On the Relativistic Concept of the Dirac's electron Spin
N. Hamdan1, A. Chamaa1 and J. López-Bonilla2 1Department of Physics, University of Aleppo, Aleppo-Syria. 2SEPI-ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. Z-4, 3er. Piso, Col. Lindavista CP 07 738 México DF
E-mail: [email protected]
(Received 3 November 2007; accepted 10 January 2008)
Abstract Despite the success of the Dirac equation, the incorporation of the special theory of relativity into quantum mechanics predicts some paradoxes, like that the spin prediction from Dirac equation can be only identified with non- relativistic approximations (Pauli and Foldy-Wouthysen), as well as the spin predication is a relativistic quantum phenomena because the spin prediction is a necessary requirement of the relativistic quantum mechanics only. In this paper we show that the derivation of the spin and its magnetic moment can be done with a pure classical treatment. Since we start from the classical physical laws and the classical relativity principle to get the linear Schrödinger equation as a result the derivation of the spin and its magnetic moment can be done with a pure classical treatment. This approach result a Schrödinger equation, in which we show that the spin of the electron is a non relativistic quantum phenomenon too.
Keywords: Dirac equation, electron spin, non relativistic Quantum Mechanics.
Resumen A pesar del éxito de la ecuación de Dirac, la incorporación de la teoría especial de relatividad en la mecánica cuántica predice algunas paradojas, como la predicción del spin de la ecuación Dirac que puede ser identificada solamente con aproximaciones no-relativistas (Pauli y Foldy-Wouthysen), así como el pronóstico del spin que es un fenómeno cuántico relativista porque la predicción del spin solo es un requerimiento necesario de la mecánica cuántica relativista. En este artículo mostramos que la derivación del spin y su momento magnético puede ser realizada con un tratamiento puramente clásico. Ya que comenzamos de las leyes físicas clásicas y del principio de relatividad clásico para obtener la ecuación de Schrödinger lineal como un resultado, se puede realizar la derivación del spin y su momento magnético con un tratamiento puramente clásico. De esta aproximación resulta una ecuación de Schrödinger, en la cual mostramos que el spin del electrón ya no es un fenómeno relativista cuántico.
Palabras clave: Ecuación de Dirac, spin del electrón, mecánica cuántica no relativista.
PACS: 03.30.+p, 03.50.-z, 03.65.-w, 03.65.Pm. ISSN 1870-9095
I. INTRODUCTION was impossible to directly write a non-relativistic equation for spin-1/2 particles and that it could therefore only be In 1926 while Schrödinger was publishing his non- derived as a non-relativistic limit of the relativistic Dirac relativistic single particle wave equation [1], Dirac [2] was equation. So it was known in standard quantum mechanics searching for a relativistic invariant form of the one- that the spin of electron has only relativistic nature. particle Schrödinger equation for electrons starting from However, in 1984, this supposition was questioned by W. the relativistic equation, which was known as Klein– Greiner [4] when he derivates the spin from the non- Gordon equation (KGE). However, at that time several relativistic