Effective theories as truncated trans-series and scale separated compactifications

We study the possibility of realizing scale-separated type IIB Anti-de Sitter and de Sitter compactifications within a controlled effective field theory regime defined by low-energy and large (but scale-separated) compactification volume. The approach we use views effective theories as truncations of the full quantum equations of motion expanded in a trans-series around this asymptotic regime. By studying the scalings of all possible perturbative and non-perturbative corrections we identify the effects that have the right scaling to allow for the desired solutions. In the case of Anti-de Sitter, we find agreement with KKLT-type scenarios, and argue that non-perturbative brane-instantons wrapping four-cycles (or similarly scaling effects) are essentially the only ingredient that allows for scale separated solutions. We also comment on the relation of these results to the AdS swampland conjectures. For the de Sitter case we find that we are forced to introduce an infinite number of relatively unsuppressed corrections to the equations of motion, leading to a breakdown of effective theory. This suggests that if de Sitter vacua exist in the string landscape, they should not be thought of as residing within the same effective theory as the AdS or Minkowski compactifications, but rather as defining a separate asymptotic regime, presumably related to the others by a duality transformation.

Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor

The computation of molecular excitation energies is essential for predicting photo-induced reactions of chemical and technological interest. While the classical computing resources needed for this task scale poorly, quantum algorithms emerge as promising alternatives. In particular, the extension of the variational quantum eigensolver algorithm to the computation of the excitation energies is an attractive option. However, there is currently a lack of such algorithms for correlated molecular systems that is amenable to near-term, noisy hardware. In this work, we propose an extension of the well-established classical equation of motion approach to a quantum algorithm for the calculation of molecular excitation energies on noisy quantum computers. In particular, we demonstrate the efficiency of this approach in the calculation of the excitation energies of the LiH molecule on an IBM Quantum computer.3 MoreReceived 22 January 2020Accepted 23 September 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043140Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum algorithmsQuantum simulationQuantum Information

Synchronization by noise for the stochastic quantization equation in dimensions 2 and 3

We prove uniform synchronization by noise with rates for the stochastic quantization equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the framework of order-preserving Markov semigroups derived in [O. Butkovsky and M. Scheutzow, Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting, preprint (2019), arXiv:1907.03725]. In particular, it is shown that this framework can be applied to the case of state spaces given in terms of Hölder spaces of negative exponent.

Madelung transform and probability densities in hybrid quantum–classical dynamics

This paper extends the Madelung–Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid quantum–classical Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincaré integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint quantum–classical density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.

Physics of the sapphire whispering-gallery-mode solid-state MASER oscillator

The Cryogenic Sapphire Oscillator (CSO) is currently the best available technology that can provide a relative frequency stability better than 10−15 with integration times between 1 s and 10,000 s. But, the CSO remains a complex instrument that requires multiple loop controls to achieve the best frequency stability. The possibility to use the sapphire resonator in a self-sustained MASER oscillator presents an elegant alternative to the CSO. Here, sustaining the amplification is achieved through the interaction between a high-Q factor whispering gallery mode and the paramagnetic Fe3+ ions, which are present in small concentration in the sapphire crystal. The Fe3+ ion exhibits three energy states enabling to realize a self-sustaining solid-state maser. Although, this principle has been already experimentally demonstrated few years ago, its development as a truly usable ultra-stable source has not yet been completed, mainly due to the lack of control of the complex physical phenomena involved. This paper complements the previous theoretical work based on the rate equations model. Here we derive the full quantum equations describing the evolution of the Fe3+ ions inside the sapphire lattice and submitted to a pump and a maser signal. The influence of the ions concentration and spin-spin relaxation time will be pointed out.

Long-lived quantum coherence and nonlinear properties of a two-dimensional semiconductor quantum well

In this paper, we present an analytical solution to the Maxwell–Bloch equations of the two-level semiconductor quantum well, G a A s / A l G a A s . In addition, we discuss the effects of coherent Rabi oscillations Ω ( t ) and the frequency of the semiconductor system ν ( t ) on atomic occupation probabilities, ρ 11 ( t ) and ρ 22 ( t ) , population inversion, ρ z ( t ) , and information entropies, H ( σ x ) , H ( σ y ) , and H ( σ z ) . We observe clearly the emergence of long-lived quantum coherence and the decay in curves for some special cases of Ω ( t ) and ν ( t ) . Also, we show that the dynamic nonlinear properties of the system can be controlled by changing the values of the coherent Rabi oscillations Ω ( t ) and the frequency of the semiconductor system ν ( t ) . Due to the lack of mathematical treatment of such systems, our study promises significant advantages for a large number of nonlinear dynamic systems, opening up a wide range of applications for semiconductor quantum wells.

