The Schrodinger equation for non-relativistic quantum systems is derived from some classical physics axioms within an ensemble hamiltonian framework. Such an approach enables one to understand the structure of the equation, in particular its linearity, in intuitive terms. Furthermore it allows for a physically motivated and systematic investigation of potential generalisations which are briefly discussed.
In this paper, three plausible axioms together with two definitions are employed to build an axiomatic framework, and then with the help of the Dirac formalism, it is demonstrated that the time-dependent Schrödinger wave equation is no longer a postulate for the whole theory, but a theorem. Subsequently, a proof for the theorem is presented. The result implies that the other remaining axioms of the picture essentially involve the Schrödinger equation, and this consequence lets the whole theory become stronger.
A fundamental problem in quantum information is to describe efficiently multipartite quantum states. An efficient representation in terms of graphs exists for several families of quantum states (graph, cluster, stabilizer states), motivating us to extend this construction to other classes. We introduce an axiomatic framework for mapping graphs to quantum states of a suitable physical system. Starting from three general axioms we derived a rich structure which includes and generalizes several classes of multipartite entangled state, like graph/stabilizer states, Gaussian cluster states, quantum…
Since its inception, many physicists have seen in quantum mechanics the possibility, if not the necessity, of bringing cognitive aspects into the play, which were instead absent, or unnoticed, in the previous classical theories. In this article, we outline the path that led us to support the hypothesis that our physical reality is fundamentally conceptual-like and cognitivistic-like. However, contrary to the 'abstract ego hypothesis' introduced by John von Neumann and further explored, in more recent times, by Henry Stapp, our approach does not rely on the measurement problem as expressing a p…
Axiomatic quantum field theory (QFT) provides a rigorous mathematical foundation for QFT, and it is the basis for proving some important general results, such as the well-known spin-statistics theorem. Free-field QFT meets the axioms of axiomatic QFT, showing they are consistent. Nevertheless, even after more than 50 years, there is still no known non-trivial theory of quantum fields with interactions in four-dimensional Minkowski spacetime that meets the same axioms. This paper provides a similar axiomatic basis for parameterized QFT, in which an invariant, fifth path parameter is added to th…
The definition of 'classical state', and how it was used in earlier work to prove a decomposition theorem internally in the language of State Property Systems, presupposes as an additional datum an orthocomplementation on the property lattice of a physical system. In this paper we argue on the basis of the epsilon-model on the Poincare sphere that a notion of 'topologicity' for states can be seen as an alternative (operationally foundable) classicality notion in the absence of an orthocomplementation, and compare it to the known and operationally founded concept of classicality.
A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation-dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a markovian semi-classical approach, due to Bedeaux and Mazur, from a non-markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation-dissipation theorem is seen to be incompatible with a markov…
The wavefunction of a particle is obtained from its intermediate states and interaction processes considered as happening concurrently. When the interaction is described by a potential, the energy of the particle is equal to its total kinetic plus potential energies. We derive the Schrodinger and Dirac equations as the unique conditions the wavefunction must satisfy at each point in order to fulfill the corresponding energy equation. PACS numbers: 03.65.-w. Keywords: Schrodinger equation; Dirac equation; Quantum dynamics.
We propose four information-theoretic axioms for the foundations of statistical mechanics in general physical theories. The axioms---Causality, Purity Preservation, Pure Sharpness, and Purification---identify a class of theories where every mixed state can be modelled as the marginal of a pure entangled state and where every unsharp measurement can be modelled as a sharp measurement on a composite system. This class of theories, called sharp theories with purification, includes quantum theory both with complex and real amplitudes, as well as a suitable extension of classical probability theory…
Ionicioiu and Spiller [Phys. Rev. A 85, 062313 (2012)] have recently presented an axiomatic framework for mapping graphs to quantum states of a suitable physical system. Based on their study, we first extend the axiomatic framework to hypergraphs by means of modifying its axioms and consistency conditions. Then we use the axiomatic approach to encode hypergraphs into a new family of quantum states, called the hypergraph states. Moreover, we also try to do the followings: (i) to show that real equally weighted states, which occur in Grover and Deutsch-Joza algorithms, are equivalent to hypergra…
In this paper we present exact solutions of Schrodinger equation (SE) for a class of non central physical potentials within the formalism of position-dependent effective mass. The energy eigenvalues and eigenfunctions of the bound-states for the Schrodinger equation are obtained analytically by means of asymptotic iteration method (AIM) and easily calculated through a new generalized decomposition of the effective potential allowing easy separation of the coordinates. Our results are in excellent agreement with other works in the literature.
