Physical basis for minimal time-energy uncertainty relation

A physical basis for the minimal time-energy uncertainty relation is formulated from basic high-energy hadronic properties such as the resonance mass spectrum, the form factor behavior, and the peculiarities of Feynman's parton picture. It is shown that the covariant oscillator formalism combines covariantly this time-energy uncertainty relation with Heisenberg's space-momentum uncertainty relation. A pictorial method is developed to describe the spacetime distribution of the localized probability density.     

Dynamical Measurement's Uncertainty in the Curved Space‐Time

The dynamical characteristics of measurement's uncertainty are investigated under two modes of Dirac field in the Garfinkle–Horowitz–Strominger dilation space‐time. It shows that the Hawking effect induced by the thermal field would result in an expansion of the entropic uncertainty with increasing dilation‐parameter value, as the systemic quantum coherence reduces, reflecting that the Hawking effect could undermine the systemic coherence. Meanwhile, the intrinsic relationship between the uncertainty and quantum coherence is obtained, and it is revealed that the uncertainty's bound is anti‐correlated with the system's quantum coherence. Furthermore, it is illustrated that the systemic mixedness is correlated with the uncertainty to a large extent. Via the information flow theory, various correlations including quantum and classical aspects, which can be used to form a physical explanation on the relationship between the uncertainty and quantum coherence, are also analyzed. Additionally, this investigation is extended to the case of multi‐component measurement, and the applications of the entropic uncertainty relation are illustrated on entanglement criterion and quantum channel capacity. Lastly, it is declared that the measurement uncertainty can be quantitatively suppressed through optimal quantum weak measurement. These investigations might pave an avenue to understand the measurement's uncertainty in the curved space‐time.     

Can Sobolev Inequality Be Written for Sharma-Mittal Entropy?

In this paper, we focus on Sobolev inequality in the context of Sharma-Mittal entropy. Using this new inequality, generalized entropic uncertainty relation in accordance with Sharma-Mittal entropy is derived and the pseudoadditivity relation has been obtained. This new entropic uncertainty relation has then been applied to physical examples such as one dimensional harmonic oscillator and Pösch-Teller potential. Finally, it has been shown that for certain values of the parameters of Sharma-Mittal measure, the present results reduce to the corresponding results of Shannon, Renyi and Tsallis measures.     

Energy levels of one-dimensional systems satisfying the minimal length uncertainty relation

The standard approach to calculating the energy levels for quantum systems satisfying the minimal length uncertainty relation is to solve an eigenvalue problem involving a fourth- or higher-order differential equation in quasiposition space. It is shown that the problem can be reformulated so that the energy levels of these systems can be obtained by solving only a second-order quasiposition eigenvalue equation. Through this formulation the energy levels are calculated for the following potentials: particle in a box, harmonic oscillator, Pöschl–Teller well, Gaussian well, and double-Gaussian well. For the particle in a box, the second-order quasiposition eigenvalue equation is a second-order differential equation with constant coefficients. For the harmonic oscillator, Pöschl–Teller well, Gaussian well, and double-Gaussian well, a method that involves using Wronskians has been used to solve the second-order quasiposition eigenvalue equation. It is observed for all of these quantum systems that the introduction of a nonzero minimal length uncertainty induces a positive shift in the energy levels. It is shown that the calculation of energy levels in systems satisfying the minimal length uncertainty relation is not limited to a small number of problems like particle in a box and the harmonic oscillator but can be extended to a wider class of problems involving potentials such as the Pöschl–Teller and Gaussian wells.     

Quantum integrals of motion and gravity wave experiment: Measurements in pure quantum states

Quantum limitations arising in measurements of a classical force acting on a quantum harmonic oscillator are studied in connection with the problem of increasing the sensitivity of gravity wave experiments. The physical nature of possible limits of sensitivity is elucidated. It originates in a degree of an uncertainty of an observable used for detecting an external force. This uncertainty can be made as small as desired for all moments of time for the observables corresponding to quantum integrals of motion. Advantages of integrals of motion with continuous spectra (like the operator of the initial coordinate) over integrals with discrete spectra (like energy) are discussed. An example of an observable suitable for exact continuous measurements of an external force independently on the initial state of the system—the difference link operator—is given. The general rule for constructing such “optimal observables” can be derived from the quantum optimal filtration theory. It is shown using Ehrenfest's theorem that no quantum limitations exist in principle for the accuracy of measurements of an external classical force acting on an arbitrary quantum system: limitations can appear only due to nonadequate measuring procedures. The general problem of finding the initial quantum states possessing the best sensitivity to an external force is formulated. The parametrically excited oscillator is briefly discussed, and it is shown that measuring the suitable integral of motion one can achieve the great gain in sensitivity. The role of quantum interference effects is emphasized.     

Remarks on the uncertainty relation for a Bose-like oscillator

The uncertainty relation is discussed for the canonical variablesq andp of a Bose-like oscillator that satisfy the general commutation relations. It is shown that the approach usually adopted for the case of the canonical commutation relation does not enable us to find the final expression of the required relation. It is possible, however, to prove a lemma concerning the possible minimum value of Δq·Δp.     

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K₀ of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.     

