Quantum Mechanical Problem Research Articles

An optical tweezer phonon laser

Phonon lasers are mechanical analogues of the ubiquitous optical laser and have been realized in a variety of contexts1–12. However, no such demonstration exists for mesoscopic levitated optomechanical systems, which are emerging as important platforms for conducting fundamental tests of quantum mechanics13–15 and gravity16, as well as for developing sensing modalities that couple mechanical motion to electron spin17–20 and charge21. Inspired by the pioneering work of Arthur Ashkin on optical tweezers22,23, we introduce a mesoscopic, frequency-tunable phonon laser based on the centre-of-mass oscillation of a silica nanosphere levitated in an optical tweezer under vacuum. Unlike previous levitated realizations, our scheme is general enough to be used on single electrons, liquid droplets or even small biological organisms24. Our device thus provides a pathway for a coherent source of phonons on the mesoscale that can be applied to both fundamental problems in quantum mechanics as well as tasks of precision metrology25–27. A phonon laser based on an optically levitated silica nanosphere is demonstrated. A lasing threshold—a phase transition from Brownian motion to coherent oscillation—is observed when the modulation depth of the trapping beam power is increased.

Квантово-механическая задача с периодическими начальными условиями в классической физике и геометродинамике

Квантово-механическая задача с периодическими начальными условиями в классической физике и геометродинамике

On the possibility of constructing relativistic quantum mechanics on the basis of the definition of the functions of differential operators

A new definition of the function of a differential operator that leads to local operators of infinite order is proposed. It allows one to obtain an expression for the square root of the differential operator (the Hamiltonian of a free spinless particle) and to determine the relativistic Schrödinger equation as a close analog of the non-relativistic Schrödinger equation. It is shown that this equation does not lead to difficulties of the Klein-Gordon equation. Boundary conditions that lead to self-adjoint boundary problems similar to Sturm-Liouville problems, periodic boundary value problems, and singular boundary value problems are determined. Some problems of relativistic quantum mechanics are solved.

ORIGINATION AND PROPAGATION OF TEMPERATURE SOLITONS WITH WAVE HEAT TRANSFER IN THE BOUNDED AREA DURING ADDITIVE TECHNOLOGICAL PROCESSES

Within this work, based on analyses of problems on wave heat transfer in bounded bodies, the theory of thermally isolated waves (solitons) is developed to investigate the heat transfer processes in the initial time vicinity and in the vicinity of the bounded body, that is the time scales are commensurate with the relaxation time (nanoseconds), and the scales of the spatial variable are measured in nanometers. A new analytical solution of the wave heat transfer based on the heat conduction equation of hyperbolic type under the action of a series of solitons was received, based on which the interaction of individual solitons with each other, absorption and reflection of the solitons from the body boundaries was analyzed. Analysis of a large number of results made clear that thermal solitons reflect not as mechanical ones, since first there is absorption of the soliton thermal energy by the heat-insulated boundary on the heat-insulated walls, and then the energy is rejected by the thermal conductivity in the opposite direction. It was found that the temperature gradient inside the soliton is negative in the forward direction and positive in the reflected direction. The results of the paper can be used in thermal interaction of high-power radiation with solid surfaces, as well as in the problems of quantum mechanics.

The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimensional physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer’s companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.

Are smooth pseudopotentials a good choice for representing short-range interactions?

When seeking a numerical representation of a quantum-mechanical multiparticle problem it is tempting to replace a singular short-range interaction by a smooth finite-range pseudopotential. Finite basis set expansions, e.g.~in Fock space, are then guaranteed to converge exponentially. The need to faithfully represent the artificial length scale of the pseudopotential, however, places a costly burden on the basis set. Here we discuss scaling relations for the required size of the basis set and demonstrate the basis set convergence on the example of a two-dimensional system of few fermions with short-range $s$-wave interactions in a harmonic trapping potential. In particular we show that the number of harmonic-oscillator basis functions needed to reach a regime of exponential convergence for a Gaussian pseudopotential scales with the fourth power of the pseudopotential length scale, which can be improved to quadratic scaling when the basis functions are re-scaled appropriately. Numerical examples for three fermions with up to a few hundred single-particle basis functions are presented and implications for the feasibility of accurate numerical multi-particle simulations of interacting ultra-cold atom systems are discussed.

