A New Wave Equation of the Electron

Home , Compton wavelength, Fine structure, Zitterbewegung

Journal of Modern Physics, 2011, 2, 1012-1016 doi:10.4236/jmp.2011.29121 Published Online September 2011 (http://www.SciRP.org/journal/jmp)

Arbab I. Arbab Department of Physics, Faculty of Science, University of Khartoum, Khartoum, Sudan E-mail: [email protected], [email protected] Received July 17, 2010; revised April 12, 2011; accepted April 28, 2011

Abstract

A new form of Dirac equation of a second order partial differential equation is found. With this wave equation the quivering motion (Zitterbewegung) is satisfactorily explained. A quaternionic analogue of Dirac equation is presented and compared with the ordinary Dirac equation. The two equations become the same if we replace the particle rest mass, m0, in the latter by im0. New space and time transformations in which these two equations represent a massless particle are found. The invariance of Klein-Gordon equation under these transformations yields the Dirac equation. The electron is found to be represented by a superposition of two waves with a group velocity equals to speed of light in vacuum.

Keywords: Dirac Equation, Zitterbewegung, Universal Quantum Wave Equation, Quaternion Quantum Mechanics

1. Introduction ties of the Dirac electron, including these problematical properties, lend themselves to a physical modeling ap- In 1928 Dirac introduced his relativistic wave equation proach, which may help resolve some of these problem- [1]. He had wanted to find an equation for the electron atical properties. that would be consistent with special relativity and de- The Zitterbewegung is a theoretical rapid motion of scribe the known fine structure frequency spectrum of elementary particles, in particular electrons, that obey the hydrogen. In technical terms the equation was Lor- Dirac equation. The existence of such motion was first entz-covariant. It correctly described the fine structure of proposed by Erwin Schrodinger in 1930 as a result of his hydrogen. When Dirac examined the solutions to his analysis of the wave packet solutions of the Dirac equa- equation, he found that, much to his surprise, it also pre- tion for relativistic electrons in free space, in which an dicted the electron’s spin and its magnetic moment. Per- interference between positive and negative energy states haps even more surprising, his equation also suggested a produces what appears to be a fluctuation (at the speed of positively charged particle with the mass of the electron light) of the position of an electron around the median, 2 could exist. Based on his equation, in 1931 Dirac pre- with a circular frequency of 2mc0  , or approximately dicted what he called the anti-electron (a positron) and 1.6 1021 Hz . This very rapid motion of the electron confirmed experimentally in 1932. In 1933 he shared means we cannot localize the electron extremely well with Schrodinger the Nobel Prize for physics for his and gives rise to the Darwin term. This motion never work in quantum mechanics. been observed for a free electron, but the behavior of In developing his equation, Dirac assumed that the such a particle has been simulated with a trapped ion, by electron was a point-like particle with an electric charge putting it in an environment such that the non-relativistic equal to the experimentally measured charge of the elec- Schrodinger equation for the ion has the same mathe- tron. Dirac however did not try to develop visualizable matical form as the Dirac equation (although the physical physical models as aids to picturing or understanding his situation is different) [2,3]. Thus, if we measure the ve- mathematical results. locity component in any direction, we should either get There are several results of Dirac’s equation which plus or minus c. This seems quite surprising, but we were problematical, despite the equation’s general suc- should note that a component of the velocity operator cess. It was mentioned that Dirac did not in this period does not commute with momentum, the Hamiltonian, or attempt to make a physical model of the electron and did even the other components of the velocity operator. If the not support such attempts. However, many of the proper- electron were massless, velocity operators would com-

