An investigation into the source of stability of the electron spin projections
aa r X i v : . [ qu a n t - ph ] J un An investigation into the source of stability of theelectron spin projections
A. M. Cetto and L. de la Pe˜naInstituto de F´ısica, Universidad Nacional Aut´onoma de M´exicoCiudad de M´exico, MexicoOctober 7, 2018
Abstract
We propose that the stability of the two projections of the electronspin along the direction of an applied magnetic field, is an effect of high-frequency vibrations acting on the spin magnetic moment. The source ofthe high-frequency vibrations is to be found in the zero-point radiationfield.
In contrast with a classical compass needle, which points invariably towards theNorth pole, in the presence of an external magnetic field the spin of an electronorients itself either parallel or antiparallel to the direction of the field. Thisis such a well-established fact in quantum theory that nobody ever seems towonder about it, let alone try to find an explanation for it. The double signof the spin projection is taken for granted, even though the parallel orientationappears as classically counterintuitive, as it is energetically less favourable thanany other orientation of the spin with respect to the field: it corresponds in theclassical case to a position of unstable equilibrium.Yet also in classical mechanics there exist systems that can have two positionsof stable equilibrium in opposite directions. The best known instance of sucha system is the inverted pendulum, which is used as a benchmark in controltheory and finds numerous technical applications, some as popular as the self-balancing scooter. Without going so far, we refer to the experience of balancinga broom that stands upside down on the palm of our hand, by making small,rapid movements with the hand to keep it upright. Such remarkable state ofmotion is the result of adding to the pendulum a vibrating motion of highfrequency applied to the supporting point [1]. This would suggest the idea thatsomething similar can be occurring in the case of the spin magnetic moment.But such an idea would seem not to work at all, if no high-frequency source isavailable. 1ere we address this enigma by appealing to the existence of the vacuumor zero-point radiation field, as suggested by stochastic electrodynamics ( sed ).In this theory, proposed as a foundation for quantum mechanics, a central roleis played by the zero-point radiation field ( zpf ), taken as a Maxwellian fieldthat fills the whole space and covers the entire frequency spectrum (see e.g .[2] and references therein). In particular, the high-frequency modes (of Comp-ton’s frequency) acting on the electron have been shown elsewhere to be relatedto the origin of de Broglie’s wave [3] and to (a nonrelativistic version of) the zitterbewegung [4].It is therefore natural within sed to consider that also the magnetic momentof the electron is subject to the action of these zpf modes. This is what wedo in the present paper, with the purpose of finding an explanation for the twopositions of stable equilibrium of the (quantum) electron spin. In section wepresent the model of the electron spin as it emerges from sed , to be used forthis purpose. In section 3 we establish the equations of motion for the electronsubject to an applied magnetic field in addition to the zpf , and solve them forthe one-dimensional case by separating the fast variables from the slow ones,which leads to a direct demonstration of the stability of the two spin states, asseen in section 4. The extra vibrations of the magnetic moment of the electrondue to its interaction with the high-frequency magnetic component of the zpf result in an effective potential with two deep minima. As shown in section5, under conditions that hold in real experiments these minima correspond tothe two positions of stable equilibrium of the spin magnetic moment: spin ’up’and spin ’down’. In section 6 the same conclusion is seen to apply in the two-dimensional case, that takes into account the Larmor precession. The paperconcludes with a brief recapitulation. Before proceeding with the calculations let us briefly present the model of theelectron spin to be used in this paper. In quantum theory the electron spin isrepresented with the aid of the Pauli matrices, which automatically introduceall the required quantum properties. The situation is quite different in sed ,a theory based on the hypothesis that quantum mechanics is the result of theaction of the radiation zpf on an otherwise classical system. This theory hasevolved in the course of time to reach a well-developed status (see e.g. ref. [2])that includes, in particular, a theory