Search

Differential quotient rule of composite function operator and its applications in quantum physics, quantum statistics, operator ordering theory, matrix theory and control theory are given. The integration problem of Wigner operator and Weyl corresponding rules are studied. Two kinds of typical operator identity formulas are proved. The differential form of Wigner operator in ordered product of operators and new differential form of important functions are obtained. Finally, a Wigner operator with parameter for unifying regular order, Weyl sequencing and abnormal order is introduced.

The right-hand inverse operator of the annihilation operators and the left-hand inverse operator of the creation operator are given by using the contour integration. Furthermore, the corresponding relation between the right-hand inverse or left-hand inverse operator derived and the column elementary transformation or row transformation of a matrix is investigated.

The new two mode coherent entangled state representation 1 p , β 〉 is proposed by the technique of integration within an ordered product of operators. Its characteristics and mechanism are analysed. The results show that, the common eigenvector | p , β 〉 of p 1 - p 2 and a 1 + a 2 exhibits both the characteristics of the coherent state and entangled state. The 1| p , β 〉 makes up a new quantum mechanical representation and possesses the completeness relation. The ideal single-mode squeezed state | p =0〉 2 =exp［1/2 a +2 2 |0〉 2 and the vacuum state |0〉 1 overlapping on a beam splitter may produce the coherent entangled state | p , β 〉at the output by the modulation of the laser field.

A two-mode integral form projection unitary operator is introduced. Using the technique of integration within an ordered product （IWOP） of operators, the transformation property of the integral form projection operator is studied, and its normal ordered expression is derived. Then the Hamiltonian of two-mode quantum harmonic oscillator system with coordinate-momentum coupling is diagonalized by virtue of the integral form projection operator, there by the eigenenergy and eigenfunction of the system are exactly solved. Lastly, a special case is discussed.

In order to solve dynamic problems of double non-identical quantum harmonic oscillators with interaction potential and movement coupling, the new two mode composite coordinate representation |η1，η2〉 is proposed by the technique of integration within an ordered product of operators. The two mode squeezing operator U is constructed, and its characteristics are analyzed. Moreover, its application to double non-identical quantum harmonic oscillators with interaction potential and movement coupling is presented for solving some the dynamic problems. Approach of solving some the dynamic problems involving complicated quantum harmonic oscillators is proposed.