Quantum Mechanical Theorem Research Articles

Encoding beyond cosmological horizons in de Sitter JT gravity

Black hole event horizons and cosmological event horizons share many properties, making it natural to ask whether our recent advances in understanding black holes generalize to cosmology. To this end, we discuss a paradox that occurs if observers can access what lies beyond their cosmological horizon in the same way that they can access what lies beyond a black hole horizon. In particular, distinct observers with distinct horizons may encode the same portion of spacetime, violating the no-cloning theorem of quantum mechanics. This paradox is due precisely to the observer-dependence of the cosmological horizon — the sharpest difference from a black hole horizon — although we will argue that the gravity path integral avoids the paradox in controlled examples.

Controlled quantized adiabatic transport in a superlattice Wannier-Stark ladder

The Born–Fock theorem is one of the most fundamental theorems of quantum mechanics and forms the basis for reliable and efficient navigation in the Hilbert space of a quantum system with a time-dependent Hamiltonian by adiabatic evolution. In the absence of level crossings, i.e. without degeneracies, and under adiabatic time evolution all eigenstates of the Hamiltonian keep their energetic order, labeled by a conserved integer quantum number. Thus, controlling the eigenstates of the Hamiltonian and their energetic order in asymptotic limits allows one to engineer a perfect adiabatic transfer between a large number of initial and target states. The fidelity of the state transfer is only limited by adiabaticity and the selection of target states is controlled by the integer invariant labeling the order of eigenstates. We show here, for the example of a finite superlattice Wannier-Stark ladder, i.e. a one-dimensional lattice with alternating hopping amplitudes and constant potential gradient, that such an adiabatic control of eigenstates can be used to induce perfectly quantized single-particle transport across a pre-determined number of lattice sites. We dedicate this paper to the memory of our late friend and colleague Bruce Shore, who was an expert in adiabatic processes and taught us much about this field.

Symmetry properties of the ground state of the system of interacting spinless bosons

We perform the symmetry analysis of the properties of the ground state of a finite system of interacting spinless bosons for the three most symmetric boundary conditions (BCs): zero BCs with spherical and circular symmetries, as well as periodic BCs. The symmetry of the system can lead to interesting properties. For instance, the density of a periodic Bose system is an exact constant: ρ(r) = const. Moreover, in the case of perfect spherical symmetry of BCs, the crystalline state cannot produce the Bragg peaks. The main result of the article is that symmetry properties and general quantum-mechanical theorems admit equally both crystalline and liquid ground state for a Bose system of any density.

Contextuality and the fundamental theorems of quantum mechanics

Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr [Albert Einstein: Philosopher-Scientist, Library of Living Philosophers Vol. VII, edited by P. A. Schilpp (Open Court, 1998), pp. 199–241] and later realized more technically by Kochen and Specker [J. Math. Mech. 17, 59 (1967)]. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen–Specker theorem in the language of presheaves in Isham and Butterfield [Int. J. Theor. Phys. 37, 2669 (1998)]. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen–Specker theorem, Wigner’s theorem, Gleason’s theorem, and Bell’s theorem also relate fundamentally to contextuality. We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. This shows that a very substantial part of the structure of quantum theory is encoded by contextuality.

Addendum: Considerations about the incompleteness of the Ehrenfest’s theorem in quantum mechanics (2021 Eur. J. Phys. 42 065405)

We describe the analytical solution of the eigenvalue problem introduced in our article mentioned in the title and relative to a punctiform electric charge confined in an one-dimensional box in the presence of an electric field. We also derive and discuss the analytical expressions of the external forces acting on the punctiform charge and associated with the boundaries of the one-dimensional box in the presence of the electric field.

Plasma Confined Ground and Excited State Helium Atom: A Comparison Theorem Study Using Variational Monte Carlo and Lagrange Mesh Method

The energy eigenvalues of the ground state helium atom and lowest two excited states corresponding to the configurations 1s2s embedded in the plasma environment using Hulthén, Debye–Hückel and exponential cosine screened Coulomb model potentials are investigated within the variational Monte Carlo method, starting with the ultracompact trial wave functions in the form of generalized Hylleraas–Kinoshita functions and Guevara–Harris–Turbiner functions. The Lagrange mesh method calculations of energy are reported for the He atom in the ground and excited 1S and 3S states, which are in excellent agreement with the variational Monte Carlo results. Interesting relative ordering of eigenvalues are reported corresponding to the different screened Coulomb potentials in the He ground and excited electronic states, which are rationalized in terms of the comparison theorem of quantum mechanics.

