This work presents a comprehensive comparison of quantum-mechanical and classical evolution for nuclear motion within a finite-dimensional quantum chemistry model. We employ a modified Tavis--Cummings--Hubbard model featuring two two-level artificial atoms in optical cavities to simulate the association and dissociation of a neutral hydrogen molecule. The initial conditions leading to the formation and decomposition of the molecule are examined. Dissipation in a Markovian open system is simulated by solving the Lindblad master equation. We compare the quantum and classical descriptions of nuclear motion: quantum mobility is characterized by nuclear tunneling, while classical motion is represented via fluctuations in interaction strengths. The emergence of dark states during dissociation and their significant impact on the evolutionary outcome are also analyzed. Key findings show that classical motion reaches the same final molecular states as quantum tunneling but requires an order-of-magnitude shorter time, with distinct patterns of population evolution. Two singlet dark states are identified that modify the branching ratio between neutral and ionic products.
We show that lattice systems, such as the Bose-Hubbard model, can be simulated on a single nano- or micro-mechanical resonator, by exploiting its many modes. The on-site Hamiltonians are engineered by coupling the mechanical modes to the modes of a pair of optical or stripline resonators, and the connections between the lattice sites are engineered in a similar way. The lattice network structure is encoded in the frequency components of the fields driving the resonators. This three-resonator configuration also allows universal quantum computing on the nano-resonator.
The behavior of a classical charged point particle under the influence of only a Coulombic binding potential and classical electromagnetic zero-point radiation, is shown to yield agreement with the probability density distribution of Schroedinger's wave equation for the ground state of hydrogen. These results, obtained without any fitting parameters, again raise the possibility that the main tenets of stochastic electrodynamics (SED) are correct, thereby potentially providing a more fundamental basis of quantum mechanics. The present methods should help propel yet deeper investigations into SE…
The variational quantum eigensolver (VQE) is one of the most appealing quantum algorithms to simulate electronic structure properties of molecules on near-term noisy intermediate-scale quantum devices. In this work, we generalize the VQE algorithm for simulating extended systems. However, the numerical study of an one-dimensional (1D) infinite hydrogen chain using existing VQE algorithms shows a remarkable deviation of the ground state energy with respect to the exact full configuration interaction (FCI) result. Here, we present two schemes to improve the accuracy of quantum simulations for ex…
Is it possible that two different transitions in the non-relativistic quantum mechanical model of the hydrogen atom give the same frequency? That is, can different energy level transitions in a hydrogen atom have the same photon radiation frequency? This question, which was asked during a Ph.D. oral exam in 1997 at the University of Colorado Boulder, is well-known among physics graduate students. We show a general solution to this question, in which all equifrequency transition pairs can be obtained from the set of solutions of a Diophantine equation. This fun puzzle is a simple yet concrete e…
We consider general theoretical models of water and in particular the nature of the motions of the hydrogen nuclei. If the motion of hydrogen nuclei is classical, then the thermodynamic pressure equation of state for heavy water wherein the hydrogen nuclei are deuterons is identical to the pressure equation of state for light water wherein the hydrogen nuclei are protons. Since the experimental thermodynamic phase diagram for light water is clearly measurably different from the experimental thermodynamic phase diagram for heavy water, one may deduce that the motions of hydrogen nuclei are quan…
Recently a new formulation of quantum mechanics has been suggested which is based on the concept of signed particles, that is, classical objects provided with a position, a momentum and a sign simultaneously. In this paper, we comment on the plausibility of simulating atomic systems beyond the Born-Oppenheimer approximation by means of the signed particle formulation of quantum mechanics. First, in order to show the new perspective offered by this new formalism, we provide an example studying quantum tunnelling through a simple Gaussian barrier in terms of the signed particle formulation. Then…
Quantum squeezing of mechanical resonator is important for studying the macroscopic quantum effects and the precision metrology of weak forces. Here we give a theoretical study of a hybrid atom-optomechanical system in which the steady-state squeezing of the mechanical resonator can be generated via the mechanical nonlinearity and cavity cooling process. The validity of the scheme is assessed by simulating the steady-state variance of the mechanical displacement quadrature numerically. The scheme is robust against dissipation of the optical cavity, and the steady-state squeezing can be effecti…
Bohr atomic model is based on the assumption that electrons on allowed quantized orbits do not radiate. Its main results include the values of the radii of circular quantized orbits and of the hydrogen atom energy levels. Quantum mechanical justifications of both his hypothesis and of the well known radii relation, which is used to compute the energy levels, are presented.
