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Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a 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, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

In the present note, we consider the annihilation and creation operators with complex friction coefficient and we find the coherent, squeezed states, the uncertainty Heisenberg relation and the behavior of the PT and, CPT symmetries. Also we demonstrated that the (CK) Hamiltonian is a special case of the complex time-dependent mass.

A nonlinear Schrodinger equation is used to replace the linear Schrodinger equation in quantum mechanics and to study further the stability of microscopic particles by using the minimum theorem of energy and variational theorem. The investigation shows that the microscopic particles described by the nonlinear Shrodinger equation are localized and have a wave–corpuscle duality, which are completely different from that depicted by the linear Schrodinger equation in quantum mechanics. The results obtained by the minimum theorem of energy and variational theorem indicate that the solutions of the nonlinear Schrodinger equation or the microscopic particles are stable. Therefore, the microscopic particles have the feature of classical particles in such a case.

The properties of microscopic particles are studied using the linear Schrödinger equation in quantum mechanics and nonlinear Schrödinger equation, respectively. The results obtained show that the microscopic particles have only a wave nature in quantum mechanics, but a wave-corpuscle duality in nonlinear systems depicted by the nonlinear Schrödinger equation, no matter the form of external potentials. Thus we know that the kinetic energy term in dynamic equations determines the wave feature of the particles; the nonlinear interaction term determines the corpuscle feature; their combination makes the microscopic particles have a wave-corpuscle duality. However the external potential term can change the phase and group velocities of motion, phase, amplitude, frequency and form of wave for the particles in both quantum mechanics and the nonlinear quantum systems, although it cannot change these fundamental natures of particles, no matter the forms. Meanwhile, we find that the changes of positions of the microscopic particles by increasing the time under action of an external potential satisfy the Newton-type equation of motion in nonlinear quantum systems. Thus the investigations make us not only see the limits and approximations of quantum mechanics but also know the necessity and importance of developing nonlinear quantum mechanics on the basis of the nonlinear Schrödinger equation.

We construct a discrete quantum mechanics (QM) using a vector space over the Galois field GF(q). We find that the correlations in our model do not violate the Clauser–Horne–Shimony–Holt (CHSH) version of Bell's inequality, despite the fact that the predictions of this discrete QM cannot be reproduced with any hidden variable theory.

We use the Fourier operator to transform a time-dependent mass quantum harmonic oscillator into a frequency-dependent one. Then we use Lewis–Ermakov invariants to solve the Schrödinger equation by using squeeze operators. Finally, we give two examples of time dependencies: quadratically and hyperbolically growing masses.

We know energy and mass of a particle can be connected by $E=m{c}^2$. What is the physical basis of this relation? Historically, it was thought to be based on the principle of relativity (PR). A careful examination of the literature, however, indicated that this understanding is not true. Einstein did not derive this relation from PR. Instead, his argument was mainly based on thought experiments, which focused on the similarity between radiation and matter. Following this hint, we suspect that the mass–energy equivalence could be based on the quantum property of wave–particle duality. We know photon and electron can behave as a particle as well as a wave. Such a wave property could make the particle behave differently from Newtonian mechanics. Indeed, using a wave model which treats particles as excitations of the vacuum, we show that the mass–energy equivalence relation can be directly derived based on the quantum relations of Planck and de Broglie. This wave hypothesis has several advantages; not only can it explain naturally why particles can be created in the vacuum; it also predicts that a particle cannot travel faster than the speed of light. This hypothesis can also be tested in experiment.

In this paper, we have studied the Maxwell radiation from the electron orbiting the nucleus in an atom such as hydrogen atom. The orbiting electron is thought as two oscillating dipoles: one on $x$-axis and the other on $y$-axis with the same frequency. We have derived the electromagnetic wave and calculated powers of the radiation from these two oscillating dipoles, and then determined the time period it takes for the electron to change its energy levels from $E_2$ to $E_1$ due to the Maxwell radiation. Compared the time period due to the Maxwell radiation with the time period from the uncertainty principle, it is found that the two time periods are different from each other by about nine orders of magnitude. So, we are led to believe that the Maxwell theory could not be used in the quantum photo emission by the electron in an atom. Thus, we suggest that the quantum photo emission by the electron in an atom may be photon tunneling from the electron $t$ in an atom.

It has been well known that there is a redshift of photon frequency due to the gravitational potential. Scott et al. [Can. J. Phys.44 (1966) 1639, https://doi.org/10.1139/p66-137] pointed out that general relativity theory predicts the gravitational redshift. However, using the quantum mechanics theory related to the photon Hamiltonian and photon Schrodinger equation, we calculate the redshift due to the gravitational potential. The result is exactly the same as that from the general relativity theory.

The role of dissipation with respect to a microscopic superposition of quantum states is investigated by means of master equations. This has implications for the study of the emergence of classicality from the quantum level. In particular, it illustrates why it is difficult to observe a macroscopic quantum state. The role of the environment is assumed by the measuring apparatus. A pure state is reduced to a mixture in the pointer basis of the system by means of the interaction with the apparatus. It is the intention that this type of analysis will have applications to experiments which are designed to better understand the environmental-assisted invariance formulation of quantum mechanics.

Quantum computers by their nature are many particle quantum systems. Both the many-particle arrangement and being quantum are necessary for the existence of the entangled states, which are responsible for the parallelism of the quantum computers. Second quantization is a very important approximate method of describing such systems. This lecture will present the general idea of the second quantization, and discuss shortly some of the most important formulations of second quantization.

