Spin-Magnetic Moment of Dirac Electron, and Role of Zitterbewegung


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.

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Sasabe, S. (2014) Spin-Magnetic Moment of Dirac Electron, and Role of Zitterbewegung. Journal of Modern Physics, 5, 534-542. doi: 10.4236/jmp.2014.57064.

1. Introduction

The spin and the spin-magnetic moment are basic and the most important concepts in the spintronics [1] that is new research field in considerable expansion. In the previous work [2] , we found that the spin-magnetic moment seems to be caused from well-known definitional equation of magnetic moment. Such a case never happen in the non-relativistic quantum mechanics. In the relativistic quantum mechanics, however, the electron has another degree of freedom called Zitterbewegung [3] -[5] that is trembling motion of relativistic electron. Some physical connection between the spin-magnetic moment and Zitterbewegung was implied in the previous work [2] .

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 [6] -[8] has been a subject of great interest in recent years, since observable Zitterbewegung-like dynamics of band electron was predicted [9] [10] for electron moving in narrow-gap semiconductors [11] , graphene sheets [12] , carbon nanotube [13] , and super conductor [14] . Our research in this paper is therefore worthwhile on both sides of science and technology. As a result, we obtain


more easily than the previous work [2] , where denotes expectation value of z-component of spin-magnetic moment of a free Dirac electron in positive energy state.

2. Relation between Spin and Spin-Magnetic Moment

It is well known that the relativistic electron put in the external magnetic field gives interaction energy with the magnetic field. This term [15] [16]


was understood as the interaction energy between an external magnetic field and the magnetic moment. Then, physicists concluded that


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 and the spin operator by the analogy with classical electrodynamics. We do not still know how the spin-magnetic moment is generated, and what the spin-magnetic moment is. In order to clarify the origin of the spin-magnetic moment, we must deduce it without the external magnetic field which always leads to the interaction energy of the form. Generally, the magnetic moment for a charged particle moving with the velocity and the charge is defined as [17] [18]


in the classical electrodynamics, where is speed of light. In the non-relativistic quantum theory, the above equation can be expressed as


by using and.

In relativistic quantum theory, however, we should mind that we can not use Equation (5) for Dirac electron, because the velocity and the momentum are independent variables to each other (i.e.) in this case [2] .

3. Zitterbewegung

As to Zitterbewegung, we briefly show all equations that are needed later. The velocity of Dirac electron is given [19] by Heisenberg equation,


where are the Dirac matrices in Dirac Hamiltonian


The matrices and are defined as


where are the Pauli matrices. These matrices satisfy the following relations:



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 as an operator. Making use of Equation (10), we easily find [4]


where is the original matrix of Equation (7) in schrödinger picture. Differentiation of both sides of Equation (11) by gives


The solution of the above differential equation is


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


Taking, we have the above relation in another form.


We finally obtain


from Equations (15) and (14).

4. Solutions of Dirac Equation

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 defined as


where is the energy of a free Dirac electron with momentum,


Two arrows and denote ‘Up Spin’ and ‘Down Spin’ respectively. The explicit forms of eigensolutions in Heisenberg picture are given by


where is normalization volume, and are expressed as follows [16] ;





with normalization factor


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


respectively. The momentum takes discrete values in normalization volume: That is for. The functions are orthonormalized:


as well as


Next relations are especially important in a frame:



Because each component of satisfies, takes the eigenvalues and. This means the velocity also has two eigenvalues, that is the speed of light. However, states are not eigen states of. An explicit form of in the next section is applicable to the states.

5. Expectation Value of Zitterbewegung

In actual calculation, we will take z-axis along the momentum of the electron:. This procedure is necessary in order to not only simplify our calculation but also exclude the z-component of angular momentum caused by orbital motion of the electron.

The velocity operator is divided into two parts.






from Equation (16). The velocity is Zitterbewegung part which includes oscillation factor, and corresponds to uniform velocity. The coordinate operator is also easily calculated from corresponding part of the above equations.




where is an integration constant, and agrees to an initial point of the electron in classical sense. Remembering that is equal to the original of Equation (7), we easily calculate the expectation value of for in our frame, by the use of relations in Sections 3 and 4.


Equation (37) leads to







where is used. Arbitrary constant can set to zero without loss of generality. The similar results occur for x and y components of both and. The contribution from uniform velocity vanish at this time because of.





where the expression of the expectation value for is simplified. Results (38)-(46) indicate that Zitterbewegung (trembling motion) phenomenon for relativistic electron is un-observable effect in the sense that the expectation values of physical quantities always agree to classically measured one in accordance with Ehrenfest’s law [20] . A question whether Zitterbewegung works or not in actual physics phenomena then arises. The answer will be shown in the next section.

6. Spin-Magnetic Moment

We calculate the magnetic moment based on Equation (4). As mentioned in Section 2, the velocity of relativistic electron is not equal to but equal to. So that the expression of magnetic moment must be


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, and to the z-component of the magnetic moment.


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 is made only by Zitterbewegung parts. We can easily obtain each element:



We then finally obtain


which is the same result as in the previous work [2] .

It is necessary to recall that there is not any z-component of magnetic moment arising from orbital motion of electron because of and in our frame. Nevertheless, magnetic moment of Equation (59) has actually appeared. Therefore, we may conclude that of Equation (59) and the spin-magnetic moment of the electron must be identified. When the momentum of the electron is small and, it becomes


indicating correct -factor because of for Dirac electron in our frame; that is


where is the spin matrix operator.

7. Concluding Remarks

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 physical meaning of these matrix elements is inferred as follows:

The initial state of electron with positive energy and up-spin undergoes transition into the state with opposite signs of energy and spin (See Equations (29) and (30)), by operating. This matrix element is a kind of “transition current” because it exactly corresponds to the electric current in the definition of magnetic moment [2] which is written in another form [18] ,


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


The electron which has undergone transition into must immediately return to within time. The period of transition cycle is about


where means the energy gap of Dirac Sea in vacuum. This agrees with the frequency of Zitterbewegung (See Equations (57) and (58)).

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 (Figure 1). It then seems that the former is the magnetic moment which corresponds to Equation (5) with

Figure 1. Spin-magnetic moment caused by transition current.

, and the latter is the spin-magnetic moment with. In other word, the spin-magnetic moment may be caused not by usual electric current but by some new current which yield when the electron undergoes transition between two states of positive and negative energies. It should be noted that even an electron at rest (i.e. and) in space is able to yield this new current.

8. Another Remarks

Such a situation as described above occurs in some kind of solid state. In the two band model of Cohen and Blount [21] , Wolff [22] indicated that the Hamiltonian takes the Dirac form after a suitable transformation, and that the resulting equations are essentially identical to those of the Dirac theory. This fact means that we could apply the method developed here to the spin-magnetic moment of electron in solid state [23] . Zawadzki [11] indeed pointed out that the energy of the electron in narrow-gap semiconductor was given as


(: effective mass), where is maximum value of the group velocity of electron in band:


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


in the frame

where and are the Dirac matrices. The above Hamiltonian agrees completely to the Dirac Hamiltonian in Equation (7) if we replace by, and by. Obeying the Heisenberg equation of motion, it is then predicted that of the equal form reproduces the same results obtained here with the replacements and. They will appear as a change of -factor of the electron in semiconductor from new equation corresponding to Equation (59).

Conflicts of Interest

The authors declare no conflicts of interest.


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