quantum mechanics, i.e., he derivates the spin objections emerged against the KGE as a single particle from the Schrödinger equation. equation because its solutions allowed negative probability In this paper we obtain additional advantage densities, besides there was the possibility of negative concerning the same result of Greiner, where we revealed energies and their solutions did not have clear spin that the spin of the electron and its magnetic moment can dependence. In 1928 Dirac published an equation [2, 3] be derived from the modified Schrödinger equation which was presented as a definite solution to the above without using any kind of approximations (non-relativistic mentioned problems where he has shown that the spin limit of Dirac equation), and that the derivation of the spin belongs to the relativistic wave equation. The integration and its magnetic moment can be done with a pure classical of the special relativity theory with quantum mechanics treatment. has yielded many paradoxes that remained unsolved that it Lat. Am. J. Phys. Educ. Vol. 2, No. 1, January 2008 65 http://www.journal.lapen.org.mx
N. Hamdan, A. Chamaa and J. López-Bonilla
II. THE RELATIVISTIC DIRAC EQUATION an intrinsic magnetic moment for the electron and gives its AND THE PAULI EQUATION AS A NON- correct value only when it is obtained as the non- RELATIVISTIC LIMIT OF THE DIRAC relativistic limit of the Dirac equation. EQUATION In the Dirac equation for the relativistic charged particle moving in a constant magnetic field The early twentieth century saw two major revolutions in ∂ ⎡ ⎛⎞e ⎤ ⎢ ⎜ ⎟ 2 ⎥ , (7) the way physicists understand the world. The first one was ic= ψ = α⎜ pAˆ − ⎟+ βψ mc0 ∂tc⎢ ⎜⎝⎠⎟ ⎥ quantum mechanics itself and the other was the theory of ⎣ ⎦ relativity. Important results also emerged when these two we can follow Pauli's approach by eliminating small theories, i.e., quantum mechanics and the theory of components to derive the Pauli equation. We consider a relativity were brought together and one of these results is two-component representation, where the four-component the Dirac equation which leads to the spin of an electron spinor is decomposed into two spinors b and s ,each that was known as a relativistic effect. one with two components When calculating kinetic energy relativistically using Lorentz transformation instead of Newtonian mechanics, ⎛⎞ψ ⎛⎞ψψ ⎛⎞ ψψ===⎜ b ⎟⎟⎟,,.⎜⎜13 ψ (8) Einstein discovered that the amount of energy is directly ⎜ ⎟ bs⎜⎜⎟⎟ ⎝⎠ψ s ⎝⎠ψψ24 ⎝⎠ proportional to the mass of body: In the non-relativistic limit, the rest energy, m c2 becomes 2 o Emc= , (1) dominant; therefore, the two component solution is approximately where E is the total energy and m the relativistic mass. 2 The energy and momentum of a particle momentum −im0 c t are then related by the principal equation governing the ψ =e = ψ 0 . (9) dynamics of a free particle: bs,, bs