The central limit theorem has been found to apply to random vectors in complex Hilbert space. This amounts to sufficient reason to study the complex valued Gaussian, looking for relevance to quantum mechanics. Here we show that the Gaussian, with all terms fully complex, acting as a propagator, leads to Schrodinger nonrelativistic equation including scalar and vector potentials, assuming only that the norm is conserved. No prior physical laws need to be postulated. It thereby presents as a process of irregular motion analogous to the real random walk but executed under the rules of the complex…
Here we present a transformation that maps the Schrodinger equation of quantum mechanics to the incompressible Euler equations of fluid mechanics. The transformation provides a wave solution and a potential function based on fluid properties that satisfy the Schrodinger equation given that the fluid velocity potential and pressure satisfy the Euler equations. Interestingly, in our transformation, the equivalent of quantum potential becomes the physical surface tension. This is contrary to the Madelung transformation that maps the Schrodinger equation to the compressible Euler equations where t…
We obtain novel nonlinear Schrödinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.
Lagrangian formulation of quantum mechanical Schrödinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein-Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrödinger equation.
Symmetry shares an entwined history with the structure of physical theory. We propose a consequence of symmetry towards the axiomatic derivation of Hilbert space quantum theory. We introduce the notion of information symmetry (IS) and show that it constraints the state-space structure in any physical theory. To this end, we study the minimal error binary state discrimination problem in the framework of generalized probabilistic theories. A theory is said to satisfy IS if the probability of incorrectly identifying each of two randomly prepared states is same for both the states. It is found tha…
Any time-dependent solution of Schrödinger equation may be always correlated to a solution of Hamilton equations or to a statistical combination of their solutions; only the set of corresponding solutions is somewhat smaller (due to existence of quantization). There is not any reason to the physical interpretation according to Copenhagen alternative as Bell's inequalities are valid in the classical physics only (and not in any alternative based on Schrödinger equation). The advantage of Schrödinger equation consists then in that it enables to represent directly the time evolution of a statisti…
In this article we present a detailed description of an electron in a uniform magnetic field evolving under the Schrödinger equation using ladder operators. Based on this analysis, we describe the same physical system using the Dirac equation, known from relativistic quantum mechanics. The main differences between these two quantum mechanical approaches are discussed and we observe specifically how the relativistic phenomena modify the description of this particular quantum system.
A new non-perturbative approach is proposed to solve time-independent Schrödinger equations in quantum mechanics and chromodynamics (QCD). It is based on the homotopy analysis method (HAM), which was developed by the author for highly nonlinear equations since 1992 and has been widely applied in many fields. Unlike perturbative methods, this HAM-based approach has nothing to do with small/large physical parameters. Besides, convergent series solution can be obtained even if the disturbance is far from the known status. A nonlinear harmonic oscillator is used as an example to illustrate the val…
A concise review of the derivation of the Born rule and Schrödinger equation from first principles is provided. The starting point is a formalization of fundamental notions of measurement and composition, leading to a general framework for physical theories known as the positive formalism. Consecutively adding notions of spacetime, locality, absolute time and causality recovers the well established convex operational framework for classical and quantum theory. Requiring the partially ordered vector space of states to be an anti-lattice one obtains quantum theory in its standard formulation. Th…
We introduce an axiomatic approach for characterizing quantum conditional entropy. Our approach relies on two physically motivated axioms: monotonicity under conditional majorization and additivity. We show that these two axioms provide sufficient structure that enable us to derive several key properties applicable to all quantum conditional entropies studied in the literature. Specifically, we prove that any quantum conditional entropy must be negative on certain entangled states and must equal -log(d) on dxd maximally entangled states. We also prove the non-negativity of conditional entropy …
According to classical non-relativistic Schrödinger equation, any local perturbation of wave function instantaneously affects all infinite region, because this equation is of parabolic type, and its solutions demonstrate infinite speed of perturbations propagation. From physical point of view, this feature of Schrödinger equation solutions is questionable. According to relativistic quantum mechanics, the perturbations propagate with speed of light. However when appropriate mathematical procedures are applied to Dirac relativistic quantum equation with finite speed of the wave function perturba…
Physics-Informed Evolution: An Evolutionary Framework for Solving Quantum Control Problems Involving the Schrödinger Equation
University of Science and Technology of China（中国科学技术大学）；Nankai University（南开大学）；National University of Singapore（新加坡国立大学）；Wenzhou University（温州大学）；Hanyang University ERICA Campus（汉阳大学ERICA校区）
