量子力学中一个自发对称性破缺的模型

IF 0.2 Q4 MATHEMATICS, APPLIED
A. Restuccia, A. Sotomayor, V. Strauss
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引用次数: 0

摘要

我们的目标是为一维量子力学问题中出现的自发对称性破缺现象找到一个模型。为此,我们考虑与实线的两个内部点有关的边值问题,它们相对于原点对称。这种方法可以看作是包含移位狄拉克函数及其导数的奇异势的存在。从数学的角度出发,我们将自伴随扩展技术应用于一个对称微分算子,该算子的定义域包含在上述两点上消失的光滑函数。我们计算了相应扩展的解,并研究了其在内部点改变位置时的行为。这些扩展的定义域可以包含一些在上述点上具有不可微性或不连续的函数,后者可以解释为在同一点上的奇异势的外观。其次,发现了破对称束缚态。更准确地说,对于边界条件的特定纠缠,存在一个基态,产生自发的对称破缺,在外部波动引起的退相干现象下稳定。我们在像NH3这样的分子的“手性”对称性破缺状态的背景下讨论这个模型。我们证明,在希尔伯特空间方法中,如果上述内部点之间的距离趋于零,则自发对称性破缺就会消失。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On a Model of Spontaneous Symmetry Breaking in Quantum Mechanics
Our goal is to find a model for the phenomenon of spontaneous symmetry breaking arising in one dimensional quantum mechanical problems. For this purpose we consider boundary value problems related with two interior points of the real line, symmetric with respect to the origin. This approach can be treated as a presence of singular potentials containing shifted Dirac delta functions and their derivatives. From mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator which has a domain containing smooth functions vanishing in two mentioned above points. We calculate the resolvent of corresponding extension and investigate its behavior if the interior points change their positions. The domain of these extensions can contain some functions that have non differentiability or discontinuity at the points mentioned above, the latter can be interpreted as an appearance of singular potentials centered at the same points. Next, broken-symmetry bound states are discovered. More precisely, for a particular entanglement of boundary conditions, there is a ground state, generating a spontaneous symmetry breaking, stable under the phenomenon of decoherence provoked from external fluctuations. We discuss the model in the context of the “chiral” broken-symmetry states of molecules like NH3. We show that within a Hilbert space approach a spontaneous symmetry breaking disappears if the distance between the mentioned above interior points tends to zero.
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来源期刊
CiteScore
1.00
自引率
50.00%
发文量
1
期刊介绍: Series «Mathematical Modelling, Programming & Computer Software» of the South Ural State University Bulletin was created in 2008. Nowadays it is published four times a year. The basic goal of the editorial board as well as the editorial commission of series «Mathematical Modelling, Programming & Computer Software» is research promotion in the sphere of mathematical modelling in natural, engineering and economic science. Priority publication right is given to: -the results of high-quality research of mathematical models, revealing less obvious properties; -the results of computational research, containing designs of new computational algorithms relating to mathematical models; -program systems, designed for computational experiments.
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