Minimal Uncertainty Relation Research Articles

Interaction of Two-Level Atoms with a Single-Mode Quantized Radiation Field

We have studied the statistical and squeezing properties of the cavity light generated by a two-level laser. This optical system contains N two-level atoms available in a cavity coupled to a single-mode vacuum reservoir. They are pumped to the top level from the bottom level by means of the electron bombardment. Applying the steady-state solutions of the equations of evolution of the expectation values of the atomic operators and the quantum Langevin equation, we obtained the global and local photon statistics of the single-mode light beam. We have found that, for the two-level laser operating well above the threshold, the uncertainties in the plus and minus quadratures are equal and satisfy the minimum uncertainty relation. In view of this, we have identified the light generated by the laser operating well above threshold to be coherent. On the other hand, the light generated by the laser operating at threshold is found to be chaotic. From the obtained results, we have also observed that a large part of the local mean photon number, the local photon number variance, and the local quadrature variance are confined in a relatively narrow frequency interval.

Minimum Length Uncertainty Relations in the Presence of Dark Energy

We introduce a dark energy-modified minimum length uncertainty relation (DE-MLUR) or dark energy uncertainty principle (DE-UP) for short. The new relation is structurally similar to the MLUR introduced by Károlyházy (1968), and reproduced by Ng and van Dam (1994) using alternative arguments, but with a number of important differences. These include a dependence on the de Sitter horizon, which may be expressed in terms of the cosmological constant as l dS ∼ 1 / Λ . Applying the DE-UP to both charged and neutral particles, we obtain estimates of two limiting mass scales, expressed in terms of the fundamental constants G , c , ℏ , Λ , e . Evaluated numerically, the charged particle limit corresponds to the order of magnitude value of the electron mass ( m e ), while the neutral particle limit is consistent with current experimental bounds on the mass of the electron neutrino ( m ν e ). Possible cosmological consequences of the DE-UP are considered and we note that these lead naturally to a holographic relation between the bulk and the boundary of the Universe. Low and high energy regimes in which dark energy effects may dominate canonical quantum behaviour are identified and the possibility of testing the model using near-future experiments is briefly discussed.

Coherent States for Supersymmetric Partners of the Infinite Well

We define linear and quadratic coherent states for the supersymmetric partners of the quantum infinite well through formal series expansions of the energy eigenfunctions of the systems and we study the appropriateness of this definitions as coherent states by means of their properties. In particular, we examine the localization in position and time evolution, minimum uncertainty relations and the behavior of the Wigner function.

The minimum mass of a charged spherically symmetric object in D dimensions, its implications for fundamental particles, and holography

We obtain bounds for the minimum and maximum mass/radius ratio of a stable, charged, spherically symmetric compact object in a $D$-dimensional space-time in the framework of general relativity, and in the presence of dark energy. The total energy, including the gravitational component, and the stability of objects with minimum mass/radius ratio is also investigated. The minimum energy condition leads to a representation of the mass and radius of the charged objects with minimum mass/radius ratio in terms of the charge and vacuum energy only. As applied to the electron in the four-dimensional case, this procedure allows one to re-obtain the classical electron radius from purely general relativistic considerations. By combining the lower mass bound, in four space-time dimensions, with minimum length uncertainty relations (MLUR) motivated by quantum gravity, we obtain an alternative bound for the maximum charge/mass ratio of a stable, gravitating, charged quantum mechanical object, expressed in terms of fundamental constants. Evaluating this limit numerically, we obtain again the correct order of magnitude value for the charge/mass ratio of the electron, as required by the stability conditions. This suggests that, if the electron were either less massive (with the same charge) or if its charge were any higher (for fixed mass), a combination of electrostatic and dark energy repulsion would destabilize the Compton radius. In other words, the electron would blow itself apart. Our results suggest the existence of a deep connection between gravity, the presence of the cosmological constant, and the stability of fundamental particles.

Phase amplitude conformal symmetry in Fourier transforms

For the Fourier transform ℑ : L2(R) → L2(R) of a complex-valued even or odd function ψ, it is found that the amplitude invariance |ℑψ| = |ψ| leads to a phase invariance or inversion as arg(ℑψ) = ±argψ + θ (θ = constant). The converse holds unless arg ψ = constant. The condition |ψ| = |ℑψ| is required in dealing with, for example, the minimum uncertainty relation between position and momentum. Without the evenness or oddness of ψ, |ℑψ| = |ψ| does not necessarily imply arg(ℑψ) = ±argψ + θ, nor is the converse.

Local Softening of Information Geometric Indicators of Chaos in Statistical Modeling in the Presence of Quantum-Like Considerations

In a previous paper (C. Cafaro et al., 2012), we compared an uncorrelated 3D Gaussian statistical model to an uncorrelated 2D Gaussian statistical model obtained from the former model by introducing a constraint that resembles the quantum mechanical canonical minimum uncertainty relation. Analysis was completed by way of the information geometry and the entropic dynamics of each system. This analysis revealed that the chaoticity of the 2D Gaussian statistical model, quantified by means of the Information Geometric Entropy (IGE), is softened or weakened with respect to the chaoticity of the 3D Gaussian statistical model, due to the accessibility of more information. In this companion work, we further constrain the system in the context of a correlation constraint among the system’s micro-variables and show that the chaoticity is further weakened, but only locally. Finally, the physicality of the constraints is briefly discussed, particularly in the context of quantum entanglement.