Starting from the apparently periodic structures of the universe, revealed in a series of recent observations, we suppose that Dark Matter is composed of a quantum particle of very low mass and is clustered around the luminous matter of galaxies. We reduce the cosmological Friedman–Einstein dynamical system to a sort of Schrödinger equation for this quantum particle with a simple first quantization scheme. Comparing the eigen-solutions of this Cosmological Schrödinger Equation with the experimental periodic large scale structure, we predict the possible value of the mass of the dark quantum particle in two remarkable cases.

A possible solution to the problem of providing a spacetime description of the transmission of signals for quantum entangled states is obtained by using a bimetric spacetime structure, in which quantum entanglement measurements alter the structure of the classical relativity spacetime. A bimetric gravity theory locally has two lightcones, one which describes classical special relativity and a larger lightcone which allows light signals to communicate quantum information between entangled states, after a measurement device detects one of the entangled quantum states. This theory would remove the tension that exists between macroscopic classical, local gravity and macroscopic nonlocal quantum mechanics.

We show that a suitably chosen position-momentum commutator can elegantly describe many features of gravity, including the IR/UV correspondence and dimensional reduction ("holography"). Using the most simplistic example based on dimensional analysis of black holes, we construct a commutator which qualitatively exhibits these novel properties of gravity. Dimensional reduction occurs because the quanta size grow quickly with momenta, and thus cannot be "packed together" as densely as naively expected. We conjecture that a more precise form of this commutator should be able to quantitatively reproduce all of these features.

It is commonly assumed that the equivalence principle can coexist without conflict with quantum mechanics. We shall argue here that, contrary to popular belief, this principle does not hold in quantum mechanics. We illustrate this point by computing the second-order correction for the scattering of a massive scalar boson by a weak gravitational field, treated as an external field. The resulting cross-section turns out to be mass-dependent. A way out of this dilemma would be, perhaps, to consider gravitation without the equivalence principle. At first sight, this seems to be a too much drastic attitude toward general relativity. Fortunately, the teleparallel version of general relativity — a description of the gravitational interaction by a force similar to the Lorentz force of electromagnetism and that, of course, dispenses with the equivalence principle — is equivalent to general relativity, thus providing a consistent theory for gravitation in the absence of the aforementioned principle.

We do not know the symmetries underlying string theory. Furthermore, there must exist an inherently quantum, and space–time independent, formulation of this theory. Independent of string theory, there should exist a description of quantum mechanics which does not refer to a classical space–time manifold. We propose such a formulation of quantum mechanics, based on noncommutative geometry. This description reduces to standard quantum mechanics, whenever an external classical space–time is available. However, near the Planck energy scale, self-gravity effects modify the Schrödinger equation to the non-linear Doebner–Goldin equation. Remarkably, this non-linear equation also arises in the quantum dynamics of D0-branes. This suggests that the noncommutative quantum dynamics proposed here is actually the quantum gravitational dynamics of D0-branes, and that automorphism invariance is a symmetry of string theory.

Pairs of Planck-mass drops of superfluid helium coated by electrons (i.e. "Millikan oil drops"), when levitated in a superconducting magnetic trap, can be efficient quantum transducers between electromagnetic (EM) and gravitational (GR) radiation. This leads to the possibility of a Hertz-like experiment, in which EM waves are converted at the source into GR waves, and then back-converted at the receiver from GR waves into EM waves. Detection of the gravitational-wave analog of the cosmic microwave background using these drops can discriminate between various theories of the early Universe.

Newton–Hooke spacetimes are the nonrelativistic limit of (anti-)de Sitter spacetimes. We investigate some peculiar facts about the Newton–Hooke spacetimes, among which the "extraordinary Newton–Hooke quantum mechanics" and the "anomalous Newton–Hooke spacetimes" are discussed in detail. Analysis on the Lagrangian/action formalism is performed in the discussion of the Newton–Hooke quantum mechanics, where the path integral point of view plays an important role, and the physically measurable density of probability is clarified.

It has been argued that when black holes are treated as quantum systems there are implications at the horizon and not just the singularity. Infalling observers will meet a firewall of high energy quanta. We argue that the question of whether an observer falling into a black hole experiences a smooth horizon or a firewall is identical to the question of whether Schrödinger's cat is either in a definite state, alive or dead, or in a superposition of the two. Since experience with real macro-systems indicate the former, the black hole state vector is seen to describe a set of decoherent alternatives each with a smooth horizon and the entanglement puzzle is thereby side stepped.

It is congruous with the quantum nature of the world to view the spacetime geometry as an emergent structure that shows classical features only at some observational level. One can thus conceive the spacetime manifold as a purely theoretical arena, where quantum states are defined, with the additional freedom of changing coordinates like any other symmetry. Observables, including positions and distances, should then be described by suitable operators acting on such quantum states. In principle, the top-down (canonical) quantization of Einstein–Hilbert gravity falls right into this picture, but is notoriously very involved. The complication stems from allowing all the classical canonical variables that appear in the (presumably) fundamental action to become quantum observables acting on the “superspace” of all metrics, regardless of whether they play any role in the description of a specific physical system. On can instead revisit the more humble “minisuperspace” approach and choose the gravitational observables not simply by imposing some symmetry, but motivated by their proven relevance in the (classical) description of a given system. In particular, this review focuses on compact, spherically symmetric, quantum mechanical sources, in order to determine the probability that they are black holes (BHs) rather than regular particles. The gravitational radius is therefore lifted to the status of a quantum mechanical operator acting on the “horizon wave function (HWF),” the latter being determined by the quantum state of the source. This formalism is then applied to several sources with a mass around the fundamental scale, which are viewed as natural candidates of quantum BHs.