# IPACS Electronic library

## Spin-orbit interaction in particles' motion as a model of quantum computation

It can be shown that the particles' kinematic model is associated to the model of the quantum harmonic oscillator. Such a kinematic model considers the particle as a 3-DOF variable (position in a cartesian coordinates frame) [1]. When the effect of the spin in the particle's motion is considered, and the particle's motion takes place under an external magnetic field then the deflection of the particles' trajector should be also taken into account [2-3].

To describe clearly the particle's motion elements of stochastic mechanics will be used [4]. In stochastic mechanics each particle follows a continuous path which is random but has a well-defined probability distribution. The fourth degree of freedom of the particle is its spin, and is related to the Stern-Gerlach experiment. In this experiment a particle passes through a magnetic field and emerges with a new average velocity in the direction of the field. Thus in the course of the experiment the spin becomes correlated with the particle's position. The component of the spin along the field becomes a discrete random variable which is correlated with the average velocity [2].

The paper examines spin-orbit interaction as a model of quantum computation. The trajectory of a particle with spin $\pm{1 \over 2}$ can be deflected by the electromagnetic field and thus the particle may follow one out of two antisymmetric paths $a$ and $b$. If an appropriate electromagnetic field, maintains the particle's trajectory between $a$ and $b$, then this denotes a spin value between $\pm{1 \over 2}$ with a certain probability and stands for a quantum state.

References:

[1] G.G. Rigatos, Cooperative behavior of mobile robots as a macro-scale analogous of the quantum harmonic oscillator, IEEE SMC 2008 Intl. Conference, Singapore, Sep. 2008.

[2] W.G. Faris, Spin correlation in stochastic mechanics, Foundations of Physics, Plenum Publishing, vol. 12, no.1, pp. 1-26, 1982.

[3] C. Cohen-Tannoudji, B. Diu and F. Lalo\"{e}, M\'{e}canique
Quantique I, Hermann, 1998.

[4] W.G. Faris, Diffusion, Quantum Theory, and Radically Elementary Mathematics, Princeton University Press, 2006.