222 4 Substituting Eq. (9) into Eq. (7), and using Eq. (6), it gives Ec=+p mc0 , (2) ⎛⎞ ∂ 00⎜ e ⎟ , (10a) where c is the speed of light, m is the rest mass of the ici==ψ bs= α⎜ ∇− A⎟ψ o ∂tc⎝⎠⎜ ⎟ particle and p is the momentum. Following Dirac, we take into account the time ⎛⎞ ∂ 0020⎜ e ⎟ , (10b) dependent of Schrödinger equation: icimc==ψ s = α⎜ ∇−A⎟ψψbs −2 0 ∂tc⎝⎠⎜ ⎟ ∂ iH= ψψ= ˆ , (3) ∂ ∂t imc=ψ 020ψ , (11) ∂t s 0 s using (2) and (3) Dirac assumed that and with this last approximation, Eq. (10b) becomes to ∂ 2242 icmc= ψ =+pˆ 0 ψ . ⎛⎞ ∂t ⎜ e ⎟ 020, 02= ciα⎜ =∇−A⎟ψ bs − mc0 ψ ⎝⎠⎜ c ⎟ One of the conditions imposed by Dirac in writing down a relativistic equation for the electron was that the ‘‘square’’ which gives of that equation will give the Klein-Gordon equation. ⎛⎞e Imposing the additional condition of linearity of in the ⎜ =∇− ⎟ α⎜i A⎟ 00⎝⎠c . (12) components of pˆ led Dirac to following relation ψ s = ψ b 2mc0 ∂ ˆ , (4) 0 iH= ψ = Dψ The lower component ψ is generally referred to as the ∂t s 'small' component of the wavefunction , relative to the where 'large' component 0 . ψ p Hcˆ =+()αpˆ mc2β , (5) 0 D 0 Substituting the expression ψ s given by Eq. (12), into Eq. (10a), we obtain ⎛⎞0 σ ⎛⎞I 0 ⎛⎞⎛⎞ee k , (6) α⎜⎜ii==∇−A⎟⎟α ∇− A αk ==,1,2,3,k β = ⎜⎜⎟⎟ ⎜⎟ ⎜⎟ ∂ 00⎝⎠⎝⎠cc ⎝⎠σ k 0 ⎝⎠0 −I i= ψ = ψ . ∂tmbb2 0 and I is the two-by-two identity matrix. Finally, by using the well-known identities In standard quantum mechanics, it is not possible to directly extend the Schrödinger equation to spinors, so the ((σa) σb) = ab + iσ(a× b) , Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. This is in particular the we deduce that, being B=∇×A the magnetic field, case for the Pauli equation which predicts the existence of Lat. Am. J. Phys. Educ. Vol. 2, No. 1, January 2008 66 http://www.journal.lapen.org.mx
On the Relativistic Concept of the Dirac's electron Spin
2 ⎛⎞0 σ ⎛⎞ ⎛⎞e ⎜ j ⎟ I 0 ⎟ ⎜i=∇− A⎟ γγ===⎜ ⎟ (1,2,3),j ⎜ ⎟ .(20) ⎜ ⎟ j ⎜σ 0 ⎟ 4 ⎜ ⎟ ∂ 00⎝⎠c e= . (13) ⎝⎠j ⎝⎠0 −I i= ψ bb=[]− σB ψ ∂tmmc2200
The Pauli equation for the theory of spin was derived as a IV. DERIVATION OF MODIFIED non-relativistic limit of the relativistic equation and it was well known in standard quantum mechanics as a direct SCHRÖDINGER EQUATION FROM THE proof of the fundamentally relativistic nature of the spin. CLASSICAL PHYSICAL LAWS AND ITS As we shall verify, the modified Schrödinger equation will SOLUTIONS lead directly to the Pauli equation and therefore to the spin of the electron without using the non- relativistic limit, In several recent papers [5, 6] we suggested another way to with the same results as in Dirac's theory of the relativistic account for the Lorentz transformation and its kinematical electron. effects in relativistic electrodynamics as well as in relativistic mechanics. And by following the same approach we derived Einstein’s equation as well as the De Broglie relation from classical physical laws such as the III. DERIVATION OF LINEAR SCHRÖDIN- Lorentz force law and Newton’s second law [7, 8, 9] GER EQUATION 2 Et = mc , (21a) It is well known that the nature of spin defies non relativistic QM. Therefore a statement was known that the h p = . (21b) spin must have to do with special relativity although its λ connection is not entirely understood. In contrast to this statement, W. Greiner [4] has followed the Dirac’s We showed also that Eq. (21a) could be written as approach where he started from the same premise: the v2 pˆ 2 Emvmc=+221,− (22) Schrödinger operator KEˆˆ= − must be linear in t 0 c2 2m 0 momentum. Then Greiner writes the free Schrödinger and from the last relation we have by definition the kinetic energy equation KˆΨ = 0 as ⎛⎞2 22 2⎜ v ⎟ ΘΨˆˆˆ=++=()AEˆˆBpˆ C Ψ 0 , (14) EEmcmvmc= − =+ ⎜11−− ⎟ , kt00⎜ 2 ⎟ ⎝⎠⎜ c ⎟ where the operator Θˆ would be linear in momentum, thus there must be an operator such that and for non-relativistic velocities, v<
