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On the basis of the Landauer-Buttiker scattering formalism and transfer matrix method, we investigated the spin-dependent shot noise in parabolic-well with two ferromagnetic contacts (F/PW/F). The quantum size and Rashba spin-orbit interaction are discussed simultaneously. The results indicate that the shot noise is periodic function of the parabolic-well width. The oscillation frequencies of the shot noise decrease with the increasing of the parabolic-well depth, and increase with the increasing of the Rashba spin-orbit coupling strength. The amplitude and peak to valley ratio of the shot noise are strongly dependent on the magnetization configuration of the junction.

The uncertainty relations are discussed on a noncommutative plane when noncommutativity of momentum spaces is considered. It is possible to construct normalizable states by simultaneously saturating two coordinate-momentum uncertainty relations. However, under the natural condition θη ≪ 4ħ2 one can not construct a normalizable state by simultaneously saturating any other pairs out of four basic nontrivial uncertainty relations.

The entangled state representation has been constructed on noncommutative phase space. Using this appropriate representation, the energy spectrum of general two-dimensional harmonic oscillator has been obtained exactly.

We proposed a scheme to implement the entanglement swapping based on cavity QED. Using this scheme, we can obtain the entanglement swapping with the success probability of 100% for a two atoms system and a three atoms system respectively, and the entanglement swapping is insensitive to both the cavity decay and thermal field.

Relativistic comparison theorems are established for discrete eigenvalues of Klein–Gordon equation with vector and scalar potentials in d -dimensions. Theorem 1 : If V ( λ ) and S ( λ ) depend on a parameter λ , ∂ S / ∂ λ ⩾ 0 , S ⩾ 0 , ∂ V / ∂ λ ⩾ 0 , V ⩽ 0 , E > 0 , then it follows that ∂ E n / ∂ λ ⩾ 0 . Theorem 2 : If S 2 ⩾ S 1 ⩾ 0 , 0 ⩾ V 2 ⩾ V 1 , E > 0 , then the corresponding eigenvalues are ordered as E n ( 2 ) ⩾ E n ( 1 ) . Theorem 3 : If 0 ⩾ V 2 ⩾ V 1 , S 2 ⩾ | S 1 | , E n ( 1 ) > 0 , E n ( 2 ) > 0 , then E n ( 2 ) ⩾ E n ( 1 ) . Some illustrative examples are given.

It is shown that the Dirac equation for a new equal scalar and vector anharmonic oscillator potentials could be separated into a solvable angular equation and a radial equation. Corresponding exact solutions of bound states for the Dirac equation have been obtained. The angular solutions are the associated-Legendre polynomial and the radial solutions are expressed in terms of the confluent hypergeometric functions. Finally, the energy equation is found from the boundary condition satisfied by the radial wavefunction.

A physical model is derived to create a biological cell separator that is based on controlling the entropy in a microfluidic system having conjoined spherical structures. A one-dimensional simplified model of this three-dimensional problem in terms of the corresponding effects of entropy on the Brownian motion of particles is presented. This dynamic mechanism is based on the Langevin equation from statistical thermodynamics and takes advantage of the characteristics of the Fokker-Planck equation. This mechanism can be applied to manipulate biological particles inside a microfluidic system with identical, conjoined, spherical compartments. This theoretical analysis is verified by performing a rapid and a simple technique for separating yeast cells in these conjoined, spherical microfluidic structures. The experimental results basically match with our theoretical model and we further analyze the parameters which can be used to control this separation mechanism. Both numerical simulations and experimental results show that the motion of the particles depends on the geometrical boundary conditions of the microfluidic system and the initial concentration of the diffusing material. This theoretical model can be implemented in future biophysics devices for the optimized design of passive cell sorters.