光学信号分析在量子力学中的研究进展和量子层析技术
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摘要
在量子力学中,物理上的可观测量使用厄米算符的形式来表达的,它们之间一般是不对易的。故而当我们将经典力学中的函数推广到量子力学范围时,其所对应的量子力学么正算符并不唯一,根据不同的对应规则或者说是算符编序形式,有不同的表达形式。故而人们在将经典函数和量子力学算符对应起来的过程中,总是面临着算符编序的问题。最常见的算符编序形式有正规乘积编序,反正规乘积编序和Weyl编序。其中Weyl编序具有在相似变换下的不变性,在实际运算中比较方便,故而应用最广泛。迄今为止,对于编序后算符的运算问题,人们主要用两种方法来解决,即李代数方法和相干态表象微分操作法。这两种方法对一个较为复杂的排序算符的计算无能为力。有鉴于此,范洪义教授基础了有序算符内的积分技术(]WOP技术)来处理排序算符的计算问题。IWOP技术将原本只适用于经典可对易函数的牛顿-莱布尼茨积分推广到了对量子力学ket-bra型非对易量子算符的运算,从而建立起了联系非对易q数和对易的c数之间的桥梁。
本文在算符编序理论的基础之上,利用IWOP技术,将经典光学中常用的分析方法引入到了量子力学范围内(其中包括层析摄像技术,Radon变换和小波分析方法)。这不仅丰富了Dirac符号法和变换论,对量子力学中光学变换的研究也有实际的应用价值。
本文分为六章,具体内容安排如下
第一章首先用IWOP技术从新审视研究了量子力学的基本表象,包括坐标表象,动量表象,粒子数表象和相干态表象。然后我们又介绍了一些用1WOP技术寻找到的新的量子力学表象,它们都具有实际的物理意义。这些表象包括坐标-动量中介表象,纠缠态表象,中介纠缠态表象,三模纠缠态表象,三模相干纠缠态表象等等。
第二章介绍了Weyl对应规则和Wigner算符,任意算符的Weyl编序展开式和Weyl编序下的积分技术。进而利用IWOP技术,给出了Wigner算符在各个量子力学基本表象中的表达式。最后推导出了Wigner算符在各个新找到的量子力学表象中的表达形式。
第三章中,我们首先介绍了一种降低维度的光学分析方法-层析摄像技术和其数学基础Radon变换在各个维度的一般表达式。然后我们将其推广到量子力学范围,给出了Wigner算符的Radon变换的一般表达式,并将其推广到了双模情形。
第四章中,我们对Wigner算符的Radon变换做了更加详细的研究,对各个光学过程下Wigner算符的Radon变换进行了计算,我们得出一个结果,各光学过程所带来的信号的变化等价于其Wigner算符的Radon变换参数的变化,这就简化了其运算过程并可以将Radon变换应用到量子光学过程中。最后我们对双模情形在中介纠缠态表象下做了相同的研究。
第五章我们介绍了另外一种提高维度的光学分析方法-小波变换。我们将小波变换推广到量子力学的过程中,利用其容许性条件,推出了一个新的小波系列-高阶墨西哥帽小波系。然后我们利用不同的信号输入,对这个新的小波系列的性质做了详细的分析,并给出了一些应用实例。
最后我们总结了一下上面的研究工作并给出了这些领域未来可能的研究方向。
In quantum mechanics, the physical observables are all represented by the Hermitian operators. These are generally not commutative between them. Therefore, when the function of classical mechanics promote to the operators of quantum mechanics, the form of corresponding is not only one. For the different operator ordering forms, or for the different corresponding rules, their corresponding operators are different. So when people correspond the function of classical mechanics to the operators of quantum mechanics, they will have a problem that how to order the operators. There are some definite operator ordering forms, such as the normal ordering form, the anti-normal ordering form and the Weyl ordering form. In addition, the Weyl ordering form has a remarkable property, i.e.,it has similar invariance when the Weyl ordered operators transform under similar transformations. To our knowledge, there are two main approaches to handle the operator ordering problems. They are the Lie algebra method and Louisell's differential operation method via the coherent state representation. However, these two methods are not very efficient in the processes which calculate the complex quantum operator ordering problems. In order to solve this problem, Prof.Fan firstly proposed the technique of integration within an ordered product of operators(IWOP). The IWOP technique generalizes the Newton-Leibniz rule to the integrations over the ket-bra operators in quantum mechanics and builds a bridge between classical mechanics and quantum mechanics.