Lat. Am. J. Phys. Educ. Vol. 2, No. 1, January 2008 67 http://www.journal.lapen.org.mx
N. Hamdan, A. Chamaa and J. López-Bonilla
⎛⎞−iI 0 For a given E from Eqs. (28) follows that AM= ⎜ ⎟ , (24a) ⎜ ⎟ ⎝⎠00 −σ p χ = jjϕ , (31) 002mi ⎛⎞00 (24b) 0 CM= ⎜ ⎟ , ⎜ ⎟ ⎝⎠02miI and from Eqs.(28) and (31) we find ⎛⎞ 0 σ j ⎟ (24c) 2 BM= ⎜ ⎟. ()σ jj⋅ p j ⎜ ⎟ Eϕ = ϕ .. (32) ⎝⎠⎜σ j 0 ⎟ 00 2m0 Setting Eqs. (24) in Eq. (17), and multiplying by M -1, we Using Eq. (30) in Eq. (32), we find that 0 can take one of get the representations ⎛⎞ ⎛⎞0 ⎛⎞ iI 0 σ j ⎟ 00⎟ (25) ⎛⎞1 ⎜ ⎟EˆΨΨΨ=+⎜ ⎟pˆ ⎜ ⎟ . ϕ = , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎜⎟ ⎝⎠00 ⎝⎠⎜σ j 0 ⎝⎠02miI0 ⎝⎠0 or We shall continue to find the eigenfunctions of the new ⎛⎞0 , linear Schrödinger equation to prove that there is ϕ0 = ⎜⎟ contradiction with Dirac's conceptions of relativistic spin ⎝⎠1 of electron. That means we will show that the new linear and from the above calculations we deduce that Schrödinger equation is an equation for describing the spin ⎛⎞ϕ0 i of electron. Solutions to Eq. (25) are plane waves which ⎛⎞ϕ(,)xt ⎜⎟()px− Et Ψ=(,)xt N = N iσ pˆ e = . (33) ⎜⎟⎜⎟ϕ can be written in the following form ⎝⎠χ(,)xt ⎜⎟0 ⎝⎠2m0 ⎛⎞⎛⎞ϕϕ(,)xt () x −iEt = (26) Ψ=(,)xt N⎜⎟⎜⎟ = N e , By calculating the normalization constant N for positive χχ(,)xt () x ⎝⎠⎝⎠ energy solution, we obtain where N is the normalization constant, and ⎛⎞ϕ()x is a four ⎜ ⎟ 2m . ⎝⎠⎜χ()x ⎟ N = 0 E + 2m component spinor. Substituting Eq. (26) in Eq. (25), and 0 considering that If we consider the free electron motion along the z- direction, then the two states that represent free moving ⎛⎞ϕ()x ⎛⎞ϕ ipx 0 ⎟ = ⎜ ⎟=⎜ ⎟e , electron are ⎜ ⎟ ⎜ ⎟ ⎝⎠χ()x ⎝⎠χ0 ⎛⎞⎛⎞1 ⎜⎟⎜⎟ where 0 (34a) 2m0 ⎜⎟⎝⎠ ipzEt()− ψ G 1 = e , p,+ ⎜⎟ ⎛⎞ 2 Em+ 2 0 ipσ ⎛⎞1 ψ1 ⎟ ⎜⎟z ϕ =⎜ ⎟ , ⎜⎟⎜⎟ 0 ⎜ ⎟ ⎝⎠2m ⎝⎠0 ⎝⎠ψ 2 ⎛⎞⎛⎞0 and ⎜⎟ ⎜⎟1 (34b) 2m0 ⎜⎟⎝⎠ ipzEt()− ψ G 1 = e . ⎛⎞ψ 3 p,− ⎜⎟ ⎜ ⎟ 2 Em+ 2 0 ipσ ⎛⎞0 χ0 = ⎜ ⎟ , ⎜⎟z ⎜ψ ⎟ ⎜⎟⎜⎟ ⎝⎠4 ⎝⎠2m ⎝⎠1 are two-component spinors, we find that These are just the wave functions that can describe the spin up, and the spin down. For each value of p there is one iI 000⎛⎞ϕ ⎛⎞0 σ ⎛⎞ϕϕ ⎛⎞ ⎛⎞000j ⎛⎞ (27) positive eigenvalue, Eq. (30), and two eigenfunctions, Eqs. Ep=+j ⎜⎟2 mi0 ⎜⎟⎜⎟σ 0 ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠00⎝⎠χ000⎝⎠j ⎝⎠χχ ⎝⎠ 0I ⎝⎠ (34), in according with eigenvalue Eq. (25). it implies the two equations
(28a) V. INTERACTION WITH THE MAGNETIC iEϕσ00− jj p χ= 0, FIELD- THE PAULI EQUATION (28b) σϕjjpmi000+=20, χ The most important result of the relativistic Dirac equation which have solution if was presenting a theoretical description of the electron iE−σ p spin and its magnetic moment, which means that the jj= 0, (29) predictions of electron spin is the property of the Dirac σ pmi2 jj 0 equation only. That is not true for many reasons: and by using the same known identities, that used above, If one derives the spin of the electron and its magnetic then Eq. (29) goes into moment from the non relativistic linear Schrödinger equation, Eq. (25). The predictions of electron spin did not 2 p (30) occur directly from the Dirac equation, but using E = . approximation like the non relativistic limit of Dirac 2m 0 Lat. Am. J. Phys. Educ. Vol. 2, No. 1, January 2008 68 http://www.journal.lapen.org.mx