Liouville-Green expansions of exponential form, with an application to modified Bessel functions

AbstractLinear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).

WKB method and quantum periods beyond genus one

We extend topological string methods in order to perform WKB approximations for quantum mechanical problems with higher order potentials efficiently. This requires techniques for the evaluation of the relevant quantum periods for Riemann surfaces beyond genus one. The basis of these quantum periods is fixed using the leading behaviour of the classical periods. The full expansion of the quantum periods is obtained using a system of Picard–Fuchs like operators for a sequence of integrals of meromorphic forms of the second kind. Discrete automorphisms of simple higher order potentials allow to view the corresponding higher genus curves as covering of a genus one curve. In this case the quantum periods can be alternatively obtained using the holomorphic anomaly solved in the holomorphic limit within the ring of quasi modular forms of a congruent subgroup of SL as we check for a symmetric sextic potential.

Pseudospectral solution of the Schrödinger equation for the Rosen-Morse and Eckart potentials

Pseudospectral methods based on non-classical quadratures are used to numerically compute the eigenvalues and eigenfunctions of the Schrodinger equation for the Rosen-Morse and Eckart potentials. The method uses a basis set of non-classical polynomials, $$\{ P_n(x) \}$$ , orthonormal with respect to a weight function, $$w(x)>0$$ , to construct an $$N \times N$$ matrix representative, $$\{ H_{nm} \}$$ , of the Hamiltonian, H. This matrix representative is transformed to an equivalent pseudospectral representative, $$\{H_{ij}\}$$ . The rate of convergence of the eigenvalues of $$\{ H_{ij} \}$$ to the eigenvalues of H, versus the grid size N, is reported for non-classical basis functions in comparison with the use of Legendre and Laguerre polynomials as well as a Fourier basis. The use of non-classical polynomials is shown to provide the fastest convergence for the eigenvalues. The pseudospectral method based on nonclassical quadratures proposed in this paper should find wide applicability to other problems in quantum and statistical mechanics. A review is provided of the use of a tensor product of one dimensional basis functions to describe two and three dimensional problems.

Ab initio predictions for polarized deuterium-tritium thermonuclear fusion

The fusion of deuterium (D) with tritium (T) is the most promising of the reactions that could power thermonuclear reactors of the future. It may lead to even more efficient energy generation if obtained in a polarized state, that is with the spin of the reactants aligned. Here, we report first-principles predictions of the polarized DT fusion using nuclear forces from effective field theory. By employing the ab initio no-core shell model with continuum reaction method to solve the quantum mechanical five-nucleon problem, we accurately determine the enhanced fusion rate and angular distribution of the emitted neutron and 4He. Our calculations demonstrate in detail the small contribution of anisotropies, placing on a firmer footing the understanding of the rate of DT fusion in a polarized plasma. In the future, analogous calculations could be used to obtain accurate values for other, more uncertain thermonuclear reaction data critical to nuclear science applications.

TBA equations and resurgent Quantum Mechanics

We derive a system of TBA equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in mathcal{N} = 2 gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of PT-symmetric cubic oscillators and quartic oscillators.

Discussion of the Book “The Quantum Challenge”

Modern but not entirely coordinated foundations of quantum physics are described in the book “The Quantum Challenge”. The difficulties and philosophical problems of this area of science are discussed. Discussions of many great scientists who paved the foundations of the physics of micro-world are described. These discussions are still urgent. The diversity of interpretations of the wave function, light interference, uncertainty principle, complementarity and completeness of micro-world description are stressed in this book. Difficulties and problems of quantum mechanics described in this book allowed the author of the present communication to propose a new approach based on the infinitely small metrics The difference of infinitesimals in two geometries allows one to explain W. Heisenberg’s uncertainty principle. Interconnection of the images in these geometries is possible with the help of Weierstrass integral transform. This approach allows one to describe the interference of light behind the screen with slits as a sum of the corpuscular component (Weierstrass transform) and the wave component (Fourier transform).

