7 pages, 2 encapsulated postscript figures, uses epsf.sty talk given at the II International Workshop on Classical and Quantum Integrable Systems, Dubna (Russia), 8-12 July (1996) to be published in Int. J. Mod. Phys. A
The exactly solvable eigenproblems in Schrödinger quantum mechanics typically involve the differential "shift operators". In the standard supersymmetric (SUSY) case, the shift operator turns out to be of first order. In this work, I discuss a technique to generate exactly solvable eigenproblems by using second order shift operators. The links between this method and SUSY are analysed. As an example, we show the existence of a two-parametric family of exactly solvable Hamiltonians, which contains the Abraham-Moses potentials as a particular case.
Solving eigenproblem of the Laplacian matrix of a fully connected weighted graph has wide applications in data science, machine learning, and image processing, etc. However, this is very challenging because it involves expensive matrix operations. Here, we propose an efficient quantum algorithm to solve it based on a assumption that the element of each vertex and its norms can be effectively accessed via a quantum random access memory data structure. Specifically, we adopt the optimal Hamiltonian simulation technique based on the block-encoding framework to implement the quantum simulation of …
Institute for Advanced Study（高等研究院）；Samsung Research（三星研究院）；Korea Institute for Advanced Study（韩国高等研究院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）
We explore the effectiveness of variational quantum circuits in simulating the ground states of quantum many-body Hamiltonians. We show that generic high-depth circuits, performing a sequence of layer unitaries of the same form, can accurately approximate the desired states. We demonstrate their universal success by using two Hamiltonian systems with very different properties: the transverse field Ising model and the Sachdev-Ye-Kitaev model. The energy landscape of the high-depth circuits has a proper structure for the gradient-based optimization, i.e. the presence of local extrema -- near any…
Many numerical methods, such as tensor network approaches including density matrix renormalization group calculations, have been developed to calculate the extreme/ground states of quantum many-body systems. However, little attention has been paid to the central states, which are exponentially close to each other in terms of system size. We propose a Delta-Davidson (DELDAV) method to effciently find such interior (including the central) states in many-spin systems. The DELDAV method utilizes Delta filter in Chebyshev polynomial expansion combined with subspace diagonalization to overcome the n…
It is shown that the eigenproblem of any $2\times 2$ matrix Hamiltonian with discrete eigenvalues is involved with a supersymmetric quantum mechanics. The energy dependence of the superalgebra marks the disparity between the deduced supersymmetry and the standard supersymmetric quantum mechanics. The components of an eigenspinor are superpartners\textemdash up to a $SU(2)$ transformation\textemdash which allows to derive two reduced eigenproblems diagonalizing the Hamiltonian in the spin subspace. As a result, each component carries all information encoded in the eigenspinor. We also discuss t…
We have subjected the planar pendulum eigenproblem to a symmetry analysis with the goal of explaining the relationship between its conditional quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces, established in our earlier work [Frontiers in Physical Chemistry and Chemical Physics 2, 1-16, (2014)]. The present analysis revealed that this relationship can be traced to the structure of the tridiagonal matrices representing the symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many more -- forty in total to be exact -- analytic solutions. Furthermor…
A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schroëdinger equation in momentum representation on the case of deformed Heisenberg algebra with minimal length. Assuming that the kernel of potential energy operator do not change in the case of deformation, we obtain exact solution of eigenproblem of a particle in delta potential as well as double delta potential. Particle in Coulomb like potential is revisited and the problem of inversibility and hermicity of inverse coordinate operator is sol…
We investigate a generalisation of the Rabi system in the Bargmann-Fock representation. In this representation the eigenproblem of the considered quantum model is described by a system of two linear differential equations with one independent variable. The system has only one irregular singular point at infinity. We show how the quantisation of the model is related to asymptotic behaviour of solutions in a vicinity of this point. The explicit formulae for the spectrum and eigenfunctions of the model follow from an analysis of the Stokes phenomenon. An interpretation of the obtained results in …
One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian, as this guarantees that the eigenvalues are real. Recently there has been an interest in asking if $H^\dagger = H$ is a necessary condition, and has lead to the development of PT-symmetric quantum mechanics. This note shows that any finite physically acceptable non-Hermitian Hamiltonian is equivalent to doing ordinary quantum mechanics in a non-orthogonal basis. In particular, this means that there is no experimental distinction between PT-symmetric quantum mechanics and ordinary quantum mechanics for finite syste…
I present a short overview of my recent achievements on the Bohmian interpretation of relativistic quantum mechanics, quantum field theory and string theory. This includes the relativistic-covariant Bohmian equations for particle trajectories, the problem of particle creation and destruction, the Bohmian interpretation of fermionic fields and the intrinsically Bohmian quantization of fields and strings based on the De Donder-Weyl covariant canonical formalism.
