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The spin-magnetic moment of the electron is revisited. In the form of the relativistic quantum mechanics, we calculate the magnetic moment of Dirac electron with no orbital angular-momentum. It is inferred that obtained magnetic moment may be the spin-magnetic moment, because it is never due to orbital motion. A transition current flowing from a positive energy state to a negative energy state in Dirac Sea is found. Application to the band structure of semiconductor is suggested.

The spin and the spin-magnetic moment are basic and the most important concepts in the spintronics [

In this paper, we investigate a question about the origin of the spin-magnetic moment of the electron and roles of Zitterbewegung relating to it. The use of Heisenberg picture will make hidden roles of Zitterbewegung more clear than previous work. On the other hand, Zitterbewegung in solid state physics [

more easily than the previous work [

It is well known that the relativistic electron put in the external magnetic field gives interaction energy with the magnetic field

was understood as the interaction energy

must be the spin-magnetic moment of the electron in comparison with Equation (2). However, Equation (3) provided merely the relation of the spin-magnetic moment

in the classical electrodynamics, where

by using

In relativistic quantum theory, however, we should mind that we can not use Equation (5) for Dirac electron, because the velocity

As to Zitterbewegung, we briefly show all equations that are needed later. The velocity

where

The

where

In order to clarify the role of Zitterbewegung, we use Heisenberg picture hereafter to calculate the time evolution of any operator. We first investigate the behavior of matrix

where

The solution of the above differential equation is

We substitute Equation (13) into Equation (11) to obtain

Taking

We finally obtain

from Equations (15) and (14).

For reader’s convenience, we summarize all equations in the following; they are necessary for our calculation. The Dirac equation

has four eigen-solutions. We name these solutions

where

Two arrows

where

with normalization factor

and eigen states of “Up Spin” and “Down Spin”,

respectively. The momentum

as well as

Next relations are especially important in a frame

Because each component of

In actual calculation, we will take z-axis along the momentum of the electron:

The velocity operator

where

and

where

Equation (37) leads to

and

where

where the expression of the expectation value for

We calculate the magnetic moment based on Equation (4). As mentioned in Section 2, the velocity

for relativistic electron. It is our advantage that we need no external magnetic field. In what follows, we pay attention to the electron in state

By the use of the completeness condition

we have an expression of the expectation value of

Each matrix elements of uniform part are calculated as follows:

We have also Zitterbewegung part of the velocity.

as well as

Substitution of Equations (51)-(55) into Equation (50) gives

We find here that

We then finally obtain

which is the same result as in the previous work [

It is necessary to recall that there is not any z-component of magnetic moment arising from orbital motion of electron because of

indicating correct

where

As seen in Sections 5, the expectation values of all Zitterbewegung parts give zero or constant, both in velocity and coordinates. This means that we can not directly measure the effect of Zitterbewegung. However, they still survive behind some kind of physical quantities. The magnetic moment is an example. Although Zitter-bewegung relating to the velocity or coordinates is un-observable in the sense of the expectation value which corresponds to classical behavior, we have shown that it exists and works through the magnetic moment. A crucial point is non-diagonal matrix elements:

The initial state

In quantum theory, the conservation law of energy may break by

The electron which has undergone transition into

where

In the classical electrodynamics, the magnetic moment is caused by periodic orbital motion of a charged particles which is equivalent to an electric current. In the relativistic quantum theory, it seems that it is possible to cause the magnetic moment also by periodic transition from the positive energy state to the negative energy state (

Such a situation as described above occurs in some kind of solid state. In the two band model of Cohen and Blount [

(

The Hamiltonian which corresponds to Equation (65) has the form

in the frame

where