Formulation Of Quantum Mechanics Research Articles

Noise Transfer Approach to GKP Quantum Circuits.

The choice between the Schrödinger and Heisenberg pictures can significantly impact the computational resources needed to solve a problem, even though they are equivalent formulations of quantum mechanics. Here, we present a method for analysing Bosonic quantum circuits based on the Heisenberg picture which allows, under certain conditions, a useful factoring of the evolution into signal and noise contributions, similar way to what can be achieved with classical communication systems. We provide examples which suggest that this approach may be particularly useful in analysing quantum computing systems based on the Gottesman-Kitaev-Preskill (GKP) qubits.

A derivation of the Schrödinger equation from Feynman’s path-integral formulation of quantum mechanics

The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.

Moyal deformation of the classical arrival time

The quantum time of arrival (TOA) problem requires the statistics of measured arrival times given only the initial state of a particle. Following the standard framework of quantum theory, the problem translates into finding an appropriate quantum image of the classical arrival time TC(q,p), usually in operator form T̂. In this paper, we consider the problem anew within the phase space formulation of quantum mechanics. The resulting quantum image is a real-valued and time-reversal symmetric function TM(q,p) in formal series of ℏ2 with the classical arrival time as the leading term. It is obtained directly from the Moyal bracket relation with the system Hamiltonian and is hence interpreted as a Moyal deformation of the classical TOA. We investigate its properties and discuss how it bypasses the known obstructions to quantization by showing the isomorphism between TM(q,p) and the rigged Hilbert space TOA operator constructed in Pablico and Galapon [Eur. Phys. J. Plus 138, 153 (2023)], which always satisfy the time-energy canonical commutation relation for arbitrary analytic potentials. We then examine TOA problems for a free particle and a quartic oscillator potential as examples.

Real-space diffusion theory from quantum mechanics using analytic continuation

We show that a physical correspondence between Brownian motion and quantum mechanics can be established by formal analytic continuation if Wick rotation (t→it) is replaced by the mirror symmetric, complex conjugate time transformation t→±it. Invariance of the square modulus of the wave function under this transformation reveals a tight connection between Born's interpretation of the probability density and the proper time t2=(−it)(+it), as being both the result of a time symmetry breaking in the informational content of the microscopic (quantum) world, which takes place as we move to a macroscopic level. We find that under the transformation t→±it, the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle. From the present analysis, Nelson's osmotic velocity arises naturally and the quantum phase function admits a classical interpretation in terms of a continuous (scalar) potential field that influences the motion of the particle. The maximum phase variation defines the direction of the osmotic and current velocity. Whereas the paper focuses on the correlation between quantum mechanics and Ficks's second law for a constant diffusion coefficient, a discussion is also provided on the possibility of a quantum mechanical formulation for anomalous diffusion, where the diffusion coefficient is a function of space and/or time.

Constructing potential energy surface for carbon-chain containing systems using the radial angular network with gradual expansion method.

Investigating molecular excitation induced by collisions requires the prior determination of accurate analytical potential energy surfaces for the colliding partners. For carbon-chain molecules, such as cyanopolyynes, this has been a longstanding challenge, resulting in the absence of rate coefficients for HC5N, HC7N, HC9N, and others, induced by collisions with He. To overcome this bottleneck, we introduce a new approach: the Radial Angular Network with Gradual Expansion (RANGE). This method jointly connects the construction of abinitio interaction potentials with the determination of their analytical forms. We use the HC3N-He molecular complex as a reference to assess the reliability of our method, given that its analytical potential has been derived using various methods. Additionally, we apply the RANGE approach to construct the analytical representation of the interaction potential for HC5N-He and HC7N-He. The analysis of the analytical potentials reveals three systematic trends: (i) the anisotropy increases with the length of the carbon chain, (ii) the number of local minima correlates with the number of carbon atoms, and (iii) the shallowest local minimum is consistently located at ∼30 cm-1 below the dissociation limit of the complex. Using the time-independent quantum mechanical close-coupling formalism, we briefly estimate the propensity rules governing the excitation of HC3N, HC5N, and HC7N induced by collisions with He. Consequently, the three collisional systems exhibit the same propensity rule, favoring Δj = 2 transitions.

