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

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Spin-Magnetic Moment of Dirac Electron, and Role of Zitterbewegung Journal of Modern Physics, 2014, 5, 534-542 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jmp http://dx.doi.org/10.4236/jmp.2014.57064 Spin-Magnetic Moment of Dirac Electron, and Role of Zitterbewegung Shigeru Sasabe Department of Science & Technology, Tokyo Metropolitan University, Tokyo, Japan Email: [email protected] Received 20 December 2013; revised 18 January 2014; accepted 15 February 2014 Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract 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-momen- tum. 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. Keywords Spin-Magnetic Moment, Zitterbewegung, G-Factor, Dirac Electron, Band Structure, Semiconductor 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 How to cite this paper: Sasabe, S. (2014) Spin-Magnetic Moment of Dirac Electron, and Role of Zitterbewegung. Journal of Modern Physics, 5, 534-542. http://dx.doi.org/10.4236/jmp.2014.57064 S. Sasabe ec µz,11 = (1) 2Ep more easily than the previous work [2], where µz,11 denotes expectation value of z-component of spin-mag- netic 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 H ext . This term [15] [16] e ext − σ ′H (2) 2mc was understood as the interaction energy −⋅µ H ext between an external magnetic field H ext and the magnetic moment µ . Then, physicists concluded that e µσ= ′ (3) 2mc 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 µ and the spin operator σ ′ 2 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 −⋅µ H . Generally, the magnetic moment for a charged particle moving with the velocity v and the charge e is defined as [17] [18] e µ =rv × , (4) 2c in the classical electrodynamics, where c is speed of light. In the non-relativistic quantum theory, the above equation can be expressed as e µ = L (5) 2mc by using vp= m and Lrp= × . In relativistic quantum theory, however, we should mind that we can not use Equation (5) for Dirac electron, because the velocity v and the momentum p are independent variables to each other (i.e. vp≠ m ) in this case [2]. 3. Zitterbewegung As to Zitterbewegung, we briefly show all equations that are needed later. The velocity v of Dirac electron is given [19] by Heisenberg equation, 1 vr≡= [ r,,Hc] =α (6) i where α = (αααxyz,,) are the Dirac matrices in Dirac Hamiltonian H= cα p + β mc2 . (7) The 44× matrices α and β are defined as 0σ 10 α = ,,β = (8) σ 0 01− where σ = (σσσxyz,,) are the Pauli matrices. These matrices satisfy the following relations: 22 αβi = =1(i = xyz ,,) (9) 535 S. Sasabe {ααi, j} = 2,,0 δ ij, { αβ i } = (10) 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 αi (t) as an operator vti ( ) c. Making use of Equation (10), we easily find [4] iααii( t) =2( t) H −= 2 cp i( i x ,, y z) , (11) where αi (0) is the original matrix αi of Equation (7) in schrödinger picture. Differentiation of both sides of Equation (11) by t gives 2 ααii(t) = ( tH) . (12) i The solution of the above differential equation is 2Ht ααii(t) = (0) exp . (13) i We substitute Equation (13) into Equation (11) to obtain i 2Ht −−11 ααii(t) = (0) expH+ cpi H . (14) 2 i Taking t = 0 , we have the above relation in another form. 2 ααi(0) = ( ii( 0.) H− cp ) (15) i We finally obtain −−112Ht ααiii(t) =−+( (0) cp H ) expcpi H (16) i 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 (cα p +=β mc2 ) ψψ E (17) has four eigen-solutions. We name these solutions Eps ( ) ( s= 1,2,3,4) defined as +↑Es,1( =) p +↓ = Esp ,2( ) Es ( p) = , (18) −↑Esp ,3( =) −↓ = Esp ,4( ) where Ep is the energy of a free Dirac electron with momentum p , 22 24 Ep = pc + mc. (19) Two arrows ↑ and ↓ denote ‘Up Spin’ and ‘Down Spin’ respectively. The explicit forms of eigen- solutions in Heisenberg picture are given by 1 ipr ψ ss(rr) ≡=Eu( p) s( p)exp , (20) V where V is normalization volume, and us ( p) are expressed as follows [16]; 536 S. Sasabe χ↑ 1 = σ u1 ( p) c p , (21) N χ↑ 2 + mc Ep χ↓ 1 = σ u2 ( p) c p , (22) N χ↓ 2 + mc Ep −cσ p χ 1 2 ↑ = mc+ E u3 ( p) p , (23) N χ↑ −cσ p χ 1 2 ↓ = mc+ E u4 ( p) p . (24) N χ↓ with normalization factor 2E = p N 2 . (25) mc+ Ep and eigen states of “Up Spin” and “Down Spin”, 10 χχ↑↓=,, = (26) 01 3 respectively. The momentum p takes discrete values in normalization volume VL= : That is pii= 2π nL (ni = ±1, ± 2, ⋅⋅⋅) for i= xyz,,. The functions us ( p) are orthonormalized: † uur( pp) s ( ) = δrs , (27) as well as EEr( pq) s ( ) = δδrs pq . (28) Next relations are especially important in a frame p = (0,0, p) : α xup14( ) = up( ) (29) α yu14( p) = iu( p). (30) 2 Because each component of α satisfies αi = 1 , αi takes the eigenvalues +1 and −1 . This means the velocity vi also has two eigenvalues vci = ± , that is the speed of light. However, states Er are not eigen states of vi . An explicit form of αi (t) in the next section is applicable to the states Er . 5. Expectation Value of Zitterbewegung In actual calculation, we will take z-axis along the momentum of the electron: p = (0,0, p) . 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 v = cα is divided into two parts. (Zit) (unif ) vtii( ) = vt( ) + vt i( ) , (31) where (Zit) 21− 2Ht vi( t) =( cα ii(0) − c pH )exp , (32) i and 537 S. Sasabe (unif ) 21− vii( t) = c pH , (33) (Zit) (i= xyz,,) from Equation (16). The velocity vi is Zitterbewegung part which includes oscillation factor, (unif ) and vi corresponds to uniform velocity. The coordinate operator r = ( xyz,,) is also easily calculated from corresponding part of the above equations. (Zit) (unif ) rtii( ) = rt( ) + rt i( ) , (34) (Zit) i 2−− 112Ht ri( t) =( cα ii(0) −+ c pH) H expRi (35) 2 i (unif ) 21− ri( t) = c pH ii t + r0, (36) where Ri is an integration constant, and r0,i agrees to an initial point of the electron in classical sense. Remembering that α (0) is equal to the original α of Equation (7), we easily calculate the expectation value of cα z for E1 in our frame, by the use of relations in Sections 3 and 4. 2 † cp Epc1( ) ααzz Ep 1( ) = cup 11( ) up( ) = . (37) Ep Equation (37) leads to (Zit) Epvt11( ) z ( ) Ep( ) = 0 (38) (Zit) Epzt11( ) ( ) Ep( ) = Rz (39) and 2 (unif ) cp EpvtEp1( ) zz( ) 11( ) = Epvt( ) ( ) Ep 1( ) = (40) Ep (unif ) EpztEp1( ) ( ) 11( ) = Epzt( ) ( ) Ep 1( ) + Rz (41) cp2 =tz ++0 Rz , (42) Ep −−11 where HE11= EEp is used.
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