On the Relativistic Concept of the Dirac's electron Spin equation to get the Pauli equation with the proper value of linear Schrödinger Hamiltonian s. From Eqs. (24) and the spin of the electron and its magnetic moment from the (25) we can define the linear Schrödinger Hamiltonian s non relativistic Pauli equation ˆ =+ˆ , (40) As we know, the momentum p is replaced in the same H s αp D way to include the effects of electromagnetic fields, and if where we only consider the effect of a magnetic field B then the e ⎛⎞00 ⎛⎞ 0σ momentum is replaced as ppˆˆ→−A , and Eq. (25) D =⎜⎜⎟⎟, α= . c ⎝⎠⎜⎜02miI⎟⎟ ⎝⎠σ 0 becomes The orbital momentum Lˆ doesn’t commutes with s ⎛⎞ ⎛⎞ ⎛⎞ ⎜⎜iI 00⎟⎟σ ⎛⎞e ⎜00⎟ (35) ⎜⎜⎟EˆΨΨΨ= ⎟−⎜pˆ A⎟ +⎜ ⎟ , , (41) ⎜⎜⎟⎟⎜⎜⎟ ⎜ ⎟ [HLs ,,()]=+[α pˆˆ DL] =− i= α×p ⎝⎠00 ⎝⎠σ 0⎝⎠c ⎝⎠02miI0 which implies however, to find an operator which commutes with s, we remind the operator B in Eq. (24c): ⎛⎞⎛⎞e ⎜⎟σ⎜pˆ − A χ(,)xt⎟ ⎛⎞ˆ ⎜ ⎜ ⎟ ⎟ ⎛⎞σ 0 ⎜iEϕ(,) x t ⎟ ⎜ ⎝⎠c ⎟ ⎛⎞0 ⎟ B =⎜ ⎟ , ⎜ ⎟=+⎜ ⎟ ⎜ ⎟ ,(36) ⎜ ⎟ ⎜ ⎟ ⎜ ⎛⎞⎟ ⎜ ⎟ ⎝⎠0 σ ⎝⎠0 ⎜ e ⎟ ⎝⎠2(,)im0 Iχ r t ⎜σ⎜pˆ − A⎟ϕ(,)xt⎟ ⎝⎠⎜ ⎝⎠⎜ c ⎟ ⎟ and its commutator with s is and []His ,2().B =×α pˆ (42) ⎛⎞e , (37a) iEˆϕχ−−σ⎜pˆ A⎟ = 0, Now if we define Dirac spin operator as ⎝⎠⎜ c ⎟ = ⎛⎞ Sˆ = Bˆ , (43) ⎜ e ⎟ (37b) σ⎜pˆ − A⎟ϕχ+=20,mi0 2 ⎝⎠⎜ c ⎟ it satisfies therefore =×= ˆ , (44) 2 [HSs ,()] iα p ⎪⎪⎧⎫⎛⎞e ⎨⎬σ⎜pˆ − A⎟ ⎪⎪⎝⎠⎜ c ⎟ and from commutators (41) and (44) we see that: Extˆϕϕ(,)= ⎩⎭⎪⎪ (,). xt (38) 2m 0 [HHiiss,LS+=] [ , J] === ()()α ×pˆˆ− α×=p 0 , Using the well-known identities, we get which means that the operator 2 ⎪⎪⎧⎫⎛⎞⎛⎞eee2 ⎪⎪⎜⎜ˆˆ− AA⎟⎟= −−= , Jˆˆ=+LSˆ , (45) ⎨⎬σ⎜⎜pp⎟⎟σB ⎩⎭⎪⎪⎝⎠⎝⎠ccc commutes with s. It shows also that the total angular we recover finally an equation for the two- component momentum, Eq. (45), is given similarly as in Dirac theory. spinor- e pˆ − A 2 ∂ϕ(,)xt() e= VI. CONCLUSION ixt= = c − σB ϕ(,), (39) [] ∂tmmc2200 which is the Pauli equation with a spin g-factor of 2, the Despite successes of the Dirac equation, there remain a same result as in Dirac’s theory, is derived from the non number of misunderstandings about this equation. The first relativistic linear Schrodinger equation, this means that the misunderstanding about Dirac equation is the spin of electron which is well known as a relativistic effect. electron has a magnetic moment –e /2mo and the magnetic moment interacts with an external magnetic Although of this the spin prediction from Dirac equation field, the corresponding contribution to the energy is - B. can not be allowed directly without approximations An important characteristic of Eq. (39) is that we did not methods i.e.; the non-relativistic limit of the relativistic use any kind of approximation to reach it, i.e., we did not Dirac equation to get the Pauli equation which predicts the existence of an intrinsic magnetic moment for the electron use the condition (11) to eliminate the lower component and gives its correct value only when it is obtained as the of the wavefunction , here the component of non-relativistic limit of the Dirac equation. eliminates itself without any approximation. The second