Information-theoretic measures for Morse and Pöschl–Teller potentials

The spreading of the quantum-mechanical probability cloud for the ground state of the Morse and modified Pöschl–Teller potentials, which controls the chemical and physical properties of some molecular systems, is studied in position and momentum space by means of global (Shannon's information entropy, variance) and local (Fisher's information) information-theoretic measures. We establish a general relation between variance and Fisher's information, proving that, in the case of a real-valued and symmetric wavefunction, the well-known Cramer–Rao and Heisenberg uncertainty inequalities are equivalent. Finally, we discuss the asymptotics of all three information measures, showing that the ground state of these potentials saturates all the uncertainty relations in an appropriate limit of the parameter.     

Non-Linear Transformations and Their Applications in Many-Body Physics

The transformations of the type which convert an exponential into a Gaussian and vice-versa and their applications in various areas of many-body physics are discussed. A new and general method of obtaining such transformations is given using the method of moments. It is compared with other methods which could be employed to obtain such transformations. In atomic physics, we have shown how such transformations can be used to obtain electron interaction energy for the ground state of Helium and Wigner transform for the ground state of H atom. It is shown how to bring angular momentum operators to linear form so that one can use the usual property of rotation operator to calculate their matrix elements. A new way of calculating the approximate eigenvalues of a Hamiltonian is given which combines the variational principles with the principle of maximum entropy. The anharmonic oscillator Hamiltonian is used to illustrate this new method. An interesting aspect of these transformations is that one could combine them with other transformations like Grassmann integration to calculate quantities of physical interest in closed form. A general matrix element of the harmonic oscillator is given which can be used to calculate usual quantities like the trace and density matrix. Some future applications are also discussed.     

Early Gedanken experiments of quantum mechanics revisited

The famous gedanken experiments of quantum mechanics have played crucial roles in developing the Copenhagen interpretation. They are studied here from the perspective of standard quantum mechanics, with no ontological interpretation involved. Bohr's investigation of these gedanken experiments, based on the uncertainty relation with his interpretation, was the origin of the Copenhagen interpretation and is still widely adopted, but is shown to be not consistent with the quantum mechanical view. We point out that in most of these gedanken experiments, entanglement plays a crucial role, while its buildup does not change the uncertainty of the concerned quantity in the way thought by Bohr. Especially, in the gamma ray microscope and recoiling double‐slit gedanken experiments, we expose the entanglement based on momentum exchange. It is shown that even in such cases, the loss of interference is only due to the entanglement with other degrees of freedom, while the uncertainty relation argument, which has not been questioned up to now, is not right.     

Probing Uncertainty Relations in Non-Commutative Space

In this paper, we compute uncertainty relations for non-commutative space and obtain a better lower bound than the standard one obtained from Heisenberg’s uncertainty relation. We also derive the reverse uncertainty relation for product and sum of uncertainties of two incompatible variables for one linear and another non-linear model of the harmonic oscillator. The non-linear model in non-commutating space yields two different expressions for Schrödinger and Heisenberg uncertainty relation. This distinction does not arise in commutative space, and even in the linear model of non-commutative space.

     

Unification theory of classical statistical uncertainty relation and quantum uncertainty relation and its applications

General classical statistical uncertainty relation is deduced and generalized to quantum uncertainty relation. We give a general unification theory of the classical statistical and quantum uncertainty relations, and prove that the classical limit of quantum mechanics is just classical statistical mechanics. It is shown that the classical limit of the general quantum uncertainty relation is the general classical uncertainty relation. Also, some specific applications show that the obtained theory is self-consistent and coincides with those from physical experiments.     

Green's function and scattering matrix in a discrete oscillator basis

Convenient analytic finite-dimensional approximations for basic operators of scattering theory-specifically, the Green's function and the off-shell T matrix—are constructed in an oscillator basis for real-and complex-valued local and nonlocal interaction potentials. It is shown that the approximate operators converge smoothly to their exact counterparts as the dimensions of the oscillator basis are increased step by step. The simple and rather accurate formulas obtained in this study can be widely used in various applications of quantum scattering theory.     

An Informational Characterization of Schrödinger's Uncertainty Relations

Heisenberg's uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement. The information concerning incompatibility (non-commutativity) of observables is well included but that concerning correlation is missing. Schrödinger's uncertainty relations remedy this defect by supplementing the correlation in terms of anti-commutators. However, both Heisenberg's uncertainty relations and Schrödinger's uncertainty relations are expressed in terms of variances, which are not good measures of uncertainty in general situations (e.g., when mixed states are involved). By virtue of the Wigner–Yanase skew information, we will establish an uncertainty relation along the spirit of Schrödinger from a statistical inference perspective and propose a conjecture. The result may be interpreted as a quantification of certain aspect of the celebrated Wigner–Araki–Yanase theorem for quantum measurement, which states that observables not commuting with a conserved quantity cannot be measured exactly.     

A rigorous uncertainty relation (Cauchy Inequality) for an electromagnetic field in quantum superpositions of coherent states and in a two-photon coherent state

It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrödinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.     

不确定度的教学实践

在５年教学实践的基础上,总结了在基础物理实验教学中引入不确定度的一些看法．对如何处理误差和不确定度的关系以及以合成标准不确定度为核心的教学方式作了说明．此外对应当注意和改进的一些相关问题也进行了讨论     

Uncertainty relation for the discrete Fourier transform

We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=e(i phi) VU. Its most important application is to constrain how much a quantum state can be localized simultaneously in two mutually unbiased bases related by a discrete fourier transform. It provides an uncertainty relation which smoothly interpolates between the well-known cases of the Pauli operators in two dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the harmonic oscillator equation.