This paper presents a global optimization approach to quantum mechanics, which describes the most fundamental dynamics of the universe. It suggests that the wave-like behavior of (sub)atomic particles could be the critical characteristic of a global optimization method deployed by nature so that (sub)atomic systems can find their ground states corresponding to the global minimum of some energy function associated with the system. The classic time-independent Schrodinger equation is shown to be derivable from the global optimization method to support this argument.
This paper presents a global optimization approach to quantum mechanics, which describes the most fundamental dynamics of the universe. It suggests that the wave-like behavior of (sub)atomic particles could be the critical characteristic of a global optimization method deployed by nature so that (sub)atomic systems can find their ground states corresponding to the global minimum of some energy function associated with the system. The classic time-independent Schrodinger equation is shown to be derivable from the global optimization method to support this argument.
One of the greatest scientific achievements of physics in the 20th century is the discovery of quantum mechanics. The Schrodinger equation is the most fundamental equation in quantum mechanics describing the time-based evolution of the quantum state of a physical system. It has been found that the time-independent version of the equation can be derived from a general optimization algorithm. Instead of arguing for a new interpretation and possible deeper principle for quantum mechanics, this paper elaborates a few points of the equation as a general global optimization algorithm. Benchmarked against randomly generated hard optimization problems, this paper shows that the algorithm significantly outperformed a classic local optimization algorithm. The former found a solution in one second with a single trial better than the best one found by the latter around one hour after one hundred thousand trials.
Quantum mechanics is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Quantum mechanics is inescapable. Nearly all fields of modern physics, from elementary particles to the big bang, from semiconductors to solar energy cells, are related to quantum physics. Particles have characteristic trait of both particles and waves. The Schrodinger equation is the basic equation of quantum mechanics. The Schrodinger equation was established based on the quantization of energy conservation using quantum operators and the de Broglie relation. Wave function plays an important role in quantum mechanics. The probability interpretation assumes that the norm of the wave function measures the probability of the system in a particular coordinate or representation, in which the total probability is normalized.
The dual Schrödinger equation is defined as replacing the imaginary number i by —1 in the original one. This paper shows that the dual equation shares the same stationary states as the original one. Different from the original one, it explicitly defines a dynamic process for a system to evolve from any state to lower energy states and eventually to the lowest one. Its power as a global optimization algorithm might be used by nature for constructing atoms and molecules. It shall be interesting to verify its existence in nature.
The dual Schrödinger equation is defined as replacing the imaginary number i by —1 in the original one. This paper shows that the dual equation shares the same stationary states as the original one. Different from the original one, it explicitly defines a dynamic process for a system to evolve from any state to lower energy states and eventually to the lowest one. Its power as a global optimization algorithm might be used by nature for constructing atoms and molecules. It shall be interesting to verify its existence in nature.
In this work we present a novel method for global optimization, exploiting the mathematics of quantum mechanics, and in particular the tunnelling phenomenon. We consider a quantum mechanical system with a single particle in a potential specified by the cost function of our optimization problem. We assume the groundstate of the system is localised to the global minimum of the potential function, a condition which can be assured by choosing the particle mass sufficiently large. We then approximate this groundstate using a variational approach to optimise an expansion in terms of a truncated basis set of harmonic oscillator wavefunctions. We show how to encode integer factoring and travelling salesman problems in terms of finding the global minimum of a quartic polynomial cost function. We demonstrate our quantum algorithm on oneand two-dimensional toy problems, and apply it to factoring biprimes with 10 bits. Introduction The state of the art in global optimization for general problems is simulated annealing and its derivatives. The algorithm can be viewed in the physical context of a classical thermal system. This inspires the novel approach presented here, in which we draw upon intuition from non-classical physical systems which obey quantum mechanics (QM), and in particular exhibit the phenomenon of quantum tunnelling. In QM, a potential function V (x) describes the energy terrain experienced by a particle, which may contain multiple minima separated by barriers. Combined with a term describing the kinetic energy of the particle, it defines the Hamiltonian operator: Ĥ = − 2 2m ∇2 + V (1) QM captures all available information on the state of the particle in a (complex) wave function ψ(x). For instance, |ψ(x)|2 is the probability distribution over the position x of the particle. To obtain ψ(x) one solves the eigenvalue problem Ĥψ = Eψ (2) known as the Schroedinger equation, where E is the energy. The solution is a set of wave functions termed eigenstates, and a matching set of energies. The eigenstate with the lowest energy is the ground state, ψ. By choosing the particle’s mass sufficiently large, the probability density associated with the ground state wave function becomes localized to the global minimum of the potential. Hence, a given optimization problem involving the minimization of a cost function can be restated as the problem of finding the ground state of the QM system with a potential specified by that cost function. In particular, computationally ‘hard’ integer problems, including factoring [1] and the travelling salesman problem, can be cast as QM problems involving a quartic polynomial potential, [3]. However, solving the Schroedinger equation exactly for these potentials is generally an analytically intractable problem. In our approach we approximate the groundstate of such a system using a method based on perturbation theory.
New insight on the principles of the quantum physics development is given. The quite different behavior of quantum microworld mechanical systems in comparison with classical mechanical macroworld systems is attributed to the drastical change of the role of the gravitation field on the dynamics of microworld mechanical systems, for which it seems to be vanishing in com- parison with the role of the gravitation field on the dynamics of macroworld mechanical systems, for which it is decisive. The conclusion on the status of the second main postulate of quantum mechanics is given. Its formulation in all textbooks has to be represented in the form of the proved statement, since the hypothesis of Schrödinger on the existence of the field scalar function, being to be observable quantity, just charge density, is strictly mathematically proved. It is shown, that the field scalar function, being to be the solution of the corresponding nonstationary in general case Schrödinger equation and being to be the function the only of coordinates and time, actually describes the state of the atomic system, more strictly its corpuscular aspect. The atomic system is considered from the positions of the field theory, that is, it is the association of elementary particles with corresponding fields. The given fields being to be the media for the particles’ propagation are responsible for the wave aspect in the characteristic of atomic systems. The wave aspect is described by the independent scalar wave-function which was also introduced in quantum physics by Schrödinger, althogh the given fact seems to be unknown to the wide circle of the readers. The second main postulate being to be mathematically strictly grounded in Schrödinger formulation of quantum nechanics, in the popular probabilistic form of quantum nechanics cannot be proved for the general case, although the probabilistic theatise, proposed by Born is true in a number of special cases, quite correctly indicated by Dirac. The possible ways of the development of quantum theory, based on clear understanding of the origin of corpuscular-wave dualism and based on the fundamental both now mathemtically grounded main postulates are analysed.