从二维正态分布密度函数的角度研究量子力学基本表象

利用二维正态分布密度函数和有序算符内的积分技术,简捷地得到了坐标本征态、动量本征态、坐标-动量中介表象和相干态在Fock表象中的表达式.     

厄米算符本征函数完备性的一般证明

从一组完备的本征函数开始,以表象原理为前提,建立一个唯象的公式,利用反证法证明了力学量算符的完备性,即这一组完备的本征函数可以投影到力学量算符的本征函数空间中.从而给出了力学量算符本征函数完备性的一般证明.     

《量子力学》自学辅导之八：狄拉克符号和占有数表象（续）

《量子力学》自学辅导之八──狄拉克符号和占有数表象（续）曾心愉，宋宇辰，裴文杰（复旦大学物理系，复旦大学理论物理骨干教师班上海２００４３３）２关于一维线性谐振子的讨论２．１坐标表象一维线性谐振子Ｈ算符及其本征函数在坐标表象中为式中的本征值为由厄米多项...     

二维耦合量子谐振子的本征值和本征函数

运用广义线性量子变换理论,给出一类二维耦合量子谐振子的能量本征值、本征函数、坐标和动量算符在能量表象中的矩阵元及演化算符.     

在局域子空间中计算给定范围内的能量本征值

通过能量算符δ函数作用于完全随机格点波函数,构造了可用于直接计算给定范围[Emin,Emax]内能量本征值和本征函数的局域子空间.在非正交局域基下详细推导了交迭积分和哈密顿算符在分立位置表象中的表示,讨论了广义本征值问题的解法.以Morse势和Henon-Heiles势的多个能量范围为例检验了算法     

各向同性谐振子径向基本算符的应用

利用基本算符D/Dρ,ρ,1/ρ对本征函数RNL的递推关系和任意力学量算符对函数RNL的作用结果,容易求得力学量的矩阵元和平均值等。     

用算符因式分解法解氢原子的径向函数

在初等量子力学的教学中,首先遇到的是求解一维线性谐振子和氢原子等基本问题,通常总是利用级数方法来解此二阶微分算符的本征方程,但往往由于数学上的原因,而使教与学二方面都遇到一定的困难.为此,若将二次量子化中运用的占有数表象及其基本算符(产生算符和湮灭算符)的概念引入后,则上述诸问题均能很简洁地求得所需的本征值和本征函数. 在占有数表象中,任何一个力学量的算符,均可用其二个基本算符来表示,将此原理应用到解具体的薛定谔方程中去时,则原来是一个二阶微分算符的本征方程,可以分解成二个相互伴随的一阶微分算符乘积的本征方程,…     

关于波函数按力学量算符本征函数系展开的若干问题

本文指出波函数按力学量算符的本征函数系展开，要求表示力学量的算符线性、厄米性、且全连续。证明了全连续的线性厄米算符有正交的完全的本征函数系，伴就正交函数系的封闭性、完全世及展开的唯一性进行了讨论。     

从泡利矩阵解析量子力学中的几类典型计算问题

通过泡利矩阵分别解析量子力学中的几类典型计算问题, 即算符的本征值和本征函数、 么正变换矩阵、 力学量的期望值及可能取值的概率     

《量子力学》自学辅导之七─—表象理论

《量子力学》自学辅导之七─—表象理论曾心愉，宋宇辰，裴文杰（复旦大学物理系，复旦大学理论物理骨干教师班，上海２００４３３）根据量子力学基本原理，量子力学的状态用波函数描述，力学量用线性厄米算符描述。但我们以前所使用的波函数及力学量算符都是以坐标这个力...     