Considerations about the incompleteness of the Ehrenfest’s theorem in quantum mechanics

We describe a study motivated by our interest to examine the incompleteness of the Ehrenfest’s theorem in quantum mechanics and to resolve a doubt regarding whether or not the hermiticity of the Hamiltonian operator is sufficient to justify a simplification of the expression of the macroscopic-observable time derivative that promotes the one usually found in quantum-mechanics textbooks. The study develops by considering the simple quantum system ‘particle in one-dimensional box’. We propose theoretical arguments to support the incompleteness of the Ehrenfest’s theorem in the formulation he gave, in agreement with similar findings already published by a few authors, and corroborate them with the numerical example of an electric charge in an electrostatic field. The contents of this study should be useful to Bachelor and Master students; the style of the discussions is tailored to stimulate, we hope, the student’s ability for independent thinking.

Multiscale Thermodynamics: Energy, Entropy, and Symmetry from Atoms to Bulk Behavior

Here, we investigate how the local properties of particles in a thermal bath may influence the thermodynamics of the bath, and consequently alter the statistical mechanics of subsystems that comprise the bath. We are guided by the theory of small-system thermodynamics, which is based on two primary postulates: that small systems can be treated self-consistently by coupling them to an ensemble of similarly small systems, and that a large ensemble of small systems forms its own thermodynamic bath. We adapt this “nanothermodynamics” to investigate how a large system may subdivide into an ensemble of smaller subsystems, causing internal heterogeneity across multiple size scales. For the semi-classical ideal gas, maximum entropy favors subdividing a large system of “atoms” into an ensemble of “regions” of variable size. The mechanism of region formation could come from quantum exchange symmetry that makes atoms in each region indistinguishable, while decoherence between regions allows atoms in separate regions to be distinguishable by their distinct locations. Combining regions reduces the total entropy, as expected when distinguishable particles become indistinguishable, and as required by a theorem in quantum mechanics for sub-additive entropy. Combining large volumes of small regions gives the usual entropy of mixing for a semi-classical ideal gas, resolving Gibbs paradox without invoking quantum symmetry for particles that may be meters apart. Other models presented here are based on Ising-like spins, which are solved analytically in one dimension. Focusing on the bonds between the spins, we find similarity in the equilibrium properties of a two-state model in the nanocanonical ensemble and a three-state model in the canonical ensemble. Thus, emergent phenomena may alter the thermal behavior of microscopic models, and the correct ensemble is necessary for fully-accurate predictions. Another result using Ising-like spins involves simulations that include a nonlinear correction to Boltzmann’s factor, which mimics the statistics of indistinguishable states by imitating the dynamics of spin exchange on intermediate lengths. These simulations exhibit 1/f-like noise at low frequencies (f), and white noise at higher f, similar to the equilibrium thermal fluctuations found in many materials.

The `Quantum Game Show': a very simple explanation of Bell's Theorem in Quantum Mechanics

In this article give a very simple presentation of Bell's inequality by comparing it to a 'quantum game show', followed by a simple description of Aspect's 1985 experiment involving entangled photons which confirms the inequality. The entire article is non-technical and requires no mathematical background other than high school mathematics and an understanding of basic concepts in probability. The physics involved in Aspect's experiment is also explained.

High Efficiency Continuous-Variable Quantum Key Distribution Based on ATSC 3.0 LDPC Codes.

Due to the rapid development of quantum computing technology, encryption systems based on computational complexity are facing serious threats. Based on the fundamental theorem of quantum mechanics, continuous-variable quantum key distribution (CVQKD) has the property of physical absolute security and can effectively overcome the dependence of the current encryption system on the computational complexity. In this paper, we construct the spatially coupled (SC)-low-density parity-check (LDPC) codes and quasi-cyclic (QC)-LDPC codes by adopting the parity-check matrices of LDPC codes in the Advanced Television Systems Committee (ATSC) 3.0 standard as base matrices and introduce these codes for information reconciliation in the CVQKD system in order to improve the performance of reconciliation efficiency, and then make further improvements to final secret key rate and transmission distance. Simulation results show that the proposed LDPC codes can achieve reconciliation efficiency of higher than 0.96. Moreover, we can obtain a high final secret key rate and a long transmission distance through using our proposed LDPC codes for information reconciliation.