On testimony in scenarios with Wigner and Friend

The paper constructs a semi-formal language suited to the analysis of Wigner’s Friend scenarios: it represents an epistemic notion of rational beliefs and perspectives, to accommodate the insights of perspectival interpretations of quantum mechanics. The language is then used to analyze a paradox put forward by Frauchiger and Renner (Nat Commun, 9(1):3711, 2018). Their argument is presented as a semi-formal derivation with specified rules of reasoning. These rules bear an affinity to some of the cherished tenets of epistemology and we argue that they are valid (one universally, and the other in experimental contexts). Since our proof is a reductio, it leaves a choice which premises are responsible for a contradiction. Our first choice is a step that appears incorrect from the point of view of the universal unitary evolution as well as the view that every measurement induces a collapse of a measured system’s state. Our second choice, brought to view by the paper’s attention to perspectives and epistemology, points to a step reporting the transmission of beliefs (testimony) about measurement results. We argue that testimony is not licensed by quantum mechanical formalism; we discuss some recent attempts to save the cogency of testimony in the context of quantum measurements.

Quantum Mechanics and Inclusive Materialism

Since its inception, the intricate mathematical formalism of quantum mechanics has empowered physicists to describe and predict specific physical events known as quantum processes. However, this success in probabilistic predictions has been accompanied by a profound challenge in the ontological interpretation of the theory. This interpretative complexity stems from two key aspects. Firstly, quantum mechanics is a fundamental theory that, so far, is not derivable from any more basic scientific theory. Secondly, it delves into a realm of invisible phenomena that often contradicts our intuitive and commonsensical notions of matter and causality. Despite its notorious difficulties of interpretation, the most widely accepted set of views of quantum phenomena has been known as the Copenhagen interpretation since the beginning of quantum mechanics. According to these views, the correct ontological interpretation of quantum mechanics is incompatible with ontological realism in general and with philosophical materialism in particular. Anti-realist and anti-materialist interpretations of quantum matter have survived until today. This paper discusses these perspectives, arguing that materialistic interpretations of quantum mechanics are compatible with its mathematical formalism, while anti-realist and anti-materialist views are based on wrong philosophical assumptions. However, although physicalism provides a better explanation for quantum phenomena than idealism, its downward reductionism prevents it from accounting for more complex forms of matter, such as biological or sociocultural systems. Thus, the paper argues that neither physicalism nor idealism can explain the universe. I propose then a non-reductionistic form of materialism called inclusive materialism. The conclusion is that the acknowledgment of the qualitative irreducibility of ontological emergent levels above the purely physical one does not deny philosophical materialism but enriches it.

Quantum Big-Bounce as a phenomenology of RQM in the Mini-superspace

We investigate the emergence of a quantum Big-Bounce in the context of an isotropic Universe, filled by a self-interacting scalar field, which plays the role of a physical clock. The bouncing cosmology is the result of a scattering process, driven by the scalar field potential, which presence breaks down the frequency separation of the Wheeler-DeWitt equation, treated in strict analogy to a relativistic quantum system. Differently from previous analyses, we consider a really perturbative self-interaction potential, affecting the dynamics in a finite range of the time labeled by the scalar clock (and in particular we remove the divergent character previously allowed). The main result of the present analysis is that, when the Relativistic Quantum Mechanics formalism is properly implemented in the Mini-superspace analogy, the probability amplitude for the bounce is, both in the standard and polymerized case, characterized by a maximum in correspondence of the quasi-classical condition of a Universe minimum volume.

Revisiting Dynamics of Quantum Causal Structures-When Can Causal Order Evolve?

