Using the technique of integration within an ordered product of operators we derive some δ-function operator formulas of photon creation and annihilation operators. Applications of these new formulas in simplifying some Louisell's operator theorems and in discussing the eigenket of creation operator are presented.
We develop a star-product scheme of symbols defined by the normally ordered powers of the creation and annihilation photon operators, . The corresponding phase space is a two-dimensional lattice with nodes ( m , n ) given by pairs of non-negative integers. The star-product kernel of symbols on the lattice and intertwining kernels to other schemes are found in an explicit form. Analysis of peculiar properties of the star-product kernel results in new sum relations for factorials. Advantages of the developed star-product scheme for describing dynamics of quantum systems are discussed and time evolution equations in terms of the ordered moments are derived.
We present a novel method for deriving some new quantum operator formulas by carrying out the integrations within a normal product symbol. The idea underlying this method is combining the properties of normal product and of coherent state together. For example, we can deduce operator ordering theorems, the Fujiwara formula, and some disentangling theorems simply and directly. By means of this method we also get the explicit operator form and coherent state form for the Wigner function with these new forms, the wigner function's eigenstate is found and its zero point value property is discussed. Furthermore some famous quantum unitary transformations and the transformed matrix element for functional obtained conveniently. integral may be also
We give the normal and anti-normal order expressions of the number operator to the power k by using the commutation relation between the annihilation and creation operators. We use those expressions to give general formulae for functions of the number operator in normal and anti-normal order.
It is pointed out that the assertion that the eigenfunction of the creation operator a + is identically zero is not rigorous. On the contrary, the eigenfunction of a + may be discussed by using the contour integral representation of δ function, the property of which and the relation with the eigenstate of the annihilation operator are discussed. Its applications, especially the application in thoroughly studying the inverse operator's properties, are presented. It is shown that the phase operator can also be represented by the inverse operator.