New Application Of Quantum Mechanics Representations And New Construction Method

Language of quantum mechanics is Dirac symbol method, also known as q-number theory, in which the one of the most important contents is quantum mechanics representation which can not only describe some fundamental law on quantum mechanics as coordinate frame, but also bring great convenience for choosing an appropriate representation to study specific problem of dynamics. Thus representation has the double meaning of kinematics and dynamics. The technique of integration within an ordered product (IWOP) of operators has successfully realized the integral over ket-bra operators, which means that it exploits a new direction for Newton-Leibniz integral, and provides a new approach to finding directly evident form of q-number for performing the natural transition from classical transformation to quantum unitary transformation. It can not only interpret quantum mechanics and develop foundation of the mathematical physics of quantum mechanics, but also exploit the potential applications of Dirac symbol method and representation theory. We find that IWOP technique can help us to explore new application for some known representations and find out some new representations.The main content in this paper includes the three aspects: 1. we shall mainly apply some usual representations to some concrete physical problems, in which we show that the IWOP technique is a very powerful tool for solving these quantum problems difficult to deal with. 2. based on the significance of quantum mechanics representation, we propose a new method or equation to obtain continuous quantum mechanics representation. 3. we construct and introduce several new representations on q deformed case and discuss their some properties and applications.The main content of this paper is arranged as follows:In chapter 1, We briefly review the Technique of Integration Within normal ordered product and the Weyl quantization scheme, as well as the Technique of Integration Within Weyl ordering.In chapter 2, by virtue of the bosonic coherent state representation and the Schwinger bosonic operator realization of angular momentum, we find the formula for the quantum Hamiltonian for SU(2) rotation, we further specify the angular velocity. Though the spin as a quantum observable has no classical correspondence, we may still mimic it as a rigid body rotation characterized by 3 Euler angles, and calculate its Pseudo-classical rotational partition function of spin one-half.In chapter 3, for the bipartite Hamiltonian system with kinetic coupling, we derive time evolution equation of Wigner functions by virtue of the EPR entangled state representation and entangled Wigner operator, which just indicates that choosing a good representation indeed provides great convenience for us to deal with the dynamics problem.In chapter 4, by virtue of the thermo entangled state representation, we exhibit a novel approach to deriving density operator for a Raman-coupled model with damping of the cavity mode. The normal ordering forms of density matrix elements can be obtained, and the corresponding Wigner functions are also derived.In chapter 5, based on the significance in quantum phase space, by extending EPR entangled state representation to multipartite case. We construct n-mode Wigner operator in the common eigenstate of the multipartite centre-of-mass coordinate and two mass-weighted relative momenta, as well as its canonical conjugate state.In chapter 6, using the completeness of Wigner operator and Weyl correspondence we construct a general equation for deriving pure state density operators, that is to say, we find a new method to derive continuous quantum mechanics representation. Several important examples in quantum mechanics and quantum optics are considered as the applications of this equation.In chapter 7, we construct a new state called excited two-mode generalized squeezed vacuum states and find that it is just regarded as a generalized squeezed two-variable Hermite polynomial excitation on the vacuum state, as well as its normalization constant is just proved as a Jacobi polynomial. Their statistical properties are investigated such as squeezing properties and the form of the corresponding Wigner function.In chapter 8, by using the contour integral representation ofδ-function and the technique of integration within an ordered product of operators, we point out that the q-deformed creation operator possesses the eigenkets. A set of new completeness and orthogonality relations composed of the kets and bras which are not mutually Hermitian conjugates are derived. Application of the completeness relation in constructing the generalized P-representation of density operator is demonstrated.In chapter 9, the q-deformed entangled states are introduced by using q deformed theory and new normal ordering of the vacuum projection operator for q-deformed boson oscillator. Similarly, the new completeness and orthogonality relations composed of the bra and ket, which are not mutually Hermitian conjugates are obtained. Furthermore, the property of squeezing operator represented by the q-deformed entangled states is exhibited.In chapter 10, similarly using the q-deformed entangled states are introduced by using q deformed theory and new normal ordering of the vacuum projection operator for q-deformed boson oscillator, we introduce the q deformed coordinate representation. Further, we utilize the eigenket and eigenbra for this representation to realize and study some important quantum gate operators for continuum variables.