Recently, there has been substantial interest in studying the dynamics of quantum theory beyond that of states, in particular, the dynamics of channels, measurements, and higher-order transformations. Castro-Ruiz et al. pursues this using the process-matrix formalism, together with a definition of the possible dynamics of such process matrices, and focusing especially on the question of evolution of causal structures. One of its major conclusions is a strong theorem saying that within the formalism, under continuous and reversible transformations, the causal order between operations must be preserved. Our result here challenges that of Castro-Ruiz et al.: if one is to take into account a full picture of the physical evolution of operations within the standard quantum-mechanical formalism, then the conclusion of Castro-Ruiz et al. does not hold. That is, we show that under certain continuous and reversible dynamics, the causal order between operations is not necessarily preserved. We moreover identify and analyse the root of this apparent contradiction, specifically, that the commonly accepted and widely applied framework of higher-order processes, whilst mathematically sound, is not always appropriate for drawing conclusions on physical dynamics. Finally, we show how to reconcile the elements of the whole picture following the intuition based on entanglement processing by local operations and classical communication.

℘ -adic quantum mechanics, the Dirac equation, and the violation of Einstein causality

This article studies the breaking of the Lorentz symmetry at the Planck length in quantum mechanics. We use three-dimensional ℘ -adic vectors as position variables, while the time remains a real number. In this setting, the Planck length is 1/℘ , where ℘ is a prime number, and the Lorentz symmetry is naturally broken. In the framework of the Dirac-von Neumann formalism for quantum mechanics, we introduce a new ℘ -adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. The discreteness of the ℘ -adic space imposes substantial restrictions on the solutions of the new equation. This equation admits localized solutions, which is impossible in the standard case. We show that an isolated quantum system whose evolution is controlled by the ℘ -adic Dirac equation does not satisfy the Einstein causality, which means that the speed of light is not the upper limit for the speed at which conventional matter or energy can travel through space. The new ℘ -adic Dirac equation is not intended to replace the standard one; it should be understood as a new version (or a limit) of the classical equation at the Planck length scale.

Quantum mechanical operator Touchard polynomials studied by virtue of operators’ normal ordering and Weyl ordering

In this paper, we propose quantum mechanical operator formalism for Touchard polynomials whose generating function is [Formula: see text] That is replacing [Formula: see text] by [Formula: see text] where [Formula: see text] is the number operator, and using the method of integration within ordered product we find that [Formula: see text] is just the normal ordering form [Formula: see text]. Then by virtue of the Weyl ordering form of quantum mechanical operator, we also introduce a new special polynomial whose generating function is [Formula: see text] With use of the Weyl ordering form of [Formula: see text] we prove [Formula: see text] where [Formula: see text] denotes Weyl ordering. It seems that quantum mechancal operator formalism presents a new and simpler approach for generalizing Touchard polynomial theory.

A Causal Net approach to Relativistic Quantum Mechanics and Gravitation

In this paper we discuss a causal network model to describe quantum mechanics and gravittation, where space–time is built solely from point events and connecting probabilities. A causal net provides a representation which combines the quantum complementary features of both partticles and waves as each vertex on the causal net represents a possible point event or particle observation. Simultaneity on the causal net is defined by hyperplanes of equivalent proper time. The causal net model when constructed in Minkowksi space–time is shown to lead to a formulation for relativistic quantum mechanics including the Dirac equation. In a curved Riemann space–time the causal paths of the causal net are geodesics and in the local Lorentz frame the net probabilities and the Dirac formalism are preserved. The variation of space–time density of events in the causal paths modifies the metric and provides a space–time curvature leading to the Hilbert action associated with general relativity. Consideration of the Schwarzchild metric shows that in classical limit the perturbation in the density of possible events is proportional to the Newtonian gravitational potential.

