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We find that the resolution of unity for the EPR (Einstein, Podolsky and Rosen) entangled state representation can be in differential form. Its application in studying normally ordered expansion of some complicated exponential operators is demonstrated.

We introduce a generalized entangled state |η,λ1,λ2>, which spans a complete and orthonormal representation. Using the technique of integration within an ordered product of operators we prove that the projection operator |η,λ1,λ2>1,λ2| is just the Radon transformation of the entangled Wigner operator. The inverse Radon transformation is also derived and the tomography theory for two-mode correlated system is established.

We derive the entangled eigenstate ‖ξ> of complex scalar fields ϕ and ϕ† in the Fock space. The ‖ξ> state is found to embed the entanglement possessed by the Einstein–Podolsky–Rosen states in quantum mechanics. The ‖ξ> set spans a complete and orthonormal representation. The advantage of the new

By using the technique of integration within an ordered product (IWOP) of operators, we derive the normal-product form of the gauge-invariant Wigner operator of an electron in a uniform magnetic field and its expression in the newly constructed |λ> representation. The virtue of working in the |λ> representation lies in the fact that it is expressed in terms of the ladder operators Π±, K±(see Eq. (10)), thus the magnetic vector potential can naturally be included. On these bases we can easily obtain Wigner functions of some important electron states.

The Weyl correspondence and Wigner operator theory for two-mode correlated system is developed on the common eigenstate basis

The relationships between the basic mechanical and wear properties of mottled enamel, especially during the mastication process, are important factors and must be explored. This study evaluated mottled enamel's micro-tribological behavior under artificial saliva conditions in vitro. The basic mechanical properties were determined by nanoindentiation testing. A conical diamond nanoindenter tip was used to scratch mottled enamel and normal enamel. The scratches were sliding with a constant normal load of 2 mN, with different cycles during the tests. The hardness, elastic modulus and friction coefficient were obtained to analyze the mechanical properties. The results showed that the hardness and elastic modulus of mottled enamel were 10% and 14.6% less, respectively, than those of normal enamel. Mottled enamel showed a lower friction coefficient and a higher wear rate, compared to normal enamel. The friction coefficient did not appear to be related to the wear rate for either type of enamel. The wear mechanism for normal enamel was plastic deformation for early wear, while the combination of plastic deformation and delamination was the main damage feature of mottled enamel.