Maximally causal quantum mechanics

We present a new causal quantum mechanics in one and two dimensions developed recently at TIFR by this author and V Singh. In this theory both position and momentum for a system point have Hamiltonian evolution in such a way that the ensemble of system points leads to position and momentum probability densities agreeing exactly with ordinary quantum mechanics     

Short distance freedom of quantum gravity

Fourth order derivative gravity in 3+1$3+1$ dimensions is perturbatively renormalizable and is shown to describe a unitary theory of gravitons in a limited coupling parameter space. The running gravitational constant which includes graviton contribution is computed. Generically, gravitational Newton?s constant vanishes at short distances in this perturbatively renormalizable and unitary theory.     

Langevin formulation of quantum mechanics

Summary We present in a rather pedagogical way a new formulation of quantum mechanics. Our starting point is the path integral representation of the quantum-mechanical propagator analytically continued to imaginary timeW(X″, s″|X′, s′). We view the set of random paths contributing toW(X″, s″|X′, s′) as the manifold of solutions of a Langevin equation with a Gaussian white noise. We thus obtainW(X″, s″|X′, s′) as the noise-average of a suitable functional of the solution of the Langevin equation. The standard quantum-mechanical propagator is finally recovered by analytically continuingW(X″, s″|X′, s′) back to real time. The present approach allows for a straightforward application of standard methods of classical stochastic processes to quantum-mechanical problems and offers a new promising way to perform computer simulations of quantum-dynamical systems. To speed up publication, the author has agreed not to receive proofs which have been supervised by the Scientific Committee.     

Supersymmetric quantum mechanics for two-dimensional disk

The infinite square well potential in one dimension has a smooth supersymmetric partner potential which is shape invariant. In this paper, we study the generalization of this to two dimensions by constructing the supersymmetric partner of the disk billiard. We find that the property of shape invariance is lost in this case. Nevertheless, the WKB results are significantly improved when SWKB calculations are performed with the square of the superpotential. We also study the effect of inserting a singular flux line through the center of the disk.     

Exactly solvable diffusion models in the framework of the extended supersymmetric quantum mechanics

This work is aimed at demonstrating the possibility to construct new exactly-solvable stochastic systems by use of the extended supersymmetric quantum mechanics (N=4SUSY QM) formalism. A feature of the proposed approach consists in the fact that obtained new potentials, which enter the Langevin equation, and so probability densities have a parametric freedom. The latter allows one to change the form of potentials without changing the temporal behavior of the probability densities.     

Supersymmetric quantum mechanics and the index theorem

We consider the classical mechanics of the spinning particle and investigate which Abelian interactions can be added without breaking supersymmetry. A quantum theory is presented. The well known index theorem for the Dirac operator is extended to take into account the effect of anti-symmetric Abelian tensor fields. Furthermore interactions with non-Abelian anti-symmetric tensor fields are investigated. It turns out in both cases that these fields do not give any non-trivial contributions to the index.     

q-Deformation of algebraic structures of quantum and classical mechanics

There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant.     

工科专业的量子力学教学方法探索

主要从量子力学发展史,量子力学概念,多媒体技术运用,以及针对工科专业的实际应用4个方面,对工科专业的量子力学教学方法进行了探索,激发学生学习量子力学的兴趣,提高教学质量.     

Functional analysis and quantum mechanics: an introduction for physicists

We give an introduction to certain topics from functional analysis which are relevant for physics in general and in particular for quantum mechanics. Starting from some examples, we discuss the theory of Hilbert spaces, spectral theory of unbounded operators, distributions and their applications and present some facts from operator algebras. We do not give proofs, but present examples and analogies from physics which should be useful to get a feeling for the topics considered.     

Kink mass quantum shifts from SUSY quantum mechanics

In this paper a new version of the DHN (Dashen–Hasslacher–Neveu) formula, which is used to compute the one-loop order kink mass correction in (1+1)-dimensional scalar field theory models, is constructed. The new expression is written in terms of the spectral data coming from the supersymmetric partner operator of the second-order small kink fluctuation operator and allows us to compute the kink mass quantum shift in new models for which this calculation was out of reach by means of the old formula.     

The Wigner function in the relativistic quantum mechanics

A detailed study is presented of the relativistic Wigner function for a quantum spinless particle evolving in time according to the Salpeter equation.     

Higher dimensional supersymmetric quantum mechanics and Dirac equation

We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering ‘mass’ as a function of coordinates. Its usefulness in solving potential problems is discussed with specific examples. We also discuss the ‘physical’ significance of the supersymmetric states in this formalism.     

量子力学双语教学研究

从量子力学的教学实践出发,从双语教材的选择、教学方法与手段、考试与考核三个方面讨论了量子力学课程进行双语教学应当注意的问题.     

量子力学诠释问题

量子力学的建立不仅奠定了当代科学的基础,而且在推动当代技术革命方面取得了惊人的成功。然而,对于量子力学诠释(interpretation of quantum mechanics)——理解波函数如何刻画微观世界,人们迄今为止并未形成共识。量子力学发展的这种二元状态不仅带来了认识论方面的误导,而且依据备受争议的哥本哈根诠释建立起来的量子技术会有许多根本性问题。
量子力学的哥本哈根诠释存在二元结构的问题：微观世界的运动用量子力学描述,是一个幺正演化,而观察或测量却依赖于量子系统外部的经典世界(仪器、观察者、环境),表现出来的波包塌缩是非幺正的。为此,包括爱因斯坦、薛定谔、温伯格等在内的一些著名学者对哥本哈根诠释提出了尖锐的批评。80年过去了,为克服量子力学的哥本哈根诠释二元论困境,人们提出各种各样的量子力学诠释,包括多世界诠释、量子退相干诠释、自洽历史诠释以及量子达尔文主义等。文章将简要介绍和评述这些量子力学诠释的基本思想、它们之间的逻辑关系及其实验检验的可能性。进一步澄清量子力学诠释中的基本概念,可以避免量子观念滥用导致的意识论上的问题和量子技术发展误入歧途。     

On reformulating quantum mechanics and stochastic theory

A stochastic theory approach is used to formulate the theory of quantum mechanical motion. Apart from giving a unifying point of view to quantum mechanics and stochastic theory, the new formulation is not limited to forces derivable from a potential. A nonlinear dynamical law is deduced in contradistinction to previous works in whichad hoc linear laws are postulated.     

关于量子力学中态叠加原理的讨论

讨论了量子力学中态叠加原理的意义,评论了它的不同表述和解说,并且以简单明了的语言来叙述这一原理．     

Bootstrapping Calabi–Yau quantum mechanics

Recently, a novel bootstrap method for numerical calculations in matrix models and quantum mechanical systems was proposed. We apply the method to certain quantum mechanical systems derived from some well-known local toric Calabi–Yau geometries, where the exact quantization conditions have been conjecturally related to topological string theory. We find that the bootstrap method provides a promising alternative for the precision numerical calculations of the energy eigenvalues. An improvement in our approach is to use a larger set of two-dimensional operators instead of one-dimensional ones. We also apply our improved bootstrap methods to some non-relativistic models in the recent literature and demonstrate better numerical accuracies.