Empirical adequacy of the time operator canonically conjugate to a Hamiltonian generating translations

To admit a canonically conjugate time operator, the Hamiltonian has to be a generator of translations (like the momentum operator generates translations in space), so its spectrum must be unbounded. But the Hamiltonian governing our world is thought to be bounded from below. Also, judging by the number of fields and parameters of the Standard Model, the Hamiltonian seems much more complicated. In this article I give examples of worlds governed by Hamiltonians generating translations. They can be expressed as a partial derivative operator just like the momentum operator, but when expressed in function of other observables they can exhibit any level of complexity. The examples include any quantum world realizing a standard ideal measurement, any quantum world containing a clock or a free massless fermion, the quantum representation of any deterministic time-reversible dynamical system without time loops, and any quantum world that cannot return to a past state. Such worlds are as sophisticated as our world, but they admit a time operator. I show that, despite having unbounded Hamiltonian, they do not decay to infinite negative energy any more than any quantum or classical world. Since two such quantum systems of the same Hilbert space dimension are unitarily equivalent even if the physical content of their observables is very different, they are concrete counterexamples to Hilbert Space Fundamentalism (HSF). Taking the observables into account removes the ambiguity of HSF and the clock ambiguity problem attributed to the Page-Wootters formalism, also caused by assuming HSF. These results provide additional motivations to restore the spacetime symmetry in the formulation of Quantum Mechanics and for the Page-Wootters formalism.

The Evolution of the Bell Notion of Beable: From Bohr to Primitive Ontology

AbstractJohn S. Bell introduced the notion of beable, as opposed to the standard notion of observable, in order to emphasize the need for an unambiguous formulation of quantum mechanics. In the paper I show that Bell formulated in fact two different theories of beables. The first is somehow reminiscent of the Bohr views on quantum mechanics but, at the same time, is curiously adopted by Bell as a critical tool against the Copenhagen interpretation, whereas the second, more mature formulation was among the sources of inspiration of the so-called Primitive Ontology (PO) approach to quantum mechanics, an approach inspired to scientific realism. In the first part of the paper it is argued that, contrary to the Bell wishes, the first formulation of the theory fails to be an effective recipe for addressing the ambiguity underlying the standard formulation of quantum mechanics, whereas it is only the second formulation that successfully paves the way to the PO approach. In the second part, I consider how the distinction between the two formulations of the Bell theory of beables fares vis-a-vis the complex relationship between the theory of beables and the details of the PO approach.

Poisson geometric formulation of quantum mechanics

We study the Poisson geometrical formulation of quantum mechanics for finite dimensional mixed and pure states. Equivalently, we show that quantum mechanics can be understood in the language of classical mechanics. We review the symplectic structure of the Hilbert space and identify its canonical coordinates. We extend the geometric picture to the space of density matrices DN+. We find it is not symplectic but admits a linear su(N) Poisson structure. We identify Casimir surfaces of DN+ and show that the space of pure states PN≡CPN−1 is one of its symplectic submanifolds which is an intersection of primitive Casimirs. We identify generic symplectic submanifolds of DN+ and calculate their dimensions. We find that DN+ is singularly foliated by the symplectic leaves of varying dimensions, also known as coadjoint orbits. We also find an ascending chain of Poisson submanifolds DNM⊂DNM+1 for 1 ≤ M ≤ N − 1. Each such Poisson submanifold DNM is obtained by tracing out the CM states from the bipartite system CN×CM and is an intersection of N − M primitive Casimirs of DN+. Their Poisson structure is induced from the symplectic structure of the bipartite system. We also show their foliations. Finally, we study the positive semi-definite geometry of the symplectic submanifold ENM consisting of the mixed states with maximum entropy in DNM.