Feynmans ideas to employ quantum systems for simulating other quantum systems gave rise to quantum simulation. While current quantum computers are still prone to decoherence and rely on error correction, the development of hybrid algorithms that employ both quantumand classical computation allow to perform quantum simulation. Among these algorithms, the Variational Quantum Eigensolver algorithm has permitted to explore the electronic structure of simple atoms and molecules by exploiting the Rayleigh Ritz variational principle. In this work we provide the implementation of the VQE algorithm for…
Stochastic Electrodynamics (SED) has had success modeling black body radiation, the harmonic oscillator, the Casimir effect, van der Waals forces, diamagnetism, and uniform acceleration of electrodynamic systems using the stochastic zero-point fluctuations of the electromagnetic field with classical mechanics. However the hydrogen atom, with its 1/r potential remains a critical challenge. Cole and Zou in 2003 and Nieuwenhuizen and Liska in 2015 found that the SED field prevented the electron orbit from collapsing into the proton but eventually the atom became ionized. We look at the issues of …
We study the spectrum of the hydrogen atom in Snyder space in a semiclassical approximation based on a generalization of the Born-Sommerfeld quantization rule. While the corrections to the standard quantum mechanical spectrum arise at first order in the Snyder parameter for the $l=0$ states, they are of second order for $l\neq 0$. This can be understood as due to the different topology of the regions of integration in phase space.
Simulating physical systems has been an important application of classical and quantum computers. In this article we present an efficient classical algorithm for simulating time-dependent quantum mechanical Hamiltonians over constant periods of time. The algorithm presented here computes the mean value of an observable over the output state of such short-time Hamiltonian evolutions. In proving the performance of this algorithm we use Lieb-Robinson type bounds to limit the evolution of local operators within a lightcone. This allows us to divide the task of simulating a large quantum system int…
It is conjectured that Stark states of excited hydrogen atom posses magnetic charge for which the quantum mechanical operator is $${\cal G}_{op} = {e\over \hbar} (\vecσ\cdot\vec A)$$ where $\vec A$ is the Runge-Lenz vector. The expectation value $g$ of this operator for Stark states is found to be $$ g = e(n_1-n_2)$$ which obeys a Dirac-Saha type quantization formula $${eg\over c} = (n_1-n_2)α$$ where $α$ is the fine structure constant and $n_1$ and $n_2$ are parabolic quantum numbers. An experimental arrangement is outlined to test this conjecture.
No physical measurement can be performed with infinite precision. This leaves a loophole in the standard no-go arguments against non-contextual hidden variables. All such arguments rely on choosing special sets of quantum-mechanical observables with measurement outcomes that cannot be simulated non-contextually. As a consequence, these arguments do not exclude the hypothesis that the class of physical measurements in fact corresponds to a dense subset of all theoretically possible measurements with outcomes and quantum probabilities that \emph{can} be recovered from a non-contextual hidden var…
We analyze dynamical consequences of a conjecture that there exists a fundamental (indivisible) quant of time. In particular we study the problem of discrete energy levels of hydrogen atom. We are able to reconstruct potential which in discrete time formalism leads to energy levels of unperturbed hydrogen atom. We also consider linear energy levels of quantum harmonic oscillator and show how they are produced in the discrete time formalism. More generally, we show that in discrete time formalism finite motion in central potential leads to discrete energy spectrum, the property which is common …
We simulate an optomechanical system via a cavity QED scenario with a movable atom and investigate its application in the tiny mass sensing. We find that the steady-state solution of the system exhibits a multiple stability behavior, which is similar to that in the optomechanical system. We explain this phenomenon by the opto-mechanical interaction term in the effective Hamiltonian. Due to the dressed states formed by the effective coupling between the vibration degree of the atom and the optical mode in the cavity, we observe a narrow transparent window in the output field. We utilize this vi…
In 1926, Erwin Schrodinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrodinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quant…
The semiclassical Kepler-Coulomb problem and the quantum-mechanical Schrödinger-Coulomb problem are compared for their predictions of quadrupole E2 transitions. The semiclassical treatment involves an extension of previous work for the electric dipole transitions (Physical Review A 71, 020501), and rates are derived for $Δ\ell= 0, \pm 2$ transitions on the basis of the multipolar properties of the emitted radiation. For the quantum case a derivation is presented within the Schrödinger framework without reference to spin. Comparison of the E2 rates shows reasonable agreement, but not as good as…
Algorithms are described for efficiently simulating quantum mechanical systems on quantum computers. A class of algorithms for simulating the Schrodinger equation for interacting many-body systems are presented in some detail. These algorithms would make it possible to simulate nonrelativistic quantum systems on a quantum computer with an exponential speedup compared to simulations on classical computers. Issues involved in simulating relativistic systems of Dirac and gauge particles are discussed.