Copyright © 2011 SciRes. JMP A. I. ARBAB 1013 mute with momentum. In more speculative theories of  ψ =0. (3) particles, electrons are actually thought to be massless, Equations (1) - (3) yield the two wave equations getting an effective mass from interactions with particles 2 present in the vacuum state. Barut and Bracken [4] ana- 1 2ψψmmc   2ψψ 2=00  0, (4) lyzed Schrodinger’s Zitterbewegung results and pro- 22    ct t   posed a spatial description of the electron where the Zit- terbewegung would produce the electron’s spin as the and orbital angular momentum of the electron’s internal sys- 2 2 1 00002 mmc   tem, while the electron’s rest mass would be the elec- 2200 2=    . (5) 0 tron’s internal energy in its rest frame. The states of ct t   definite momentum are not eigen-states of velocity for a Using the transformation massive electron. The velocity eigen-states mix positive mc2 and “negative energy” states equally. = 0 , (6) Thus, while momentum is a constant of the motion for  t  a free electron and behaves as it did in non-Quantum Equations (1) and (2) become Mechanics, the velocity behaves very strangely in the 1  ψ Dirac theory, even for a free electron. While Dirac equa- 0 ψ =,=2  0 . (7) tion is a first-order differential equation in space and c   time, Klein-Gordon equations is a second-order in space Employing Equation (6), Equations (4) and (5) are and time. The two equations are invariant under Lorentz transformed into the wave equations transformations. 2 We would like in this paper to write Dirac equation as 2221  2   0 =0,ψ =0, =22  . (8) a second - order differential equation in space and time c  mimicking the standard wave equation. In doing so, we The physical meaning of the vector ψ is still not have found new space and time transformations under clear. But I argue that it can be associated with funda- which Dirac equation represents a particle with zero mental particles (eg., quarks) that making up all hadrons, mass. Moreover, the ordinary Dirac equation is invariant since it has three components. In this way  0 may rep- under these transformations. Interestingly, the invariance resent a scalar particle (boson). Notice that Equations (4) of Klein-Gordon equation under these transformation and (5) can be obtained from the energy equation yields Dirac equation. The new Dirac equation has many 222 2 Epc = , where EEimc =  0 , E is the total en- interesting consequences. As in the de Broglie theory, ergy of the particle, and using the familiar quantum me- the electron is described by a wavepacket whose group chanical operator replacements, viz., piˆ =   and velocity is equal to the speed of light in vacuum. The ˆ  nature of this wave may explain the Zitterbewegung ex- Ei=  . Ecp = is an energy equation for a massless t hibited by the electron as found by Schrodiner. particle. Thus, it is interesting that a massive particle can be transformed into a massless particle using Equation 2. Quaternionic Quantum Mechanics: (6). Since energy is a real quantity, this equation is Dirac-Like Equation physically acceptable if it describes a particle with imaginary mass. In this case the energy equations split Consider a particle described by the quaternion wave- 2 into two parts; one with EEmc =  0 and the other i 2  with energy EEm =  0c. Such energies can describe function  =, 0 ψ . This is equivalent to spinor c the state of a particle and antiparticle. A hypothetical representation of ordinary quantum mechanics which we particle with an imaginary mass moving at a speed have recently developed [5]. The evolution of this qua- higher than speed of light in vacuum is known as ternion wavefunction is defined by the three equations tachyon [6]. Hence, our above equation can be used to [5] treat the motion of tachyons. This implies that our equations, Equations (4) and (5), can be applied to tachyons. 1  m 00 Some scientists propose that neutrino can be a tachyonic  ψ 2   0 =0, (1) c t  fermion [7]. This is in favor of the experimental finding

2 that the squared mass of neutriono is negative [8]. We ψ mc0  0  ψ =0, (2) know that Cherenkov radiation is emitted from a particle t  moving in a medium with speed greater than speed of and light in vacuum. When the speed exceeds the speed of

Copyright © 2011 SciRes. JMP 1014 A. I. ARBAB light in vacuum, the extra energy acquired by the particle This is a wave equation for a massless particle. Hence, is transformed in radiation. This can happened momen-  can also represents a particle with zero mass when tarily for a particle keeping its total energy conserved. taken in  —time. Equations (12) and (16) show that the Thus, the excess energy (speed) is such that it compen- particle is in state of continuous creation and annihi- sate the dissipations.  lation. This happens after a time of and a dis- mc2 3. Ordinary Quantum Mechanics: Dirac’s 0  Equation tance of which is traversed with speed c , as evi- mc0 Dirac’s equation can be written in the form [1] dent from Equations (13) and (15). 1  im c It is interesting to see that the invariance of Klein-    0 =0. (9) Gordon equation under the transformations defined by ct  Equations (13) and (15) yields Dirac equation. Moreover,