of the electron spin, just the one we takehere as the basis for our proposal.The most immediate effect of the presence of the zpf is that an electron,which normally is part of an atomic system, acquires a stochastic motion. The zpf covers the entire spectrum; but not all frequencies have the same importancefor the atomic system. As is well known, Dirac’s equation for the free electronreveals the existence of the zitterbewegung , a helicoidal motion of frequency ofthe order of ω C and amplitude of the order of λ C , where ω C = mc / ~ representsthe Compton frequency of the electron of mass m , and λ C the corresponding2avelength λ C = h/mc. In the textbooks on quantum mechanics, λ C appearsnormally only in association with the Compton effect and related considerations.When looking beyond the quantum formalism, λ C reappears, with an importantthough indirect role as the source of de Broglie’s wavelength ([2], [3]). In sed —as in nonrelativistic qed — ω C is used frequently as a convenient cutoff toregularize integrals that determine e.g. the average properties of dynamicalvariables. This leads to the emergence of particle oscillations of frequencies oforder ω C . The introduction of the cutoff is thus of physical significance ratherthan a simple mathematical device, meaning that the electron (as all matter)becomes transparent to the radiation field of very high frequencies.The physical meaning assigned to the cutoff leads to an also physically mean-ingful finite effective size for the electron, of the order of λ C . Such effectivesize has only a statistical (or coarse-grained) sense, since it is the result of thefluctuations or rapid oscillations impressed upon the particle by the field, theoriginal electron being still a point particle; it is the interaction with the fieldthat dresses it. This view coincides with the qed picture, where the electronacquires through its electromagnetic interactions an effective size of the orderof λ C (see e.g. [4]).We thus arrive at a consistent picture of the electron as a small sphere ofeffective radius of order λ C realizing a kind of nonrelativistic zitterbewegung offrequency of order ω C . As a consequence of the torque exerted by the electricfield modes of a given circular polarization, the electron describes a helicoidalmotion, which results in a mean intrinsic angular momentum of value ~ /
2, meansquare angular momentum 3 ~ / , and an associated magnetic moment with g -factor of 2. The spin is thus identified as an emergent property generated bythe action of the zpf , as shown in detail in refs. [5]-[7].The approach used in sed may seem (to some) too classical to reproducethe quantum properties of matter. However, as shown e.g. in chapter 5 of ref.[2], when the sed system transits to a state in which it has acquired ergodicproperties (and also energy balance holds), the appropriate description of thedynamics becomes one in which the variables are represented by matrices andthe whole Hilbert-space formalism of quantum mechanics applies —includingthe electron spin with its two projections. In brief, a qualitative change ofstate occurs in the behaviour of the system (as a sort of phase transition), thatdemands a leap in its description from classical to quantum. The present paperis intended to shed some light on the physical mechanism leading precisely tothe two spin projections.Such abrupt changes in the description are not unknown to physics. Atraditional example is the transition (in both, classical and quantum physics) inthe description of a system of a few particles to a system with a huge number ofthem, which requires a statistical treatment. A more recent example is that ofnonlinear dissipative systems, in which the phenomenon of deterministic chaoscharacteristically takes place, contrary in principle to the traditional classicalbehaviour. In both cases new notions become indispensable in replacement ofthe older ones. 3 The effective potential
Let us consider an electron with its spin and magnetic moment µ forming anangle θ with the z -axis; the applied magnetic field is B = ˆ e z B, with B constantfor simplicity. The corresponding interaction energy is V = − µ · B . Since µ = e ~ / mc = − µ < , the energetically stable orientation of the spin is oppositeto the direction of the magnetic field. However, quantum theory considers alsothe direction along the field to correspond to a stable solution, as has beenexperimentally proven again and again.With the purpose of finding an explanation for the stability of the spinorientation in both directions, we take into account that the above system issubject to the action of the zpf of high frequency. For our