From light polarization to quantum physics: Supporting lower secondary school students’ transition from gestalt to functional thinking

In this paper, we present a new minimal mathematical conceptual approach to quantum mechanics using light polarization for lower secondary school students with the aim of bringing students closer to the so-called quantum mechanical way of thinking. We investigated how students think about some of the basic concepts and fundamental laws and we found that certain concepts are quite well-understandable in younger grades too. We studied the introduction of the so-called state circle, which can faithfully represent quantum mechanical formalism without involving students in abstract algebraic calculations. We then categorized and analyzed students’ thoughts on the superposition principle and the lack of trajectory, finding that the concept of measurement and the lack of trajectory were problematic. We explored that younger students tend to hold gestalt-like mental models of quantum concepts, while at the same time being able to use visualizations correctly for reasoning in the quantum realm. Overall, this paper provides evidence in favor of introducing basic features of quantum mechanics as early as in lower secondary school.

Power and efficient power optimization of one-qubit Novikov quantum heat engines with an external dissipative heat leak

Abstract We optimize one-qubit Novikov quantum heat engines with a dissipative heat leak using the formalisms of open-system Quantum Mechanics and the Finite-Time Thermodynamics. We show that the leak changes the power-efficiency and the efficient-power-efficiency curves, reduces the maximum efficiency of the machine and does not change its power. We also discuss the effects of the leak on the power that is rejected from the machine to the environment. Finally, we study the high-temperature limit to check that the engine reduces in this limit to a classic Novikov heat engine.

An algebraic formulation of nonassociative quantum mechanics

We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop Gel’fand–Naimark–Segal constructions. We describe Heisenberg and Schrödinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra.

Quantum Mechanics Based on an Extended Least Action Principle and Information Metrics of Vacuum Fluctuations

We show that the formulations of non-relativistic quantum mechanics can be derived from an extended least action principle. The principle can be considered as an extension of the least action principle from classical mechanics by factoring in two assumptions. First, the Planck constant defines the minimal amount of action a physical system needs to exhibit during its dynamics in order to be observable. Second, there is constant vacuum fluctuation along a classical trajectory. A novel method is introduced to define the information metrics to measure additional observability due to vacuum fluctuations, which is then converted to an additional action through the first assumption. Applying the variational principle to minimize the total actions allows us to recover the basic quantum formulations including the uncertainty relation and the Schrödinger equation in the position representation. In the momentum representation, the same method can be applied to obtain the Schrödinger equation for a free particle while further investigation is still needed for a particle with an external potential. Furthermore, the principle brings in new results on two fronts. At the conceptual level, we find that the information metrics for vacuum fluctuations are responsible for the origin of the Bohm quantum potential. Even though the Bohm potential for a bipartite system is inseparable, the underlying vacuum fluctuations are local. Thus, inseparability of the Bohm potential does not justify a non-local causal relation between the two subsystems. At the mathematical level, quantifying the information metrics for vacuum fluctuations using more general definitions of relative entropy results in a generalized Schrödinger equation that depends on the order of relative entropy. The extended least action principle is a new mathematical tool. It can be applied to derive other quantum formalisms such as quantum scalar field theory.

Higher-order nonlocal approach to classical and quantum mechanics through non-standard Lagrangians: The case of quantum wells and the quantum states of neutron in earth’s gravitational field

It is well-known that any dynamical system governed by a differential equation containing time derivatives higher than second order unavoidably holds unbounded energy solutions, dubbed ghosts that appear in the Hamiltonian. They correspond to instabilities displayed at the classical level. In this study, we show first that it is possible to construct in classical mechanics, characterized by non-standard Lagrangians and nonlocal-in-time kinetic energy, higher-order derivative theories that avoid the Ostrogradsky ghost. Second, we show that in the realm of quantum mechanics, higher-order discretized energies emerge in the theory which may lead to extended quantum mechanics formalism. The problem of quantum wells has been treated where we showed that negative energy and negative action, complexified energies and complexified actions may emerge in our formalism. We have also discussed the quantum motion of a neutron in the Earth’s gravitational field. It was observed that within the realm of higher-order non-standard Lagrangians, the quantum energies of the neutrons are higher than the energy levels obtained in the basic formalism.