Философские основания моделирования квантовых компьютеров

The source of some problems of the quantum mechanics is the observer’s influence on the system. In particular, such problems include the reduction wave function, which forces physicists to talk about “hidden parameters” and the incompleteness of quantum mechanics. Measurements of a quantum system violate its internal state and make it impossible to obtain information about its other parameters (Heisenberg’s uncertainty principle). In 1980 there appeared the thesis that since modeling the behavior of a quantum system on a classical computer cannot provide sufficient accuracy for reproducing all its parameters, there is a need for a quantum computer. The question arises: to what degree can a quantum computer help to solve traditional epistemological problems of quantum mechanics? Can modelling the behavior of elementary particles on a quantum computer “bypass” the problem of the observer’s influence on the system? In other words, is it possible to obtain information about the behavior of a quantum system without observation? Will the internal state of the simulated system be preserved?

Dissolving the measurement problem is not an option for the realist

This paper critically assesses the proposal that scientific realists do not need to search for a solution of the measurement problem in quantum mechanics, but should instead dismiss the problem as ill-posed. James Ladyman and Don Ross have sought to support this proposal with arguments drawn from their naturalized metaphysics and from a Bohr-inspired approach to quantum mechanics. I show that the first class of arguments is unsuccessful, because formulating the measurement problem does not depend on the metaphysical commitments which are undermined by ontic structural realism, rainforest realism, or naturalism in general. The second class of arguments is problematic due to its refusal to provide an analysis of the term “measurement”. It turns out that the proposed dissolution of the measurement problem is in conflict not only with traditional forms of scientific realism but even with the rather minimal realism that Ladyman and Ross themselves defend. The paper concludes with a brief discussion of two related proposals: Healey's pragmatist approach and Bub's information-theoretic interpretation.

Confinement of Lévy flights in a parabolic potential and fractional quantum oscillator

We study L\'evy flights confined in a parabolic potential. This has to do with a fractional generalization of ordinary quantum-mechanical oscillator problem. To solve the spectral problem for the fractional quantum oscillator, we pass to the momentum space, where we apply the variational method. This permits to obtain approximate analytical expressions for eigenvalues and eigenfunctions with very good accuracy. Latter fact has been checked by numerical solution of the problem. We point to the realistic physical systems ranging from multiferroics and oxide heterostructures to quantum chaotic excitons, where obtained results can be used.

Oscillations in the mean transition time of a particle scattered on a double slit potential.

Scattering through a double slit potential is one of the most fundamental problems in quantum mechanics. It is well understood that due to the superposition of amplitudes, one observes a spatial interference pattern in the scattered wavefunction reflecting the superposition of amplitudes coming from both slits. However, the effect of the double slit on the mean time it takes to traverse the slit has not been considered previously. Using a transition path time formalism, we show that when a single Gaussian wavepacket is scattered through a double slit potential, one finds not only oscillations in the scattered density resulting from the spatial interference created by the splitting of the wavepacket but also an oscillatory pattern in the mean scattering time. Long times are associated with low values of a suitably defined momentum, and short times with higher values. The double slit thus serves as a momentum filtering device. We also find an interference pattern in the time averaged momentum weak value profile of the scattered particle implying that the double slit also acts as a weak momentum filter. These results not only demonstrate the value of considering transition path time distributions in their quantum mechanical context but also present a challenge to semiclassical approximations-can they account for temporal interference?