Based on the operator ordering theory, we introduced the analysis methods of classical optics to th quantum optics by the IWOP technique. The analysis methods involved the Tomography technique, the Radon transform and the Wavelet transform. These efforts are very useful to study the optical transformations in quantum mechanics and enrich the Dirac's symbolic method and the transformation theory.
The whole thesis is divided into six chapters, arranged in details as follows:
In Chapter.1, based on the IWOP technique, we resurvey the preliminary quantum representations such as the coordinate representation, the momentum representation, the particle number representation and the coherent representation. And then, we introduced some new quantum representations such as the entangled state representation, the coordinate-momentum intermediate representation, the entangled coherent representation, the three-mode entangled state representation and the three-mode entangled coherent representation. They have clear physical meaning.
In Chapter.2, we introduced the Weyl ordering form and the Wigner operator, Weyl ordering operator formula, the technique of integration within the Weyl ordering product of operators and the order-invariance of Weyl ordered operators under similar transformations. Using the IWOP technique, we derived the form of the Wigner operator in different basic quantum representations. At last, we derived the form of the Wigner operator in different new quantum representations we introduced in Chapter.1.
In Chapter.3, at first we introduced a dimension reduced optical analysis method which is the Tomography technique. The basis of its mathematical is the Radon transform. And then we presented the mathematical formulation of Radon transforms in different dimensions. Then we promote them to the quantum field, and derived the Radon transformation of Wigner operators in both single-mode and two-mode by the coordinate-momentum intermediate representation and the entangled intermediate representation.
In Chapter.4, we did more detailed research on the Radon transformation of Wigner operators and calculated the Radon transformation of Wigner operators with various optical processes. We come to a conclusion that the alteration of Radon transformation of signal's Wigner function through these optical processes can be ascribed to the variation of Radon transformation parameters. This method simplify the calculation processes and can be promote the Radon transformation to the quantum optics. At last, we did the same works in two-mode situation by the entangled intermediate representation.
In Chapter.5, we introduced a dimension enhanced optical analysis method which is the Wavelet transformation. In the process when we promote the Wavelet transform to the quantum field, we find a general formula of qualified continuum mother wavelets---High-order Mexican Hat wavelets by the Admissible condition. And then, we make comparisons among different wavelet transforms computed with our new mother wavelets and the classical Mexican Hat wavelet by several special optic pulses and find the character of the High-order Mexican Hat wavelets.
In the last chapter, we summarized the research works of the above and some possible future research directions in these fields are given.
本文在算符编序理论的基础之上,利用IWOP技术,将经典光学中常用的分析方法引入到了量子力学范围内(其中包括层析摄像技术,Radon变换和小波分析方法)。这不仅丰富了Dirac符号法和变换论,对量子力学中光学变换的研究也有实际的应用价值。
本文分为六章,具体内容安排如下
第一章首先用IWOP技术从新审视研究了量子力学的基本表象,包括坐标表象,动量表象,粒子数表象和相干态表象。然后我们又介绍了一些用1WOP技术寻找到的新的量子力学表象,它们都具有实际的物理意义。这些表象包括坐标-动量中介表象,纠缠态表象,中介纠缠态表象,三模纠缠态表象,三模相干纠缠态表象等等。
第二章介绍了Weyl对应规则和Wigner算符,任意算符的Weyl编序展开式和Weyl编序下的积分技术。进而利用IWOP技术,给出了Wigner算符在各个量子力学基本表象中的表达式。最后推导出了Wigner算符在各个新找到的量子力学表象中的表达形式。
第三章中,我们首先介绍了一种降低维度的光学分析方法-层析摄像技术和其数学基础Radon变换在各个维度的一般表达式。然后我们将其推广到量子力学范围,给出了Wigner算符的Radon变换的一般表达式,并将其推广到了双模情形。
第四章中,我们对Wigner算符的Radon变换做了更加详细的研究,对各个光学过程下Wigner算符的Radon变换进行了计算,我们得出一个结果,各光学过程所带来的信号的变化等价于其Wigner算符的Radon变换参数的变化,这就简化了其运算过程并可以将Radon变换应用到量子光学过程中。最后我们对双模情形在中介纠缠态表象下做了相同的研究。
第五章我们介绍了另外一种提高维度的光学分析方法-小波变换。我们将小波变换推广到量子力学的过程中,利用其容许性条件,推出了一个新的小波系列-高阶墨西哥帽小波系。然后我们利用不同的信号输入,对这个新的小波系列的性质做了详细的分析,并给出了一些应用实例。
最后我们总结了一下上面的研究工作并给出了这些领域未来可能的研究方向。