Optical potential discrete variable representation method in the adiabatic representation

The optical potential discrete variable representation method (OP-DVR) has been applied recently to calculate resonances in the framework of the diabatic representation [J. Chem. Phys. 101, 7580 (1994)]. This method is based on the conjoint use of the discrete variable representation (DVR) method and the properties of a complex absorbing potential (CAP). The OP-DVR method is the DVR version of the CAP stabilization method initially proposed by Jolicard and Austin [Chem. Phys. Lett. 121, 106 (1985)]. In the present study, we show that this efficient and accurate method can also be applied within the adiabatic representation since it allows one to overcome in a simple way, numerical difficulties associated with the first derivative operator which appears in the expression of non adiabatic couplings. Within the OP-DVR method, the choice of the representation (diabatic or adiabatic) is governed by physical arguments and by the fact that the potentials and the couplings are known in one or the other of these two representations. In the case where the potentials and the couplings are obtained in the adiabatic representation, we show in this paper that the transformation into the diabatic framework is not necessary. We demonstrate that the discrete variable representation can be a simple and an efficient way to deal with the adiabatic representation. Received: 30 April 1998 / Revised: 29 September 1998 /Accepted: 21 October 1998     

Topological representation of leptons

It is proposed that on the quantum surface of a topological-expansion surface pair the lepton be associated with a 2-triangle ? \(\bar I\bar Y\) ? band whose perimeter is built from the same two peripheral \(\bar I\) -triangle edges that constitute an antiquark's contribution to a hadron-disk perimeter. Although all lepton “internal” degrees of freedom are in one to one correspondence with antiquark attributes, it is shown that the simplest representation of quarkgeneration (Cabibbo) mixing cannot be invoked for leptons. The proposed lepton contains a neutral \(\bar Y\) triangle and permits a topological representation of baryon-lepton mixing. Described in parallel is a quantum-classical surface pair to represent the (standard) electroweak leptonic current.     

The unitary representation operators

A representation (called theU-representation) which remains unitary for all spins and for all ranges of velocities was obtained by us in a recent paper. We obtain here relevant expressions for the boosts operator and the observables in such a representation.     

Coherent-mode representation of supercontinuum

We apply the coherent-mode expansion to correlation functions used to describe the coherence properties of supercontinuum generated in nonlinear ?bers. We show that the leading term of the expansion represents the quasi-coherent part of the ?eld while the quasi-stationary part is embedded into the higher-order modes. The evolution of the modal expansion and the number of modes needed to describe the supercontinuum ?eld are also discussed.     

Fast transform from an adaptive multi-wavelet representation to a partial Fourier representation

We present a fast algorithm to compute the partial transformation of a function represented in an adaptive pseudo-spectral multi-wavelet representation to a partial Fourier representation. Such fast transformations are useful in many contexts in physics and engineering, where changes of representation from a piece wise polynomial basis to a Fourier basis. The algorithm is demonstrated for a Gaussian in one and in three dimensions. For 2D, we apply this approach to a Gaussian in a periodic domain. The accuracy and the performance of this method is compared with direct summation.     

On the spinor representation

A systematic study of the spinor representation by means of the fermionic physical space is accomplished and implemented. The spinor representation space is shown to be constrained by the Fierz–Pauli–Kofink identities among the spinor bilinear covariants. A robust geometric and topological structure can be manifested from the spinor space, wherein the first and second homotopy groups play prominent roles on the underlying physical properties, associated to fermionic fields. The mapping that changes spinor fields classes is then exemplified, in an Einstein–Dirac system that provides the spacetime generated by a fermion.     

RPA in wavefunction representation

The RPA is formulated in subspaces of coordinate-like and momentum-like 1 ph operators. This allows to embed a large class of approximative schemes into a generalized RPA treatment. We give a detailed formulation in terms of wavefunctions in coordinate space which is ideally suited to practical programming. In particular, we work out the reduction to spherical tensors in the case of spherical symmetry which is most often the starting point in finite Fermion systems.     

Holomorphic representation of coherent states of a singular oscillator and their Darboux representation

We consider the coherent states of a singular oscillator; these states are defined to be the characteristic states of an operator that reduces the number of basis functions for a discrete spectrum to one. We use the Darboux transform to study the coherent states of a transformed Hamiltonian. We obtain an expression for measures that can be used to decompose unity. We construct a holomorphic representation for the state vectors in the space of functions holomorphic everywhere in the complex plane, including vectors for discrete spectra and coherent states. We obtain a holomorphic representation of the Darboux transformation operators. Tomsk State University. Institute of High-Energy Electronics. Siberian Division of the Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 46–53, February, 1998.