Quantization of Underdamped, Critically Damped, and Overdamped Electric Circuits with a Power Source

We have investigated the quantum mechanical effect of the underdamped, critically damped, and overdamped electric circuits with a power source. The charge of the underdamped circuit oscillates while those of the critically damped and overdamped ones don't. The wave function of the system of overdamped circuit represented parabolic cylinder function while underdamped circuit was represented by well-known Hermite polynomial. The eigenvalues of underdamped circuit is discrete while those of the critically damped and overdamped ones are given as continuously.     

WKB Wave Function of the General Time-Dependent Quadratic Hamiltonian System

We derived the WKB wave function for the general time-dependent quadratic Hamiltonian system using a unitary transformation method. We applied our research to sinusodially drived Caldirola–Kanai oscillator and confirmed that the time evolution of our approximated WKB wave function is similar to that of the exact one. This wave function can be used to analyze the interference between the probability amplitudes contributed by the area of overlap in phase space of quantum states.     

Group Representation of Relativistic Quantum Crystal Optics

A group representation of radiation propagation in an anisotropic medium is developed. The system of wave equations for electromagnetic potentials, obtained from the Maxwell equations with account for the constitutive equations, has been factorized. It is shown that the linear differential operator of the factorized system is orthogonal in transparent crystals and unitary in gyrotropic ones and is represented through the momentum operator. On the basis of the commutation relations for the components of this operator, the eigenvalue problem has been solved and the expression for the change in the radiant energy in the crystal in the form of spherical waves has been obtained. The dependences of the ray and phase velocities and the polarization vectors of waves on the birefringence anisotropy and gyrotropy as well as on the angular momentum, displacement current, and bound charge determining them have been analyzed. It has been established that in the general case of gyrotropic crystals where the nonreciprocity phenomenon takes place and in magnetoelectrics Maxwell equations are represented in a form similar to the Dirac equations and the electromagnetic radiation is correctly described by means of bispinors and is quantized as fermions.     

Hilbert Space of Probability Density Functions Based on Aitchison Geometry

The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison＇s ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space. A Hilbert space of densities, whose logarithm is square-integrable, is obtained as the natural completion of the pre-Hilbert space.     

Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

Let U_m be an m×m Haar unitary matrix and U_[m,n] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U_[m,n] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U_[m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.     

Quantum Dynamics of Systems Connected by a Canonical Transformation

Quantum Hamiltonian systems corresponding to classical systems related by a general canonical transformation are considered. The differential equation to find the unitary operator, which corresponds to the canonical transformation and connects quantum states of the original and transformed systems, is obtained. The propagator associated with their wave functions is found by the unitary operator. Quantum systems related by a linear canonical point transformation are analyzed. The results are tested by finding the wave functions of the under-, critical-, and over-damped harmonic oscillator from the wave functions of the harmonic oscillator, free-particle system, and negative harmonic potential system, using the unitary operator to connect them, respectively.     

Probabilistic Teleportation of a Three-Particle State

A scheme for teleporting a three-particle state is proposed when three pairs of entangled particles are used as quantum channels. Quantum teleportation can be successfully realized with a certain probability if the receiver adopts an appropriate unitary-reduction strategy. The probability of successful teleportation is determined by the smaller coetficients of the three entangled pairs.     

一类矩阵迹不等式的推广

利用平均值不等式 ,得到关于矩阵迹的不等式 :如果 A1 ,A2 ,… ,Am 皆为 n阶 Hermite半正定矩阵 ,且乘法两两可交换 ,0 相似文献   

广义酉矩阵与广义Hermite矩阵

给出了广义酉矩阵与广义(斜)Hermite矩阵的概念，研究了它们的性质及其与酉阵、共轭辛阵、Hermite阵、Hamilton及广义逆矩阵之间的联系；取得了许多新的结果；推广了酉矩阵、Hermite阵与斜Hermite阵间的相应结果，特别将正交阵的广义Cayley分解推广到了广义酉矩阵上；将各类酉矩阵、Hermite矩阵及广义逆矩阵统一了起来．