Two Flavors of Hydrogen Atoms: A Possible Explanation of Dark Matter

In one of our previous papers, it was shown that for the ground state of hydrogenic atoms/ions, it is possible to match the interior (inside the nucleus) solution of the Dirac equation with the singular exterior solution of the Dirac equation, so that the singular solution should not be rejected for the ground state of hydrogenic atoms/ions. In that paper, there was presented also the first experimental proof of the existence of this Alternative Kind of Hydrogen Atoms (AKHA)—by showing that the presence of the AKHA solves a long-standing mystery of the huge discrepancy between the experimental and previous theoretical results concerning the high-energy tail of the linear momentum distribution in the ground state of hydrogen atoms. In another paper, we showed that for hydrogen atoms, the singular solution of the Dirac equation outside the proton is legitimate not just for the ground state 12S1/2, but also for the states 22S1/2, 32S1/2 and so on: it is legitimate for all the discrete states n2S1/2. Moreover, the singular exterior solution is legitimate also for the l = 0 states of the continuous spectrum. In that paper, we demonstrated that the AKHA can be the basis for explaining the recent puzzling astrophysical observational results concerning the redshifted radio line 21 cm from the early Universe. Thus, there seems to be the astrophysical evidence of the existence of the AKHA—in addition to the already available observational proof of their existence from atomic experiments. In the present paper, we point out that the AKHA provide an alternative view on dark matter—without resorting to new subatomic particles or dramatically changing the existing physical laws. This is because due to the selection rules, the AKHA do not have state that can be coupled by the electric dipole radiation. We also reformulate the above theoretical results in terms that hydrogen atoms can have two flavors: one flavor corresponding to the regular solution outside the proton, another—to the singular solution outside the proton, both solutions corresponding to the same energy. Since this means the additional degeneracy, then according to the fundamental theorem of quantum mechanics, there should be an additional conserved quantity, which we call isohydrogen spin (isohyspin). Further atomic experiments for accurately measuring the high-energy tail of the linear momentum distribution in the ground state of hydrogen atoms, as well as further observational studies of the redshifted 21 cm radio line from the early Universe, could provide a further proof that dark matter or a part of it is the AKHA.

A 1d Up Approach to Conformal Geometric Algebra: Applications in Line Fitting and Quantum Mechanics

We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell’s Theorem in quantum mechanics, and its relation to the sphere S^7 and the exceptional group E_8, and proposed a new associative version of the division algebra normally thought to require the octonians. We show that what is being discussed by Christian is mathematically the same as our 1d up approach to 3d geometry, but that after the removal of some incorrect mathematical assertions, the results he proves in the first part of the paper, and bases the application to Bell’s Theorem on, amount to no more than the statement that the combination of two rotors from the Clifford Algebra Cl(4, 0) is also a rotor.

Particle confined in modified ring-shaped potential

The spectrum of a particle confined in Hulthén plus ring-shaped potential is obtained by solving the time-independent Schrödinger equation numerically. The effect of potential parameters on various properties of the particle have been investigated in detail. The energy levels, radial matrix elements, oscillator strengths and polarizabilities of the particle have been found to show strong dependence on the confining potential parameters. The presence of the ring potential is found to appreciably alter the angular part of dipole matrix elements. Also, it is shown that the comparison theorem of Quantum Mechanics for energy eigenvalues for four different potentials, viz., Coulomb, Hulthén, Yukawa and Hulthén2 is independent of the presence of ring potential.

A new concept of QM-models and truth in quantum mechanics (QM), the new hybrid-epistemic model of QM and its observer independence, the proof of the invalidity of no-go theorems

We define a concept of a QM-model. The true theorem in QM is a statement which is true in all QM-models. In some QM-models the Bell’s theorem can be proved (e.g. in the standard model of QM) while in other QM-models the Bell’s theorem cannot be proved (e.g. in the hybrid-epistemic model of QM). The same situation is true when other no-go theorems are considered. Thus no-go theorems are not true in QM and the proof of the non-locality of QM is invalid.Then the axiomatic definition of the hybrid-epistemic (HE) model of QM is presented in all details. At the end the recent proof of the inconsistency of the standard QM-model is discussed.Our main program is to start the axiomatic study of QM (in the sense of the Hilbert’s sixth problem), to prove the invalidity of no-go theorems and to identify the right (acceptable) QM-model.