A Quantum Model of the Distribution of Prime Numbers and the Riemann Hypothesis

After proving that numerical sequences such as Fibonacci numbers and prime numbers, can be generated as sequences of equilibrium points of an ideal half-infinite one-dimensional distribution of electric charges, a model of the distribution of primes on the x-axis is proposed, where primes ρ(n) are considered as quantum particles oscillating around the sequence of stationary points r(n) of the Lennard-Jones-like potential of the single-particle Hamiltonian. A particle-counting function πQ(x) is defined over the many-particle system, in the same way as the prime-counting function π(x). Through the application of the Hellmann-Feynman theorem, the existence of a solution n(x) ≈ li(x), coinciding asymptotically with the logarithmic integral function, is proved for the counting function of stationary points, together with the quantum equivalent of the Prime Number Theorem for the particle-counting function πQ(x). The conditions on the sequence of energy eigenvalues E(n) of the system, so that the Riemann hypothesis holds for the function πQ(x), are derived, thus proving the quantum theory results in the same bound of the position of the prime-particle in the n-th quantum state as that derived through analytical methods for the n-th prime. The model suggests new conjectures and statistical procedures which are applied to study the sign of the difference π(x)-li(x) and to explore the region beyond the Skewes number. The data predicted by the theory find confirmation when compared with those known in the literature. A new statistical local test of the Riemann hypothesis, based on the difference π(x)-li(x), is proposed.

Quantum polariton trigger

We present a device model of the polariton logic element that is able to work both with classical and quantum signals. This device uses the nonlinear and quantum properties of exciton-polaritons. The mathematical description of our model is based on stochastic quantum equations with a complex diffusion in the Ito formalism with the corresponding quasi-probability in the quantum phase space. It is shown that the stationary states in the bistable mode can be controlled by both classical and quantum signals through the stochastic resonance within the region of quantum phase transition. Based on these properties, the model of the quantum logic elements has been developed.

Hyers–Ulam stability for quantum equations

We introduce and study the Hyers–Ulam stability (HUS) of a Cayley quantum (q-difference) equation of first order, where the constant coefficient is allowed to range over the complex numbers. In particular, if this coefficient is non-zero, then the quantum equation has Hyers–Ulam stability for certain values of the Cayley parameter, and we establish the best (minimal) HUS constant in terms of the coefficient only, independent of q and the Cayley parameter. If the Cayley parameter equals one half, then there is no Hyers–Ulam stability for any coefficient value in the complex plane.

Stochastic quantization associated with the $$\exp (\Phi )_2$$-quantum field model driven by space-time white noise on the torus

We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $$\exp (\Phi )_{2}$$ -quantum field model or Høegh-Krohn’s model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation and identify it with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.

Hyers–Ulam Stability for Quantum Equations of Euler Type

Many applications using discrete dynamics employ either q-difference equations or h-difference equations. In this work, we introduce and study the Hyers–Ulam stability (HUS) of a quantum (q-difference) equation of Euler type. In particular, we show a direct connection between quantum equations of Euler type and h-difference equations of constant step size h with constant coefficients and an arbitrary integer order. For equation orders greater than two, the h-difference results extend first-order and second-order results found in the literature, and the Euler-type q-difference results are completely novel for any order. In many cases, the best HUS constant is found.

Explicit Integrators Based on a Bipartite Lattice and a Pair of Affine Transformations to Solve Quantum Equations with Gauge Fields

An explicit integrator is proposed to solve the time-dependent Ginzburg–Landau equation and the time-dependent Gross–Pitaevskii equation, which describe quantum dynamics with gauge fields. We const...

Dynamics of quantum systems driven by time-varying Hamiltonians: Solution for the Bloch-Siegert Hamiltonian and applications to NMR

Comprehending the dynamical behaviour of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood as the usual methods including diagonalisation of the Hamiltonian do not work in this setting. In fact, since the inception of Magnus' expansion in 1954, no fundamentally novel mathematical approach capable of solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here of an entirely different non-perturbative approach, termed path-sum, which is always guaranteed to converge, yields the exact analytical solution in a finite number of steps for finite systems and is invariant under scale transformations of the quantum state space. Path-sum can be combined with any state-space reduction technique and can exactly reconstruct the dynamics of a many-body quantum system from the separate, isolated, evolutions of any chosen collection of its sub-systems. As examples of application, we solve analytically for the dynamics of all two-level systems as well as of a many-body Hamiltonian with a particular emphasis on NMR (Nuclear Magnetic Resonance) applications: Bloch-Siegert effect, coherent destruction of tunneling and $N$-spin systems involving the dipolar Hamiltonian and spin diffusion.