In quantum mechanics, the physical observables are all represented by the Hermitian operators. These are generally not commutative between them. Therefore, when the function of classical mechanics promote to the operators of quantum mechanics, the form of corresponding is not only one. For the different operator ordering forms, or for the different corresponding rules, their corresponding operators are different. So when people correspond the function of classical mechanics to the operators of quantum mechanics, they will have a problem that how to order the operators. There are some definite operator ordering forms, such as the normal ordering form, the anti-normal ordering form and the Weyl ordering form. In addition, the Weyl ordering form has a remarkable property, i.e.,it has similar invariance when the Weyl ordered operators transform under similar transformations. To our knowledge, there are two main approaches to handle the operator ordering problems. They are the Lie algebra method and Louisell's differential operation method via the coherent state representation. However, these two methods are not very efficient in the processes which calculate the complex quantum operator ordering problems. In order to solve this problem, Prof.Fan firstly proposed the technique of integration within an ordered product of operators(IWOP). The IWOP technique generalizes the Newton-Leibniz rule to the integrations over the ket-bra operators in quantum mechanics and builds a bridge between classical mechanics and quantum mechanics.
Based on the operator ordering theory, we introduced the analysis methods of classical optics to th quantum optics by the IWOP technique. The analysis methods involved the Tomography technique, the Radon transform and the Wavelet transform. These efforts are very useful to study the optical transformations in quantum mechanics and enrich the Dirac's symbolic method and the transformation theory.
The whole thesis is divided into six chapters, arranged in details as follows:
In Chapter.1, based on the IWOP technique, we resurvey the preliminary quantum representations such as the coordinate representation, the momentum representation, the particle number representation and the coherent representation. And then, we introduced some new quantum representations such as the entangled state representation, the coordinate-momentum intermediate representation, the entangled coherent representation, the three-mode entangled state representation and the three-mode entangled coherent representation. They have clear physical meaning.
In Chapter.2, we introduced the Weyl ordering form and the Wigner operator, Weyl ordering operator formula, the technique of integration within the Weyl ordering product of operators and the order-invariance of Weyl ordered operators under similar transformations. Using the IWOP technique, we derived the form of the Wigner operator in different basic quantum representations. At last, we derived the form of the Wigner operator in different new quantum representations we introduced in Chapter.1.
In Chapter.3, at first we introduced a dimension reduced optical analysis method which is the Tomography technique. The basis of its mathematical is the Radon transform. And then we presented the mathematical formulation of Radon transforms in different dimensions. Then we promote them to the quantum field, and derived the Radon transformation of Wigner operators in both single-mode and two-mode by the coordinate-momentum intermediate representation and the entangled intermediate representation.
In Chapter.4, we did more detailed research on the Radon transformation of Wigner operators and calculated the Radon transformation of Wigner operators with various optical processes. We come to a conclusion that the alteration of Radon transformation of signal's Wigner function through these optical processes can be ascribed to the variation of Radon transformation parameters. This method simplify the calculation processes and can be promote the Radon transformation to the quantum optics. At last, we did the same works in two-mode situation by the entangled intermediate representation.
In Chapter.5, we introduced a dimension enhanced optical analysis method which is the Wavelet transformation. In the process when we promote the Wavelet transform to the quantum field, we find a general formula of qualified continuum mother wavelets---High-order Mexican Hat wavelets by the Admissible condition. And then, we make comparisons among different wavelet transforms computed with our new mother wavelets and the classical Mexican Hat wavelet by several special optic pulses and find the character of the High-order Mexican Hat wavelets.
In the last chapter, we summarized the research works of the above and some possible future research directions in these fields are given.
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