Dynamics Underlying the Gaussian Distribution of the Classical Harmonic Oscillator in Zero-Point Radiation

Stochastic electrodynamics (SED) predicts a Gaussian probability distribution for a classical harmonic oscillator in the vacuum field. This probability distribution is identical to that of the ground state quantum harmonic oscillator. Thus, the Heisenberg minimum uncertainty relation is recovered in SED. To understand the dynamics that give rise to the uncertainty relation and the Gaussian probability distribution, we perform a numerical simulation and follow the motion of the oscillator. The dynamical information obtained through the simulation provides insight to the connection between the classic double-peak probability distribution and the Gaussian probability distribution. A main objective for SED research is to establish to what extent the results of quantum mechanics can be obtained. The present simulation method can be applied to other physical systems, and it may assist in evaluating the validity range of SED.

Quantum description of a time-dependent mesoscopic RLC circuit

In this paper, we present a comprehensive quantum description of a mesoscopic RLC circuit with time-dependent resistance, inductance and capacitance. Based on the dynamical invariant method and using quadratic invariants, we derive exact nonstationary quantum states for this circuit and write them in terms of solutions of the Milne–Pinney equation. Afterwards, we use quadratic invariants to construct coherent states for this quantized system and employ them to investigate quantum properties of the RLC circuit. In particular, we show that the product of the quantum fluctuations of the charge and the magnetic flux does not satisfy the minimum uncertainty relation.

Softening the Complexity of Entropic Motion on Curved Statistical Manifolds

We study the information geometry and the entropic dynamics of a three-dimensional Gaussian statistical model. We then compare our analysis to that of a two-dimensional Gaussian statistical model obtained from the higher-dimensional model via introduction of an additional information constraint that resembles the quantum mechanical canonical minimum uncertainty relation. We show that the chaoticity (temporal complexity) of the two-dimensional Gaussian statistical model, quantified by means of the information geometric entropy (IGE) and the Jacobi vector field intensity, is softened with respect to the chaoticity of the three-dimensional Gaussian statistical model.

Uncertainty relations for a q-deformed coherent spin state

A Coherent Spin State (CSS) is defined as an eigenstate of the spin component in the direction specified by angles (θ0,ϕ0). This state satisfies minimum uncertainty relation, with uncertainties equally distributed on any two orthogonal components normal to the direction of the total spin vector 〈S〉. Starting from this concept, we apply the notion of CSS to quantum groups and discuss the properties of q-deformed CSS and the associated uncertainty relations. We show that these states behave as Intelligent Spin States (ISS) on two orthogonal components normal to the direction of the mean value of the spin operator.

The wave-corpuscle properties of microscopic particlesin the nonlinear quantum-mechanical systems

We debate first the properties of quantum mechanics and its difficulties and the reasons resulting in these diffuculties and its direction of development. The fundamental principles of nonlinear quantum mechanics are proposed and established based on these shortcomings of quantum mechanics and real motions and interactions of microscopic particles and backgound field in physical systems. Subsequently, the motion laws and wave-corpuscle duality of microscopic particles described by nonlinear Schr?dinger equation are studied completely in detail using these elementary principles and theories. Concretely speaking, we investigate the wave-particle duality of the solution of the nonlinear Schr?dinger equation, the mechanism and rules of particle collision and the uncertainty relation of particle’s momentum and position, and so on. We obtained that the microscopic particles obey the classical rules of collision of motion and satisfy the minimum uncertainty relation of position and momentum, etc. From these studies we see clearly that the moved rules and features of microscopic particle in nonlinear quantum mechanics is different from those in linear quantum mechanics. Therefore, nolinear quantum mechanics is a necessary result of development of quantum mechanics and represents correctly the properties of microscopic particles in nonlinear systems, which can solve difficulties and problems disputed for about a century by scientists in linear quantum mechanics field.

On the construction of coherent states of position dependent mass Schrödinger equation endowed with effective potential

In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrödinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schrödinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new.

Minimum uncertainty wave packet in relativistic quantum mechanics

In nonrelativistic quantum mechanics in one dimension a wave packet can be constructed for which the minimum uncertainty relation for position and momentum, ΔxΔp=ℏ/2, holds exactly. The wave function of the wave packet is Gaussian and satisfies the Schrödinger equation for a harmonic oscillator potential. We illustrate a similar situation in relativistic quantum mechanics for the one-dimensional Dirac equation.

Entangled state for constructing a generalized phase-space representation and its statistical behavior

We construct a generalized phase-space representation (GPSR) based on the idea of Einstein-Podolsky-Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to the entangled case, which is characteristic of the features when two particles' relative coordinate, total momentum operators, and their conjugative variables, respectively, operate on the GPSR. This representation is complete and nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identification with the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimum uncertainty relation obeyed by the GPSR is also demonstrated.

