The Bloch-vector space for N-level systems -- the spherical-coordinate point of view

球坐标系中自由狄拉克传播子的路径积分形式

Path integral formalism for the free Dirac propagator in spherical coordinates

本文在球坐标系中用费曼路径积分方法推导自由自旋1/2费米子的相对论格林函数，并通过坐标变换将其化为可精确求解的形式，结果用球贝塞尔函数和球旋量表示。

Explicit Spin Coordinates

The recently established existence of spherical harmonic functions, $Y_\ell^{m}(θ,ϕ)$ for half-odd-integer values of $\ell$ and $m$, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles $θ_s, ϕ_s$, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, $S^2, S_z$, are represented by the usual differential operators in spherical polar coordinates (commonly used for $L^2, L_z$), and their electron-spin eigenfunctions are $\sqrt{\sinθ_s} \exp(…

Delta-like singularity in the Reduction of the Laplace Equation for Spherical Coordinates and the Validity of Radial Schrodinger Equation

By careful exploration of separation of variables into the Laplacian in spherical coordinates, we obtain the extra delta-like singularity, elimination of which restricts the radial wave function at the origin. This constraint has the form of boundary condition for the radial Schrodinger equation.

Singular Behavior of the Laplace Operator in Polar Spherical Coordinates and Some of Its Consequences for the Radial Wave Function at the Origin of Coordinates

Singular behavior of the Laplace operator in spherical coordinates is investigated. It is shown that in course of transition to the reduced radial wave function in the Schrodinger equation there appears additional term consisting the Dirac delta function, which was unnoted during the full history of physics and mathematics. The possibility of avoiding this contribution from the reduced radial equation is discussed. It is demonstrated that for this aim the necessary and sufficient condition is requirement the fast enough falling of the wave function at the origin. The result does not depend on …

开放量子Rabi模型中的量子比特-振荡器关系：耗散的作用

Qubit-oscillator relationships in the open quantum Rabi model: the role of dissipation

该研究针对耗散开放量子Rabi模型，推导量子比特布洛赫坐标与振荡器可观测量的函数关系，通过多类模拟验证其在强耦合内的有效性，分析了浴耦合对计算的简化作用及超强耦合下非马尔可夫效应带来的计算困难。

Some intricacies of the momentum operator in quantum mechanics

In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in general curvilinear coordinates arises. We have tried to elucidate the points related to the definition of the momentum operator taking spherical polar coordinates as our specimen coordinate system and proposed an…

Hypervirial and Ehrenfest theorems in spherical coordinates: systematic approach

Elaboration of some fundamental relations in three dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator time derivative and the time derivative of mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which …

球坐标下Schrödinger-Dunkl框架中的信息论度量

Information theoretic measures within Schrödinger-Dunkl framework in spherical coordinates

本文在球坐标下解析求解含Deng-Fan势的Schrödinger-Dunkl方程，获得双原子分子振转能级及多种信息论度量，解析与数值偏差极小。

Robot Affect: the Amygdala as Bloch Sphere

In the design of artificially sentient robots, an obstacle always has been that conventional computers cannot really process information in parallel, whereas the human affective system is capable of producing experiences of emotional concurrency (e.g., happy and sad). Another schism that has been in the way is the persistent Cartesian divide between cognition and affect, whereas people easily can reflect on their emotions or have feelings about a thought. As an essentially theoretical exercise, we posit that quantum physics at the basis of neurology explains observations in cognitive emotion p…

The generalized MIC-Kepler system

This paper deals with dynamical system that generalizes the MIC-Kepler system. It is shown that the Schrödinger equation for this generalized MIC-Kepler system can be separated in spherical and parabolic coordinates. The spectral problem in spherical and parabolic coordinates is solved.