On the quantum adiabatic evolution with the most general system Hamiltonian

In this paper, we study the problem that when quantum adiabatic evolution with the most general form of system Hamiltonian will get failed. Here the most general form means that the initial and final Hamiltonians are just designed according to the adiabatic theorem in quantum mechanics. As we will see, even in this most general model of quantum adiabatic evolution, it still exists the possibility that the quantum adiabatic computation can fail totally if some condition is satisfied, which implies the time complexity of the quantum algorithm is infinity. That is, here we propose a rather general criterion for judging whether a quantum adiabatic evolution is successful. This result largely extends the authors’ previous research on this topic, and it may be seen as a further important clue for us when designing quantum algorithms in the framework of adiabatic evolution for some practical problems.

Comments on the adiabatic theorem

We consider the simplest example of a nonstationary quantum system which is quantum mechanical oscillator with varying frequency and [Formula: see text] self-interaction. We calculate loop corrections to the Keldysh, retarded/advanced propagators and vertices using Schwinger–Keldysh diagrammatic technique and show that there is no physical secular growth of the loop corrections in the cases of constant and adiabatically varying frequency. This fact corresponds to the well-known adiabatic theorem in quantum mechanics. However, in the case of nonadiabatically varying frequency we obtain strong IR corrections to the Keldysh propagator which come from the “sunset” diagrams, grow with time indefinitely and indicate energy pumping into the system. It reveals itself via the change in time of the level population and of the anomalous quantum average.

Three theorems of quantum mechanics and their classical counterparts

The Hellmann–Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is shown here that these three theorems are valid for classical mechanics with a nonstandard kinetic energy. This brings some information about the connections between the quantum and classical worlds. Constraints about the functional form of the kinetic energy are also discussed.

Expectation values of general observables in the Vlasov formalism

Collisionless plasmas in an arbitrary dynamical state are described by the Vlasov equation, which gives the time evolution of the probability density ρ(x, v). In this work we introduce a new analytical procedure to generate particular partial differential equations (PDEs) for an arbitrary macroscopic observable w(x, v) that can be expressed as a function of positions and velocities, without solving the time evolution of the probability itself. This technique, which we will call the “Ehrenfest procedure” (as it produces relations that are analogous to Ehrenfest’s theorem in Quantum Mechanics), is based on the iterative application of the fluctuation-dissipation theorem and the recently proposed conjugate variables theorem (CVT) in order to eliminate the explicit dependence on ρ. In particular, we show how this formalism is applied to the Vlasov equation for collisionless plasmas, and derive a general evolution equation for the fluctuations of any macroscopic property w in this kind of plasma.

Generalized quantum no-go theorems of pure states

Various results of the no-cloning theorem, no-deleting theorem and no-superposing theorem in quantum mechanics have been proved using the superposition principle and the linearity of quantum operations. In this paper, we investigate general transformations forbidden by quantum mechanics in order to unify these theorems. First, we prove that any useful information cannot be created from an unknown pure state which is randomly chosen from a Hilbert space according to the Harr measure. And then, we propose a unified no-go theorem based on a generalized no-superposing result. The new theorem includes the no-cloning theorem, no-anticloning theorem, no-partial-erasure theorem, no-splitting theorem, no-superposing theorem or no-encoding theorem as a special case. Moreover, it implies various new results. Third, we extend the new theorem into another form that includes the no-deleting theorem as a special case.

Algorithmic analysis of termination problems for quantum programs

We introduce the notion of linear ranking super-martingale (LRSM) for quantum programs (with nondeterministic choices, namely angelic and demonic choices). Several termination theorems are established showing that the existence of the LRSMs of a quantum program implies its termination. Thus, the termination problems of quantum programs is reduced to realisability and synthesis of LRSMs. We further show that the realisability and synthesis problem of LRSMs for quantum programs can be reduced to an SDP (Semi-Definite Programming) problem, which can be settled with the existing SDP solvers. The techniques developed in this paper are used to analyse the termination of several example quantum programs, including quantum random walks and quantum Bernoulli factory for random number generation. This work is essentially a generalisation of constraint-based approach to the corresponding problems for probabilistic programs developed in the recent literature by adding two novel ideas: (1) employing the fundamental Gleason's theorem in quantum mechanics to guide the choices of templates; and (2) a generalised Farkas' lemma in terms of observables (Hermitian operators) in quantum physics.