Time-Dependent Variational Approach to Ground-State Phase Transition and Phonon Dispersion Relation of the Quantum Double-Well Model

The ground-state phase transition and the phonon dispersion relation of the quantum double-well model are studied by means of the time-dependent variational approach combined with a Hartree-type many-body trial wavefunction. The single-particle state is taken to be a frozen Jackiw–Kerman wavefunction. Under the condition of minimum uncertainty relation, we obtain an effective classical Hamiltonian for the system and equations of motion for the particle's expectation values. It is shown that the effective substrate potential transits from a symmetric double-well potential to a symmetric single-well potential, and the ground state exhibits a transition from a broken symmetry phase to a restored symmetry phase as increasing the strength of quantum fluctuations. We also obtain the phonon dispersion relations and the phonon gaps at the two phases.

Features and states of microscopic particles in nonlinear quantum-mechanics systems

In this paper, we present the elementary principles of nonlinear quantum mechanics (NLQM), which is based on some problems in quantum mechanics. We investigate in detail the motion laws and some main properties of microscopic particles in nonlinear quantum systems using these elementary principles. Concretely speaking, we study in this paper the wave-particle duality of the solution of the nonlinear Schrodinger equation, the stability of microscopic particles described by NLQM, invariances and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, the mechanism and rules of particle collision, the features of reflection and the transmission of particles at interfaces, and the uncertainty relation of particle motion as well as the eigenvalue and eigenequations of particles, and so on. We obtained the invariance and conservation laws of mass, energy and momentum and angular momentum for the microscopic particles, which are also some elementary and universal laws of matter in the NLQM and give further the methods and ways of solving the above questions. We also find that the laws of motion of microscopic particles in such a case are completely different from that in the linear quantum mechanics (LQM). They have a lot of new properties; for example, the particles possess the real wave-corpuscle duality, obey the classical rule of motion and conservation laws of energy, momentum and mass, satisfy minimum uncertainty relation, can be localized due to the nonlinear interaction, and its position and momentum can also be determined, etc. From these studies, we see clearly that rules and features of microscopic particle motion in NLQM is different from that in LQM. Therefore, the NLQM is a new physical theory, and a necessary result of the development of quantum mechanics and has a correct representation of describing microscopic particles in nonlinear systems, which can solve problems disputed for about a century by scientists in the LQM field. Hence, the NLQM built is very necessary and correct. The NLQM established can promote the development of physics and can enhance and raise the knowledge and recognition levels to the essences of microscopic matter. We can predict that nonlinear quantum mechanics has extensive applications in physics, chemistry, biology and polymers, etc.

Explicit state vector for Torres-Vega–Frederick phase space representation and its statistical behavior

We find the explicit state vector for Torres-Vega-Frederick phase space representation [Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103 (1993)], denoted by Gamma. This set of states make up a complete and nonorthogonal representation. The Weyl ordered form of Gamma Gamma [see text for the sign] is derived, which can clearly exhibit the statistical behavior of marginal distribution of Gamma Gamma [see text for the sign]. The minimum uncertainty relation for mid R:Gamma is demonstrated, which shows it being a coherent squeezed state.

PROPERTIES OF SO(2, 1) COHERENT STATE FOR THE COULOMB PROBLEM

The physical properties of the SO(2, 1) coherent state for the Coulomb problem are discussed in this paper. We find that the coherent state, for which the minimum uncertainty relation holds, has a nonvanishing geometric phase factor, and also is approximately nonspreading in the classical limit of the high excitation of angular momentum.

CHANGES IN THE PHYSICAL PROPERTIES OF NONADIABATICALLY COUPLED ELECTRON–PHONON SYSTEMS ARISING FROM SQUEEZING–ANTISQUEEZING EFFECT

Changes in the physical properties such as the ground state properties, charge density wave ordering, binding energy and energy bandwidth of polaron and quantum fluctuation, and minimum uncertainty relation of phonons and nonadiabatically coupled electron–phonon systems with spin-1/2 have been investigated by our new state ansatz which can account for correlation among the phononic displacement, squeezing and polaron effects using variational method in one-dimensional Holstein model. The investigation here shows that the squeezing–antisqueezing effect (correlated) results in a decrease of the ground state energy, an increase of the binding energy of polarons, the reduction of the uncertainty and quantum fluctuation of the phonons, a decrease of polaron narrowing of electron bandwidth, an increase of tunneling effect of the polarons and an increase of CDW ordering and phonon staggered ordering when compared with the uncorrelated case. Therefore, this shows that the ground state determined by the new state ansatz is the most stable. The new ansatz which include the squeezing–antisqueezing (correlated) effect is very relevant for the coupled electron–phonon systems, especially in strongly coupled and highly squeezed cases.

Quantum Fluctuations of a Mesoscopic Capacitance CoupledCircuit with Power Source

Starting from classical equations of motion of the charges, we obtain thequantum fluctuations of the charges, and the currents of a mesoscopiccapacitance coupled circuit with power source by using a canonicaltransformation and a unitary transformation. The uncertainty relationsbetween charges and conjugate currents do not satisfy the minimumuncertainty relation, and is independent of the power source.