misunderstanding about Dirac equation is ˆ In Schrödinger theory the orbital angular momentum L zitterbewegung. In the Dirac relativistic equation for the commutes with the Hamiltonian ˆ ˆ 2 , this is not H = p /2m spin 1/2 particle, there is a velocity operator vˆ = cαˆ . It is the case in Dirac theory, since the Dirac Hamiltonian is j linear in momentum and the total momentum Jˆˆ=+LSˆ believed that this operator is inadequate in two aspects: commutes with it. For this reason, it is conventional to The first one is that its eigenvalues are +c and -c with c choose an operator similar to J which commutes with the being the light speed in a vacuum. The other is that it is not proportional to the linear momentum. In the papers [12, Lat. Am. J. Phys. Educ. Vol. 2, No. 1, January 2008 69 http://www.journal.lapen.org.mx
N. Hamdan, A. Chamaa and J. López-Bonilla
13] it had been shown that the velocity of the particle [5] Hamdan, N., Abandoning the Ideas of Length obtained from the modified Dirac equation always moves Contraction and Time Dilation, Galilean Electrodynamics with velocity ±v as observed in the laboratory, and this 14, 83-88 (2003). result eliminate the problem of the zitterbewegung. [6] Hamdan, N., Newton’s Second Law is a Relativistic In contrast to it stands the wrong statement, which law without Einstein’s Relativity, Galilean attribute spin to relativistic characteristics and that non- Electrodynamics 16, 71-74 (2005). relativistic quantum mechanics is a theory of spinless [7] Hamdan, N., Derivation of the de Broglie's Relations particle. Recently, many Authors have argued that the from the Newton Second Law, Galilean Electrodynamics Schrödinger equation of non-relativistic quantum 18,108-111 (2007). mechanics describes not a spinless particle as universally [8] Hamdan, N., The Dynamical de Broglie Theory, assumed, but a particle in a spin eigenstate [11, 12, 13]. Annales Fondation Louis de Broglie 32, 1-13 (2007). The spin of electron can be derived gradually from the non [9] Hamdan, N., K., Hariri, A. and López-Bonilla, J., relativistic linear Schrodinger equation, and everything Derivation of Einstein’s Equation, E=mc2, from the result automatically as it is shown in the present paper. Classical Force Laws, Apeiron 14, 435-453 (2007). [10] Ward, D. and Volkmer, S., How to Derive the Schrödinger Equation, arXiv:physics/0610121, (2006). REFERENCES [11] Gurtler, R. and Hestenes, D. Consistency in the formulation of the Dirac, Pauli and Schrödinger Theories, [1] Schrödinger, E. Quantisierung als Eigenwertproblem, J. Math. Phys. 16, 573–584 (1975). Ann. der Physik 384, 361-376 (1926); 384, 489-527 [12] Bakhoum, E. G., Fundamental Disagreement of Wave (1926); 385, 437-490 (1926); 386, 109-139 (1926). Mechanics with Relativity, Physics Essays 15, (2002). [2] Dirac, P. A. M., The Quantum Theory of Electron, [13] Hamdan, N., Altorra, A. and Salman, H. A., The Proc. R. Soc. London A 117, 610–624 (1928). Classical Zitterbewegung, accepted for publication in [3] Moyer, D. F., Origins of Dirac’s Electron, 1925–1928, Galilean Electrodynamics. Am. J. Phys. 49, 944–949 (1981). [4] Greiner, W., Quantenmechanik I, (Band 4) (Verlag Harmi Deutsch, Thun, 1984) pp. 303–312.
Lat. Am. J. Phys. Educ. Vol. 2, No. 1, January 2008 70 http://www.journal.lapen.org.mx