Uncertainty of the Uncertainty Principle(上)

许多人以为uncertainty principle是量子理论的一个基本原理,但它不足是.它来自对任何正定空间都成立的Schwarz不等式,不过反映物理学中变量通过乘法或者某个方程耦合到一起的一个推论,同各种境遇中的两难选择产生了联想.量子力学中的uncertainty relation基于算符的非对易性,没有任何测量可以精确到能强化其正确性的地步.从经典扩散方程也能得出uncertainty relation关系.关于uncertainty principle有大量的错误诠释和滥用,中文语境下是翻译成“测不准原理”还是“不确定性原理”也值得商榷.     

Experimental Realization of Popper's Experiment: Violation of the Uncertainty Principle?

An entangled pair of photons (1 and 2) are emitted in opposite directions. A narrow slit is placed in the path of photon 1 to provide the precise knowledge of its position on the y-axis and this also determines the precise y-position of its twin, photon 2, due to quantum entanglement. Is photon 2 going to experience a greater uncertainty in momentum, that is, a greater Δp_y because of the precise knowledge of its position y? The experimental data show Δy Δ p_y < h for photon 2. Can this recent realization of the thought experiment of Karl Popper signal a violation of the uncertainty principle?     

Generalized Uncertainty Principle and the Quantum Entropy of Rotating Black Hole

The entropy of rotating Kerr-Newman-Kasuya black hole due to massive charged fields （bosons and fermions） is calculated by using the new equation of state density motivated by the generalized uncertainty relation. The result shows the entropy does not depend on the mass and the charge but the parameter A, the area A and the spin of the fields. Moreover, an improved approximation is provided to calculate the scalar entropy.     

物理学咬文嚼字之四十四 Uncertainty of the Uncertainty Principle（上）

许多人以为uncertainty principle是量子理论的一个基本原理,但它不足是．它来自对任何正定空间都成立的Schwarz不等式,不过反映物理学中变量通过乘法或者某个方程耦合到一起的一个推论,同各种境遇中的两难选择产生了联想．量子力学中的uncertainty relation基于算符的非对易性,没有任何测量可以精确到能强化其正确性的地步．从经典扩散方程也能得出uncertainty relation关系．关于uncertainty principle有大量的错误诠释和滥用,中文语境下是翻译成“测不准原理”还是“不确定性原理”也值得商榷．     

The Inception of Symplectic Geometry: the Works of Lagrange and Poisson During the Years 1808–1810

We analyse articles by Lagrange and Poisson written two 200 years ago which are the foundation of present-day symplectic and Poisson geometry.     

The Generalized Uncertainty Relation and The Entropy of Non-stationary and Slowly Changing Reissner-Nordstr?m Black Hole

Applying the generalized uncertainty relation to the thin film brick-wall model, the entropy of Dirac Field in Non-stationary and Slowly Changing Reissner-Nordstr?m Black Hole is obtained. The result shows that the entropy is still proportional to the horizon area of the black hole, and black hole entropy is just identical to the entropy of the quantum state near the event horizon, in addition, the divergence of state density without any cut-off parameter is avoided during black hole entropy calculation.     

Λ and the Heisenberg Principle

A time dependent “cosmological constant” Λ(t) is conjectured, in terms of the Gaussian curvature of the causal horizon. It is nonvanishing even in Minkowski space because of the lack of informations beyond the light cone. Using the Heisenberg Principle, the corresponding energy of the quantum fluctuations localized on the past or future null horizons is proportional to Λ^1/2. We compute Λ(t) for the (Lorenzian version) of the (conformally flat) Hawking wormhole geometry (written in static spherical Rindler coordinates) and for the de Sitter spacetime. A possible explanation of the Hawking temperature is proposed, in terms of a constant Λ.     

Symplectic Schemes and the Shooting Method for Eigenvalues of the Schrodinger Equation

The one-dimensional time-independent Schr6dinger equation is transformed into a Hamiltonian canonical equation by means of the Legendre transformation, then the symplectic schemes and a new shooting method extended to the eigenvalues of the Schr6dinger equation. The method is applied to the calculations of one-dimensional harmonic oscillator, an anharmonic oscillator and the hydrogen atom. The numerical results are in good agreement with the exact ones.     