Based on some results on reparmetrisation of time in Hamiltonian path integral formalism, a pseudo time formulation of operator formalism of quantum mechanics is presented. Relation of reparametrisation of time in quantum with super symmetric quantum mechanics is established. We show how some important concepts such as shape invariance and tools like isospectral deformation appear in pseudo time quantum mechanics.
On the basis of a 5-dimensional form of space-time transformations non-relativistic quantum mechanics is reformulated in a manifestly covariant manner. The resulting covariance resembles that of the conventional relativistic quantum mechanics.
All measurable predictions of classical mechanics can be reproduced from a quantum-like interpretation of a nonlinear Schrodinger equation. The key observation leading to classical physics is the fact that a wave function that satisfies a linear equation is real and positive, rather than complex. This has profound implications on the role of the Bohmian classical-like interpretation of linear quantum mechanics, as well as on the possibilities to find a consistent interpretation of arbitrary nonlinear generalizations of quantum mechanics.
With a q-deformed quantum mechanical framework, features of the uncertainty relation and a novel formulation of the Schrödinger equation are considered.
We describe a scheme for constructing quantum mechanics in which a quantum system is considered as a collection of open classical subsystems. This allows using the formal classical logic and classical probability theory in quantum mechanics. Our approach nevertheless allows completely reproducing the standard mathematical formalism of quantum mechanics and identifying its applicability limits. We especially attend to the quantum state reduction problem.
Notes from the lectures by the author at the 7th Jerusalem Winter School 1990 on Quantum Cosmology and Baby Universes. The lectures covered quantum mechanics for closed systems like the universe, generalized quantum mechanics, time in quantum mechanics, the quantum mechanics spacetime, and practical quantum cosmology. References have not been updated.
Discrete quantum mechanics is here defined to be a quantum theory of wave functions defined on integers P_i and Q_i, while canonical quantum mechanics is assumed to be based on wave functions on the real numbers, R^n. We study reversible mappings from the position operators q_i and their quantum canonical operators p_i of a canonical theory, onto the discrete, commuting operators Q_i and P_i. In this paper we are particularly interested in harmonic oscillators. In the discrete system, these turn into deterministic models, which is our motivation for this study. We regard the procedure worked o…
Standard Quantum Mechanics (QM) predicts an anti-intuitive fenomenon here referred to as "quantum autoscattering", which is excluded by Bhomian Mechanics. The scheme of a gedanken experiment testing the QM prediction is briefly discussed.
In his recent paper arXiv:1312.7738, the author discussed $J$-Hermitian quantum mechanics and showed that $PT$-symmetric quantum mechanics is essentially $J$-Hermitian quantum mechanics. In this paper, the author discusses finite $J$-Hermitian quantum mechanics which is derived naturally from its continuum one and its relationship with finite $PT$-symmetric quantum mechanics.
In this paper we develop a discussion about PT Symmetric Quantum Mechanics, working with basics elements of this theory. In a simple case of two body system, we developed the Quantum Brachistochrone problem. Comparing the results obtained through the PT Symmetric Quantum Mechanics with that ones obtained using the standard formalism, we conclude that this new approach is not able to reveal any new effect.
This is an introductory chapter of the book in progress on quantum foundations and incompleteness of quantum mechanics. Quantum mechanics is represented as statistical mechanics of classical fields.
In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative point of view in which quantum mechanics emerges as a limiting case of classical mechanics in which the classical system is decoupled from its environment.
Quantum computation teaches us that quantum mechanics exhibits exponential complexity. We argue that the standard scientific paradigm of "predict and verify" cannot be applied to testing quantum mechanics in this limit of high complexity. We describe how QM can be tested in this regime by extending the usual scientific paradigm to include {\it interactive experiments}.
Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.
We discuss foundation of quantum mechanics (interpretations, superposition, principle of complementarity, locality, hidden variables) and quantum information theory.
The correspondence of a new form of quantum mechanics based on a quantum version of the action principle, which was proposed earlier [arXiv:0807.3508], with the ordinary quantum mechanics is established. New potentialities of the quantum action principle in the interpretation of quantum mechanics are considered.
It is shown that the apparent incompatibility of Bohmian mechanics with standard quantum mechanics, found by Akhavan and Golshani quant-ph/0305020, is an artefact of the fact that the initial wavefunction they use, being proportional to a $δ$-function, is not a regular wavefunction.
A motivation is given for expressing classical mechanics in terms of diagonal projection matrices and diagonal density matrices. Then quantum mechanics is seen to be a simple generalization in which one replaces the diagonal real matrices with suitable Hermitian matrices.