10 0 σ 22 Dirac equation, viz., Equation (9), is also invariant under where   ,   , ==1 and the transformations in Equations (13) and (15). 01 σ 0 Since  is a four components spinor, we can write it σ are the Pauli matrices. From Equation (9) one has  im c 1   in terms of two components doublets, viz.,    .   0  . Squaring this equation yields,      ct Substituting these decomposed spinors in Equation (10), 2 2 one obtains the two equations 1 2 mi00 mc 22 2=  0. (10) ct t 2 2 1 2 mi00  mc  22 2=    0, (17) Employing Equation (9) once again yields, ct t   2 2 and 1   2 mci00 mc 22 2=α    (11) 0. ct   2 2 1 2 mi00  mc  22 2=   0, (18) This can be written as ct t   1 2 Equations (17) and (18) imply two energy solutions, 2 22  =0, (12) 2 ct one with EEmc =  0 and the other with energy 2 where EE =  mc0 . Equation (10) can be obtained from Eq- mc uations (4) or (5) by replacing m0 by im0 . Therefore,  = i 0 . (13) Equations (4) and (5) can represent imaginary massive  bodies (tachyons). De Broglie hypothesis that all mi- Equation (10) can be obtained from Equation (9) by croparticles exhibit a wave nature. This is possible if we squaring it. These forms of Dirac equation, Equations (9) describe a particle by a wavepacket. In this way the and (10), have not been obtained before. They exhibit group velocity of the wavepacket will be the velocity of clearly the wave nature of the electron. It can be com- the moving particle. However, a single wave can’t de- pared with the Klein-Gordon equation of spin-0 particles scribe a particle. In Dirac’s theory the electron is found 2 2 to move with speed of light and owing to Einstein’s rela- 1   2 mc0 22 =0. (14) tivity it must have a zero mass. Hence, this motion is ct  problematic in Dirac’s theory. The negative energy equa- Equations (10) and (11) are another form of Dirac’s tion, i.e., Equation (18), can be written as 1 222mc2 equation exhibiting the wave nature of spin- particles i =,2 0  2 tm222mc22 t explicitly. Using the transformation 0 0 which can be seen as a generalized Schrodinger’s equa- mc2 = i 0 , (15) tion. It can thus represent the motion of a positron. It is  t  interesting to notice that if we rotate space and time, viz., Equation (10) can be written as ti t and r ir, in Equation (4) we will obtain the

2 Dirac positive energy equation, i.e., Equation (17). Simi- 1   2 larly, for anticlockwise rotation, viz., tit and 22 =0. (16) c  rir, Equation (4) yields the Dirac negative energy

Copyright © 2011 SciRes. JMP A. I. ARBAB 1015 equation, i.e., Equation (18). Hence, our Equation (4) is h Compton wavelength,  = , (  >  ), and in a equivalent to Dirac equation in imaginary space-time. c mc c The transformation equations, Equations (13) and (15), 0 negative energy state (antiparticle) when its wavelength can be compared with the covariant transformation that is smaller than the Compton wavelength (  <  ). is defined by c However, a particle existing in the negative energy state i (antiparticle) having  =  will be in a state of rest D= eA, (19) c    (trapped). Thus, the particle is in essence a superposition between its matter and antimatter states, these two con- where A is the gauge potential. Hence, one concludes that tradictory sides of its personality should interfere, setting the particle quivering (Zitterbewegung). However, this mc0 mc0 phenomenon has never been observed experimentally A α,A0 . (20) ee because the motion is too small and too fast to detect in real quantum systems. This analogy dictates that A to be a matrix. It is re- lated to Dirac matrices by the relation The group velocity of a (non)-relativistic particle is equal to its velocity. Equations (19) and (20) implies that mc0 mc00 the group velocity of the wavepacket is given by A γ,A0  . (21) ee  v==, c. (25) The plane wave solution of Equation (5) takes the g k form This implies that not the electron as a whole that os-  rt,= Aeit kr , A = const . (22) cillates but its constituents that having zero mass. Hence real particles and tachyons are particles move with speed Substituting this solution in Equation (10) yields the of light. However, the phase velocity of each wave com- dispersion relation prising the wave packet is less than speed of light in 2 vacuum, as evident from Equations (20) and (21). The mc0   =ck, (23) group velocity v= c does not necessarily require a  g particle to have m0 =0, as evident from Equation (21). and in Equations (17) and (18) yield the positive and It is a transient property of the particle when seen in a negative frequencies different time frame ( ). Hence, the Dirac’s velocity 22 dilemma is now resolved. mc00mc =,=ck ck, (24)  4. Electromagnetic Wave in Conducting respectively. This implies that this plane wave is a col- Medium lection of two waves of two different frequencies but same wavelength. It is thus a wavepacket. We may as- For a conducting medium with conductivity  , Max- sume that a particle is a combination of two states: one well equations read for electric and magnetic fields, E , moving with momentum k in one directions and the B the wave equation other with momentum k is the opposite direction 1 2 BB with velocity of light, c . This may imply an internal 2 22B  =0, (26) circular motion resulting from the two states. Hence, in ct t the electron rest frame the total momentum is zero. Does and this circular motion give rise to the spin of the electron? 2 Does this imply that the electron is not fundamental and 1 EE2  22E   =0. (27) consists of some substructure? Dirac’s theory of the rela- ct t  tivistic electron did not include a model of the electron Comparing this with Dirac equation, Equation (4), one itself, and assumed the electron was a point-like particle. obtains the relation Notice that string theory postulated that the electrons and 221 222 quarks within an atom are not 0-dimensional objects, but mm0* =  , (28) rather 3-dimensional oscillating lines (strings). The the- 4 ory advocates that these strings can vibrate, thus giving defining the effective mass ( m* ) of the magnetic field the observed particles their charge, mass and spin. when propagating in a medium. This is acceptable if we According to Equation (21) a particle is in a positive employ the wave-particle hypothesis of de Broglie. It is energy state when its wavelength is greater than the evident that this mass depends on the conductivity of the

