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Journal of Qinzhou University

On the Solution of Spin 1/2 Particle Polarization Vector in Schrodinger Representation and Heisenberg Representation in Magnetic Field

Cai Zhao-yun

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The results of the resolution of Spin 1/2 Particle Polarization Vector in Schrodinger Representation and Heisenberg Representation in Magnetic Field are the same.However,it is simpler by applying the Heisenberg Movement Equation.Therefore,it is more convenient to apply Heisenberg Movement Equation when solving the mechanics.

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We study the Heisenberg model under the influence of a rotating magnetic field. By using a time-dependent unitary transformation, the time evolution operator for the Schrodinger equation is obtained, which involves no chronological product. The spin vectors (mean values of the spin operators) are obtained as explicit functions of time in the most general case. A series of cyclic solutions are presented. The nonadiabatic geometric phases of these cyclic solutions are calculated, and are expressed in terms of the solid angle subtended by the closed trace of the total spin vector, as well as in terms of those of the individual spins.
We study the Heisenberg model under the influence of a rotating magnetic field. By using a time-dependent unitary transformation, the time evolution operator for the Schrödinger equation is obtained, which involves no chronological product. The spin vectors (mean values of the spin operators) are obtained as explicit functions of time in the most general case. A series of cyclic solutions are presented. The nonadiabatic geometric phases of these cyclic solutions are calculated, and are expressed in terms of the solid angle subtended by the closed trace of the total spin vector, as well as in terms of those of the individual spins.
In the quantum field,due to Bohr correspondence principle,quantum mechanics go to classical mechanics in large quantum number.Based upon Heisenberg correspondence principle,quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Using Heisenberg correspondence principle,quantum-classical correspondence of particles in magnetic is studied.Applying Heisenberg correspondence principle in relativistic realm,the exact wave functions of Klein-Gordon equation are obtained for particles in magnetic with quantum matrix element in a rectangular coordinate system of new representation.It turns out that in the classical limit matrix elements of quantum operators reduces to the classical solutions and with classical approach,the coordinate is cyclical variation with time,the orbit of particles is a circle,and the movement is uniform circular motion in uniform magnetic field.
A thorough investigation was conducted for the proof process of Heisenberg’s famous inequality. It is apparent that any particle, no matter a classical or a quantum particle, as long as in wave motion, its dp always has an upper limit and a lower limit, which results in the product of dp and dx has both upper and lower limits. The Heisenberg’s inequality is nothing to do with measurement accuracy but related to energy conservation. A new deduction method for a spinning electron revolving on an orbit around a nucleus was developed based on our recently developed theory of spin vector in motion behaving particle-wave duality. The electron’s motion equation is same as Schrodinger equation while with a different energy constant j which is related to the spin vector’s motion features such as the mass of the object, the spin period and revolution period, the orbit shape and size. The new deduction process of Schrodinger equation will help explain the dilemma of the quantum mechanics.
The classical equations of motion of a charged point-particle with intrinsic spin under the influence of an external electromagnetic field are restated and compared with the Heisenberg equations of motion derived from the Dirac theory. The partition of angular momentum between particle and field in the classical theory is contrasted to the Dirac theory of electron spin. The analogy between the Dirac equation and the theory of parametric amplification is pointed out.A free spinning point particle moving according to the laws of classical relativistic point-particle mechanics may move along a helix. The sum of the intrinsic spin $\ensuremath{\sigma}$ and the angular momentum of the helical motion in this classical analog of zitterbewegung is an effective spin vector S which is a constant of the motion. Because of this internal motion, the effective mass MMM of the particle differs from the mass mmm which is ascribed to it in the equations of motion. Solutions are found in which S is parallel or antiparallel to the momentum, and the sign of MMM is determined by the helicity. When placed in a uniform electromagnetic field, the particle behaves as if it had a rest mass MMM and a magnetic moment $\frac{e\ensuremath{\sigma}}{\mathrm{Mc}}$, in addition to any explicit magnetic moment that may be ascribed to it.