Fan Hong-Yi

Weyl ordering quantum mechanical operators by virtue of the IWWP technique

The properties of Weyl ordered products of operators are investigated and the technique of integration within Weyl ordered product (IWWP) is introduced. The overcompleteness relation of the coherent state is then recast into Weyl ordered form. In so doing, a new approach for Weyl ordering quantum mechanical operators is presented.

Similarity transformations in one- and two-mode Fock space

The normally ordered Hilbert space image of the general (complex) linear similarity transformation in phase space is obtained in a coherent state representation. Although preserving the commutator of a and a+, these quantum mechanical images of classical transformations are generally not unitary. Remarkably, although the kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, squeezed state analogues produced using the similarity transformation still satisfy an overcompleteness relation. The results are extended to two-mode Fock space and simple examples of the utility of the similarity transformation are presented. The evaluation of the normally ordered operators is greatly facilitated by the use of integration within ordered products.

Calculating Density Operator from Diagonal Coherent State Representation

By means of combining the coherents properties with the normal products properties we put forward to a formulation for obtaining density operators from Glauber-Suaaishan P representation and other non-diagonal representations directly in quantum optics.

Canonical coherent-state representation of some squeeze operators

Using the IWOP technique (integration within an ordered product), the authors find canonical coherent-state representations of two kinds of squeeze operators. These representations manifestly show that they are quantum maps imaged by certain symplectic transformations in coordinate-momentum phase space. In terms of this formalism the coherent-state propagator of parametric amplifiers is easily obtained.

Permutation operators in Hilbert space gained via the IWOP technique-fermion case

For the case of fermions, permutation operators in Hilbert space are derived in terms of the fermion coherent state and the technique of integration within ordered products with the integration variables being Grassmann numbers. These operators are shown to be quantum maps imaged by a certain permutation transformation of the numbers. Some new Fermi operator identities are also obtained with the use of the same technique.

Squeezing in the triatomic linear molecule model revealed by virtue of the IWOP technique

For a typical triatomic linear molecule with different atomic masses one finds a unitary operator which is closely related to squeezing transformations caused by frequency jumps. The coordinate representation of the unitary operator is identified, which provides an alternative approach for calculating the wavefunction of internal motion of the molecule. The author also shows that the unitary operator is equivalent to a two-mode squeezing-rotating combination operator multiplied from the right by two single-mode squeezing operators. The discussion is mainly based on the technique of integration within an ordered product (IWOP) of operators.

Generalized Bogolyubov transformation-bosonic case

The formalism of the generalized fermionic Bogolyubov transformation, previously by Fan and VanderLinde (1990), is extended to the bosonic case. The corresponding bosonic quasi-particle and quasi-vacuum state are derived; the bosonic generalized Bogolyubov operator is decomposed as a normal product form by the technique of integration within an ordered product (IWOP).

A new two-mode fermion coherent state representation

A technique of integration within the ordered product for a fermion system is introduced. In terms of the technique, a new two-mode fermion coherent state representation is found. Some properties and applications of the new representation are derived.

Permutation operators in Hilbert space gained via IWOP technique

A kind of permutation operators in Hilbert space are derived in terms of the technique of integration within ordered product (IWOP). These operators are shown to be quantum maps imaged by certain permutation transformations in classical space. Some operator identities about the relationship between permutation operators and transposition operators are also derived.

A new completeness relation in the q-deformed two-mode Fock space

In two-mode Fock space a new completeness relation for the q-analogue of the single-complex-variable-state is proved by virtue of some q-deformed combinatorial identities, which are also derived here for the first time.