A quantum system can be driven by either sinusoidal, rectangular, or noisy signals. In the literature, these regimes are referred to as Landau-Zener-Stuckelberg-Majorana (LZSM) interferometry, latching modulation, and motional averaging, respectively. We demonstrate that these pronounced and interesting effects are also inherent in the dynamics of classical two-state systems. We discuss how such classical systems are realized using either mechanical, electrical, or optical resonators. In addition to the fundamental interest of such dynamical phenomena linking classical and quantum physics, we …
A famous pre-Newtonian formula for $π$ is obtained directly from the variational approach to the spectrum of the hydrogen atom in spaces of arbitrary dimensions greater than one, including the physical three dimensions.
A topological $θ$-term in gauge theories, including quantum chromodynamics in 3+1 dimensions, gives rise to a sign problem that makes classical Monte Carlo simulations impractical. Quantum simulations are not subject to such sign problems and are a promising approach to studying these theories in the future. In the near term, it is interesting to study simpler models that retain some of the physical phenomena of interest and their implementation on quantum hardware. For example, dimensionally-reducing gauge theories on small spatial tori produces quantum mechanical models which, despite being …
We study the Hydrogen atom as a quantum mechanical system with a Coulomb like potential, with a semiclassical approach based on an effective description of quantum mechanics. This treatment allows us to describe the quantum state of the system as a system of infinite many classical equations for expectation values of configuration variables, their moments and quantum dispersions. It also provides a semiclassical description of the orbits and the evolution of observables and spreadings and their back-reaction on the evolution.
Accurate and predictive computations of the quantum-mechanical behavior of many interacting electrons in realistic atomic environments are critical for the theoretical design of materials with desired properties, and require solving the grand-challenge problem of the many-electron Schrodinger equation. An infinite chain of equispaced hydrogen atoms is perhaps the simplest realistic model for a bulk material, embodying several central themes of modern condensed matter physics and chemistry, while retaining a connection to the paradigmatic Hubbard model. Here we report a combined application of …
Freely falling point-like objects converge toward the center of the Earth. Hence the gravitational field of the Earth is inhomogeneous, and possesses a tidal component. The free fall of an extended quantum mechanical object such as a hydrogen atom prepared in a high principal-quantum-number state, i.e. a circular Rydberg atom, is predicted to fall more slowly than a classical point-like object, when both objects are dropped from the same height above the Earth's surface. This indicates that, apart from transitions between quantum states, the atom exhibits a kind of quantum mechanical incompres…
We present exact energy spectrum and eigenfunctions of the one-dimensional hydrogen atom in the presence of the minimal length uncertainty. By requiring the self-adjointness property of the Hamiltonian, we completely determine the quantization condition. We indicate that the single-valuedness criteria of the eigenfunctions in non-deformed case is an emergent condition and the semiclassical solutions exactly coincide with the quantum mechanical results. The behavior of the wave functions at the origin in coordinate space and in quasiposition space is discussed finally.
We present the first purely semiclassical calculation of the resonance spectrum in the Diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with $L_z =0$. The classical system is unbound and completely chaotic for a scaled energy $ε\sim E B^{-2/3}$ larger than a critical value $ε_c >0$. The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the poten…
