Some New Hermite-polynomial-operator Identities And Their Applications

We use two methods to derive the wave functions of Fock states, one is the normal ordering expansion of the coordinate projector x x , and another is the new Hermite-polynomial-operator identities, which are derived by using the technique of integration within an ordered product (IWOP) of operators. We also obtain the wave function of squeezed number state, the moment of the quadrature (the matrix element of the power of coordinate operator in the number state)of the quantum oscillator by virtue of the new Hermite-polynomial-operator identities. Application of the new operator identities in some perturbation calculations of energy shift is also presented. On the other hand, we introduce the entangled-state representation and coordinate-momentum intermediate representation and show that the IWOP technique of operator is effective for finding new quantum mechanical representations.The dissertation is arranged as following:In chapter one, we introduce some background knowledge of our works, including the development history of the Integral technique within ordered product (IWOP) of operators and its significance and applications.In chapter two, we first list several properties of operators'normal product and then we recall some basic representations like the coordinate, momentum, particle number and the coherent state representations. We illustrate the IWOP technique through a specific example. Last we give a simple exponential operator decomposition method.In chapter three, we introduce two types of entangled state representations and coordinate-momentum intermediate representation which are established by the IWOP technique. We also show that the IWOP technique is effective in building the new representation and finding new unitary operators.In chapter four, we present two new methods of deriving the particle wave function of the number state. Specifically, we derive the expression of wave function of number state in coordinate representations by the normal product of the coordinate projection operation.. Similarly, we can obtain the wave function in momentum, coordinate-momentum intermediate representations. We also find the wave function of the dual-particle state in the entangled state representation and the new formula of double-subscript Hermite polynomials'generating function by the entangled state projection operator.In chapter five, we give some new Hermite polynomial operator identities with the IWOP technique and derive the squeezed state wave function of particle number with them. We also give the new methods of calculating the moments of the quadrature and the perturbative change of energy level.The ending section involves conclusions and outlook.