Abstract
We introduced a new model to present the states of a two-state quantum system. The space is the complexified Minkowski space. The Lorentz group acts by the linear extension of its action on the four-vectors. We applied this model to represent the spin state of an electron or any relativistic spin 1/2 particle. The spin state of such particle is of the form U+iS, where U is the four-velocity of the particle in the lab frame, and S is the 4D spin in this frame. Under this description, the transition probability between two pure spin states ϱ1 and ϱ2 of particles moving with the same velocity are defined by use of Minkowski dot product as 12<ϱ2|ϱ1>. This transition probability is Lorentz invariant, coincide with the quantum mechanics prediction and thus agree with the experimental results testing quantum mechanics predictions based on Bell’s inequality. For a a particle of mass m and charge q with the spin state ϱ, the total momentum is mcϱ and the electromagnetic momentum is qϱ. This imply that the Landé g factor for such particles must be g=2. We obtain an evolution equation of the spin state in an electromagnetic field which defines correctly the anomalous Zeeman effect and the fine structure splitting.
Highlights
Two-level quantum mechanical systems, or two-state systems, are the simplest among quantum systems, yet they are enormously useful and have a wide range of applications.They can explain the most fundamental quantum phenomena, such as interference patterns.they are used to model complicated processes, such as nuclear magnetic resonance and neutrino oscillation
The Bloch sphere is a unit 2-sphere, representing the pure state space of a two-level quantum mechanical system, with antipodal points corresponding to a pair of mutually orthogonal state vectors
Using the Lorentz invariance of the dot product, the fact that spin measurements do change the velocity of the measured particle, and the definition of the transition probability, it follows that the transition probability between two relativistic pure spin states $1 and $2 of particles moving with the same velocity are defined by use of Minkowski dot product as
Summary
Two-level quantum mechanical systems, or two-state systems, are the simplest among quantum systems, yet they are enormously useful and have a wide range of applications. Thomas [7] has shown that if one uses relativistic kinematics to describe the precession of the frame comoving to the electron, g = 2 explains both the anomalous Zeeman effect and the observed fine structure interval. This is an indication that spin should be treated within relativity theory. By replacing the Euclidean geometry in the 4D spin domain with Minkowski one leads to a complexified Minkowski space endowed with a triple product We call it the Relativistic Two-State Space and denote it by Mc. We obtain explicit description for pure states in Mc and show that it can be used as a state space of two-state quantum systems. We obtain an evolution equation of the spin state in an electromagnetic field which defines correctly the anomalous Zeeman effect and the fine structure splitting
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