In this comment we answer to the recent critique of our article [arXiv:2208.11793] about Relational Quantum Mechanics (RQM) by Aurélien Drezet [arXiv:2209.01237]. Here we point out that our critical analysis of RQM was precisely based on the most recent formulation of RQM, and that the theses found in the critique are based on neither RQM assumptions nor on our arguments.
There is a certainty that the modern (Copenhagen's) interpretation of quantum mechanics is correct. However, the some physicist had the opinion that the modern quantum mechanics is a phenomenological theory. The suggested theory is the new quantum mechanics interpretation that is entirely according to the modern interpretation and gives a number of results, which naturally explain the postulates of the modern quantum mechanics.
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket, and a quasidensity operator. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Classical mechanics can now be viewed as a deformation of quantum mechanics. The forms of semiquantum approximations to classical mechanics are indicated.
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general…
It is shown that adopting the \emph{Quantum Field} ---extended entity in space-time build by dynamic appearance propagation and annihilation of virtual particles--- as the primary ontology the astonishing features of quantum mechanics can be rendered intuitive. This interpretation of quantum mechanics follows from the formalism of the most successful theory in physics: quantum field theory.
Supmech, an algebraic scheme of mechanics integrating noncommutative symplectic geometry and noncommutative probability, subsumes quantum and classical mechanics and permits consistent treatment of interaction of quantum and classical systems. Quantum measurements are treated in this framework; the von Neumann reduction rule (generally postulated) is derived and interpreted in physical terms.
It is argued on the basis of certain mathematical characteristics that classical mechanics is not constitutionally suited to accomodate consciousness, whereas quantum mechanics is. These mathematical characteristics pertain to the nature of the information represented in the state of the brain, and the way this information enters into the dynamics.
Two questions are suggested as having priority when trying to bring together Quantum Mechanics and General Relativity. Both questions have a scope which goes well beyond Physics, and in particular Quantum Mechanics and General Relativity.
The concept of number is fundamental to the formulation of any physical theory. We give a heuristic motivation for the reformulation of Quantum Mechanics in terms of non-standard real numbers called Quantum Real Numbers. The standard axioms of quantum mechanics are re-interpreted. Our aim is to show that, when formulated in the language of quantum real numbers, the laws of quantum mechanics appear more natural, less counterintuitive than when they are presented in terms of standard numbers.
Many-Worlds quantum mechanics differs from standard quantum mechanics in that in Many-Worlds, the wave function is a relative density of universes in the multiverse amplitude rather than a probability amplitude. This means that in Many-Worlds, the Born frequencies are approached rather than given a priori. Thus in Many-Worlds the rate of approach to the final frequencies can be calculated and compared with observation. I use Many-Worlds to derive the rate of approach in the double slit experiment, and show that it agrees with observation. Standard quantum theory has never been used to derive a…
The quantum mechanical probability densities are compared with the probability densities treated by the theory of random variables. The relevance of their difference for the interpretation of quantum mechanics is commented.
In this paper we discuss and analyse the idea of trying to see (non-relativistic) quantum mechanics as a ``space-time statistical mechanics'', by using the classical statistical mechanical method on objective microscopic space-time configurations. It is argued that this could perhaps be accomplished by giving up the assumption that the objective ``state'' of a system is independent of a future measurement performed on the system. This idea is then applied in an example of quantum state estimation on a qubit system.
It is shown that probabilistic treatment of quantum mechanics can be coordinated with causality of all physical processes. The physical interpretation of quantum-mechanical phenomena such as process of measurement and collapse of quantum state is given.
The basics of the Wigner formulation of Quantum-Mechanics and few related interpretational issues are presented in a simple language. This formulation has extensive applications in Quantum Optics and in Mixed Quantum-Classical formulations.
We bring together two topics (quantum mechanics and time passage) with the goal of clarifying questions about each. Specifically, we claim that the formalism of quantum mechanics provides an answer to the question: "What is time passage?".
The star product formalism has proved to be an alternative formulation for nonrelativistic quantum mechanics. We want introduce here a covariant star product in order to extend the star product formalism to relativistic quantum mechanics in the proper time formulation.
A necessary and natural change in our application of quantum mechanics to separated systems is shown to reconcile quantum mechanics and local realism. An analysis of separation and localization justifies the proposed change in application of quantum mechanics. An important EPRB experiment is reconsidered and it is seen that when it is correctly interpreted it supports local realism. This reconciliation of quantum mechanics with local realism allows the axiom sets of quantum mechanics, probability, and special relativity to be joined in a consistent global axiom set for physics.
The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kaehler manifold. The classical principles of Quantum Mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non linear geometrical context.