物理学咬文嚼字之四十四 Uncertaintyof the Uncertainty Principle（下）

经典不确定性所谓的＂不能够无限精确地确定一个物理量＂给欧洲的物理学家们带来了惶恐,但是经典世界里存在exact positioning的说法似乎也是毫无根据的.确定公路上一辆汽车的位置,精确到毫米量级都是没必要的、原则上不可能的（用汽车的哪个部位标定它的位置呢,部位又如何定位呢？）,遑论无限精     

Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups

The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials with the weight exp tr (X Ω Y Ω ^† ) are computed for the orthogonal and symplectic groups. We proceed in two steps. First, the integral over the compact group is recast into a Gaussian integral over strictly upper triangular complex matrices (with some additional symmetries), supplemented by a summation over the Weyl group. This result follows from the study of loop equations in an associated two-matrix integral and may be viewed as the adequate version of Duistermaat–Heckman’s theorem for our correlation function integrals. Secondly, the Gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions.     

（Anti-）de Sitter Black Hole Entropy and Generalized Uncertainty Principle

We generalize the method that is used to study corrections to Cardy-Verlinde formula due to generalized uncertainty principle and discuss corrections to Cardy-Verlinde formula due to generalized uncertainty principle in （anti）- de Sitter space. Because in de Sitter black hole spacetime the radiation temperature of the black hole horizon is different from the one of the cosmological horizon, this spacetime is a thermodynamical non-equilibrium spacetime.     

The Definition of Mach’s Principle

Two definitions of Mach’s principle are proposed. Both are related to gauge theory, are universal in scope and amount to formulations of causality that take into account the relational nature of position, time, and size. One of them leads directly to general relativity and may have relevance to the problem of creating a quantum theory of gravity.     

High-Order Hamilton＇s Principle and the Hamilton＇s Principle of High-Order Lagrangian Function

In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equations and four-order Lagrangian equations are obtained from the high-order Hamilton＇s principle. Finally, the Hamilton＇s principle of high-order Lagrangian function is given.     

Internal Structure of the Heisenberg and Robertson-Schrödinger Uncertainty Relations

It is known that the Heisenberg and Robertson-Schrödinger uncertainty relations can be replaced by sharper uncertainty relations in which the “classical” (depending on the gradient of the phase of the wave function) and “quantum” (depending on the gradient of the envelope of the wave function) parts of the variances 〈(Δx)²〉 and 〈(Δp)²〉 are separated. In this paper, three types of uncertainty relations for a different number of classical parts (2, 1 or 0) with different time behaviour of their left-hand and right-hand sides are discussed. For the Gaussian wave packet and two classical parts, the left-hand side of the corresponding relations increases for t→∞ as t ² and is much larger than ? ²/4. For one classical part, the left-hand side of the corresponding relation goes to the right-hand side equal to ? ²/4. For no classical part, both the right-hand and left-hand sides of the corresponding relation go quickly to zero. Therefore, the well-known limitations following from the usual uncertainty relations can be overcome in the corresponding measurements.     

A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications

Symplectic and symmetry analysis for studying MHD superfluid flows is devised, a new version of the Z. Peradzyński (Int. J. Theor. Phys. 29(11):1277–1284, [1990]) helicity theorem based on differential-geometric and group-theoretical methods is derived. Having reanalyzed the Peradzyński helicity theorem within the modern symplectic theory of differential-geometric structures on manifolds, a new unified proof and a new generalization of this theorem for the case of compressible MHD superfluid flow are proposed. As a by-product, a sequence of nontrivial helicity type local and global conservation laws for the case of incompressible superfluid flow, playing a crucial role for studying the stability problem under suitable boundary conditions, is constructed.     

The effect of hydrogen-bonding cooperativity on the strength and properties of σ-hole interactions: an ab initio study

Ab initio calculations at the MP2/aug-cc-pVTZ level of theory are performed to investigate the effect of hydrogen-bonding cooperativity on the strength and bonding properties of σ-hole interaction in linear FCl???(NCH)_n=2–5, FHS???(NCH)_n=2–5, FH₂P???(NCH)_n=2–5 and FH₃Si???(NCH)_n=2–5 clusters. It is found that the cooperative effects in the hydrogen-bonding tend to strengthen the σ-hole interaction. However, these effects are almost saturated in the larger clusters (n > 5). For a given cluster, the amount of bond contraction in FCl???(NCH)_n is more important than other systems. A nice linear relationship is found between the σ-hole bond energies and absolute ¹⁵N chemical shieldings or spin–spin coupling constants across the σ-hole bond.