Pre-Born–Oppenheimer molecular structure theory

ABSTRACTIn pre-Born–Oppenheimer (pre-BO) theory a molecule is considered as a quantum system as a whole, including the electrons and the atomic nuclei on the same footing. This approach is fundamentally different from the traditional quantum chemistry treatment, which relies on the separation of the motion of the electrons and the atomic nuclei. A fully quantum mechanical description of molecules has a great promise for future developments and applications. Its most accurate versions may contribute to the definition of new schemes for metrology and testing fundamental physical theories; its approximate versions can provide an efficient theoretical description for molecule-positron interactions and, in general, it would circumvent the tedious computation and fitting of potential energy surfaces and non-adiabatic coupling vectors while it includes also the quantum nuclear motion also often called ‘molecular quantum dynamics’. To achieve these goals, the review points out important technical and fundamental open questions. Most interestingly, the reconciliation of pre-BO theory with the classical chemistry knowledge touches upon fundamental problems related to the measurement problem of quantum mechanics.

From parabolic to loxodromic BMS transformations

Half of the Bondi–Metzner–Sachs (BMS) transformations consist of orientation-preserving conformal homeomorphisms of the extended complex plane known as fractional linear (or Möbius) transformations. These can be of 4 kinds, i.e. they are classified as being parabolic, or hyperbolic, or elliptic, or loxodromic, depending on the number of fixed points and on the value of the trace of the associated $$2 \times 2$$ matrix in the projective version of the $$SL(2,\mathbb {C})$$ group. The resulting particular forms of $$SL(2,\mathbb {C})$$ matrices affect also the other half of BMS transformations, and are used here to propose 4 realizations of the asymptotic symmetry group that we call, again, parabolic, or hyperbolic, or elliptic, or loxodromic. In the second part of the paper, we prove that a subset of hyperbolic and loxodromic transformations, those having trace that approaches $$\infty $$ , correspond to the fulfillment of limit-point condition for singular Sturm–Liouville problems. Thus, a profound link may exist between the language for describing asymptotically flat space-times and the world of complex analysis and self-adjoint problems in ordinary quantum mechanics.

Integrable Potentials by Darboux Transformations in Rings and Quantum and Classical Problems

We study a problem in associative rings of left and right factorization of a polynomial differential operator regarded as an evolution operator. In a direct sum of rings, the polynomial arising in the problem of dividing an operator by an operator for two commuting operators leads to a time-dependent left/right Darboux transformation based on an intertwining relation and either Miura maps or generalized Burgers equations. The intertwining relations lead to a differential equation including differentiations in a weak sense. In view of applications to operator problems in quantum and classical mechanics, we derive the direct quasideterminant or dressing chain formulas. We consider the transformation of creation and annihilation operators for specified matrix rings and study an example of the Dicke model.

Low-energy excitations in helium-like dimer within an exact diagonalization approach

The non-relativistic quantum-mechanical problem on bound states of four interacting spinless electrons moving in the Coulomb field of two attractive point centers with Z0=2 spaced at fixed distance R0 (4He2 dimer within Born–Oppenheimer–Heitler–London approximation) has been solved rigorously through exact diagonalization (expansion on truncated orthonormalized basis) method. The four-spin-conditioned relativistic corrections (of order ∼1/c2) to the ground state level of the dimer have been calculated with exact diagonalization on spin cluster, the spectrum and eigenvectors of the spin problem have been obtained. It is shown that pair spin coupling is antiferromagnetic with exchange constant of 12 K (it provides antisymmetry of the spin-singlet ground-state wave function for isolated 4He atom with two fermions coupled within the unitary spatially symmetric shell), but within four-electron shell of 4He2 dimer this fact yields the quintet ground state of the four-spin cluster totally antisymmetric relative to pair permutations. The exchange within the interatomic bond depends on the interatomic distance, so that there is a coupling between spin and phonon degrees of freedom which leads to renormalization of phonon spectrum in condensed phase as compared to the corresponding spinless medium. This effect can be interpreted as a direct analog of translation-rotation interaction in molecular cryocrystals.