Wavelet Transform of Quantum States and Some Representation Transformation 

Author  SongJun 
Tutor  FanHongYi 
School  University of Science and Technology of China 
Course  Theoretical Physics 
Keywords  representation theory entangled states representation the IWOPtechnique wavelet transform complex wavelet transform Henkeltransform Wigner function 
CLC  O413.1 
Type  PhD thesis 
Year  2012 
Downloads  154 
Quotes  0 
The representation and transformation theory in quantum mechanics was first introduced by Dirac during19261930, which has become the standard "language" of quantum theory, to formulate general rules for modern quantum mechanics. Representation is the effective carrier to solve dynamic schrodinger equation, and appropriate representation will make for tackle the diagonalization of operators. As Dirac said:"When one has a particular problem to work out in quantum mechanics, one can minimize the labour by using a representation in which the representatives of the more important abstract quantities occurring in that problem are as simple as possible".In this paper, by virtue of the techniques of integration within an ordered product (IWOP) of operators and the representation and transformation theory in quantum mechanics, especially a variety of continuous variables entanglement states representation, we promotion wavelet transform and Hankel transform often used in signal analysis to quantum mechanics. We put forward the concept of wavelet transform of quantum states and explore its applications in identification of quantum states. This paper also studied phasespace distributions (Wigner function) of some quantum states. Through the above researches, we developed the representation and transformation theory, and have more profound understanding to it.We mainly do the following three aspects research:1. The representation and transformation theory in quantum mechanics was applied to wavelet transform. By using the coordinate and entangled states representation, wavelet transform and complex wavelet transform of quantum mechanics are constructed. Wavelet transform operators correspond with the squeezeddisplaced operators in quantum mechanics. Based on this, we present a general method to look for mother wavelet, and then properties of wavelet transform of some quantum states was studied.2. We introduce the representation and transformation theory in quantum mechanics into Hankel transform. By means of some properties of induced entangled states, we constitute the quantum Hankel transform and discuss its application.3. With the help of the entangled state representation and the Wigner operator in such representation, we derive the Wigner quasiprobability function of twomode squeezed states and spin coherent states based on the Schwinger Bose operators realization of angular momentum, and then discuss its properties.The main content of this paper is arranged as follows:In chapter1, we briefly introduce some general representation in quantum mechanics, such as coordinate, momentum, Fock, coherent state representation, as well as the entangled states representation, the IWOP techniques and its application.In chapter2, by means of the coordinate representation, we give the definition about one dimensional continuous wavelet transform of quantum mechanics, we structure a squeezeddisplaced operator corresponding to the wavelet transform, which normal product can be derived by the IWOP technique. Then, there is a general method looking for mother wavelet. Based on this, wavelet transform properties of the coherent states, Fock states and binomial states have been studied.In chapter3, we also structure a quantum mechanical perator corresponding to the complex wavelet transform by using the entangled states representation, and give the quantum mechanics definition about two dimensional continuous complex wavelet transform, normal product of the squeezeddisplaced operator, and the general method looking for mother wavelet. Based on this, the properties of complex wavelet transform of the twomode coherent states, twomode Fock states and Bell states have been studied.In chapter4, we introduce the representation and transformation theory in quantum mechanics into Hankel transform. With the aid of the properties of the induction entangled states, we structure the quantum Hankel transform, thus deduce some quantum mechanics Hankel transform recursive formula, and discussed its application in quantum mechanics.In chapter5, we derive the Wigner function of twomode squeezed states and spin coherent states based on the Schwinger Bose operators realization of angular momentum of employing the entangled state representation and the Wigner operator in such representation, and discuss its phasespace quasiprobability distributions.