Copyright © 2011 SciRes. JMP 1016 A. I. ARBAB medium. Hence, the magnetic field inside a conducting ond-order wave equation. Specific space and time trans- medium behaves like a particle with mass of m* . formations allow the mass term to disappear. When these Therefore, we see that when an electromagnetic field transformations are applied to Klein-Gordon equation, propagates in a conducting medium, it loses part of its Dirac equation is obtained. The solution of the Dirac energy so that the photon acquires a mass that is directly equation yields two oscillating states with positive and proportional to the conductivity of the medium. More- negative energies. These states move with a group veloc- over, if one now writes the last term in Equation (27) as ity equals to the speed of light in vacuum. The existence 22 of such states resolved the quivery motion of the electron  mc0  = 2 E, (29) (Zitterbewegung) found by Schrodinger. This is because  not the electron as a whole that oscillates but its con- so that Equation (27) will be similar to Equation (17). If stituents that having zero mass. It is quite interesting to we take the divergence of both sides of Equation (29) notice that the quaternion analogue of Dirac equation and use Gauss law, we will obtain yields the ordinary Dirac equation if we replace m0 22 with im0 . 2 mc* 2 =0. (30)  6. References Consider the reflection of positively charged spin-0 particle from a high Coulomb barrier for which the [1] J. D. Bjorken and S. D. Drell, “Relativistic Quantum Klein-Gordon equation is [9] Mechanics,” McGraw-Hill, Boston, 1964. [2] L. Lamata, J. León, T. Schätz and E. Solano, “Dirac 2 2 2 1 2 iVmc0  V Equation and Quantum Relativistic Effects in a Single 22 2 2   =0, ct  ctc  Trapped Ion,” Physical Review Letters, Vol. 98, No. 25,  2007, Article ID: 253005. (31) doi:10.1103/PhysRevLett.98.253005 where the potential V is zero to the left of the barrier [3] D. Walter and H. Gies, “Probing the Quantum Vacuum: and a constant to the right. Hence, comparison with Perturbative Effective Action Approach,” Springer Ver- lang, Berlin, 2000. Equation (17) reveals that when m0  0 (massless boson) in Equation (31) the resulting equation will be [4] A. O. Barut and A. J. Bracken, “Zitterbewegung and the 1 Internal Geometry of the Electron,” Physical Review D, similar to Equation (17) for spin- particle. In this case, Vol. 23, No. 10, 1981, pp. 2454-2463. 2 doi:10.1103/PhysRevD.23.2454 1 one finds the mass of the spin- particle will be [5] A. I. Arbab, “The Quaternionic Quantum Mechanics,” 2 arXiv: 1003.0075v1, 2010. V [6] G. Feinberg, “Possibility of Faster-Than-Light Particles,” m = . Hence, a massless boson behaves as a massive c2 Physical Review, Vol. 159, No. 5, 1967, pp. 1089-1105. doi:10.1103/PhysRev.159.1089 V Dirac particle of an effective mass of m = 2 . Thus, a [7] J. Ciborowski, “Hypothesis of Tachyonic Neutrinos,” c Acta Physicsa Polonica B, Vol. 29, No. 1-2, 1998, pp. massless meson encountering a potential of 0.511 MeV 113-121. will acquire a mass equals to the electron mass, and [8] R. G. H. Robertson, et al., “Limit on e ν Mass Observa- therefore, will behave as an electron. tion of the β Decay of Molecular Tritium,” Physical Re- view Letters, Vol. 67, 1991, pp. 957-960. 5. Concluding Remarks doi:10.1103/PhysRevLett.67.957 [9] F. Gross, “Relativistic Quantum Mechanics and Field We have written Dirac equation in terms of a sec- Theory,” John Wiley & Sons, Inc., Hoboken, 1993, p. 97.