finite temperature we construct a new thermal excitation representation |E,g>, which is the common eigenvector of Hamiltonian operator of total system (including the reservoir) and the operator (α(β)−ã†(β))(α†(β)−ã(β)), where α†(β) (ã†(β)) creates a quantum of energy hω(β) (−hω(β)), respectively, β=ãh/kT, k is the Boltzmann constant. The explicit from of |E,g> is derived by virtue of the common eigenvector α(β)− ã†(β) and α†(β)− ã(β). The thermo excitation and de-excitation operations are manifestly displayed in |E,g> representation. As an application, the |E,g> representation of thermal squeezing operator is obtained.

In similar to the relation between the coherent state and the Bargmann transform, we find a generalized Bargmann transformation naturally accompanying Einstein–Podolsky–Rosen (EPR) entangled state of continuous variables. The new Bargmann transform kernel is introduced which can naturally lead to the entangled state. The generalized Bargmann representation of two-mode Fock state is a two-variable Hermite polynomial.

We propose a concise quantization scheme for mesoscopic electric circuits which is in accord with the discreteness of electric charge. Our scheme is based on finding the correct explicit form of charge ascending and lowering operators. The charge eigenstate are derived from the appropriate entangled state representation. We show how the entangled state representation can play the role of constructing a concise and natural quantization scheme for mesoelectronics.

We develop Dirac's representation theory in quantum mechanics by constructing the nonlinear entangled state |η nl and its non-Hermite conjugate state nl η| with continuum variable. By virtue of the technique of integration within an ordered product of operators we show that |η nl and nl η| make up an orthonormal and complete representation. From |η nl we also deduce another kind of entangled states. Application of |η nl in studying two-mode squeezed state is demonstrated.

We propose the conception of invariant 'eigen-operator' of the square of the Schr