present purposesit is enough to consider a mode of this field of a sufficiently high frequency γ ,uniformly distributed on the xy -plane and directed along the z -axis; we thereforewrite the magnetic component of this mode as B ( t ) = ˆ e z B cos γt. In sphericalcoordinates, the Lagrangian for the problem is L = 12 I (cid:16) ˙ θ + ˙ ϕ sin θ (cid:17) − µ ( B cos θ + B cos θ cos γt ) , (1)with I an effective moment of inertia to be determined below. The correspondingequations of motion are¨ θ − ˙ ϕ sin θ cos θ = µ BI sin θ + µ B I sin θ cos γt, (2)˙ ϕ sin θ = constant . (3)Note that the second term on the l.h.s. is a kinematic term due to the use ofspherical variables; it does not disappear in the absence of the magnetic field.Further, the variable ϕ is ignorable, and the angular velocity about the z -axis,given by ˙ ϕ , is arbitrary. Let us, for clarity, assume in a first instance thatinitially ˙ ϕ = 0 and ϕ = 0; then according to eq. (3) the magnetic momentvector remains on the xz -plane, and (2) reduces to¨ θ = µ BI sin θ + µ B I sin θ cos γt. (4)Now we follow the usual procedure [1] to determine the effect of the high-frequency term on the orientation of µ , by separating the fast terms from theslowly varying component of the motion. This procedure applies when γ ≫ ω ,where ω is a frequency parameter associated with the external field, ω ≡ r µ BI . (5)Under this condition, a first-order determination of the effects of the high-frequency vibration is sufficient, as will become clear below (see also [1], [8]).Thus we write θ = Θ + ξ, with Θ the slow (dominant) motion component and4 the fast (small, high-frequency) correction. A Taylor series expansion of eq.(4) gives to first order in ξ ¨Θ + ¨ ξ = ω (sin Θ + ξ cos Θ) + ω B B (sin Θ cos γt + ξ cos Θ cos γt ) . (6)This equation contains both the smooth and the rapidly vibrating terms, whichmust be separately equal. For the latter we get¨ ξ = ω B B sin Θ cos γt + O ( ξ ) , (7)which gives to first order ξ = − ω B γ B sin Θ cos γt. (8)Substituting this in (6) and averaging over the short period 2 π/γ, an intervalduring which Θ remains virtually the same, we obtain an equation for the slowmotion, ¨Θ = ω sin Θ − BB sin Θ cos Θcos γt t + ω B B sin Θcos γt t − sin Θ cos Θcos γt t , (9)which simplifies into ¨Θ = ω sin Θ − Ω sin Θ cos Θ , (10)with the frequency parameter Ω defined asΩ ≡ µ B √ Iγ . (11)Since in what follows we are interested in the slow motion only, describedby eq. (10), we shall go back to the original notation for the angle, i.e. , Θ → θ . Equating now the angular acceleration I ¨ θ to (minus) the derivative of aneffective potential energy V eff associated with the magnetic moment, we obtain V eff = Iω cos θ + 12 I Ω sin θ. (12)This potential is represented in fig. 1, for two values of a/ ω / Ω < As is well known from the theory of high-frequency excitations ([1]), a first-orderconsequence of these excitations is an additional (mean) force applied on the5
Figure 1: Effective potential U eff = (2 /I Ω ) V eff = a cos θ + sin θ , for a =10 − (solid line) and a = 10 − (dashed line). Current experimental values are a ≪ − . The energy gap is ∆E = 2 a .system, able to produce peculiar effects. Here the additional force derives fromthe second term in V eff (see eq. (10)), and the effect of it is the emergence,under appropriate conditions, of two positions of stable equilibrium for µ at theminima of the potential curve represented in fig. 1. This situation is closelyanalogous to the behaviour of the inverted pendulum (see e.g. [1]), [8], [9]).Indeed, a comparison of these two theories shows that eq. (10) is common toboth, with the appropriate parameters and dynamical variables in each case.To determine the stable equilibrium positions we look for the solutions ofthe equation dV eff dθ = 0 or (cid:0) − ω + Ω cos θ (cid:1) sin θ = 0 , (13)whence both θ = 0 , π are equilibrium solutions. Since d V eff dθ = I (cid:0) − ω cos θ + Ω (cid:0) cos θ − sin θ (cid:1)(cid:1) , (14)it is clear from the outset that θ = π is a stable solution for any value of theparameters. This represents the more energetically favourable equilibrium posi-tion (the only stable one in the classical case). For θ = 0 the second derivative d V eff /dθ has positive values if and only if ω < Ω . (15)When this condition is satisfied, also the solution θ = π is stable. In terms ofthe magnetic fields, (15) rewrites as 6 < µ B Iγ , (16)which means that for a very strong applied magnetic field B the stability of thissolution may be lost.To estimate the frequencies ω ± of oscillation around the equilibrium pointswe make the usual small-amplitude approximation d V eff dθ (cid:12)(cid:12)(cid:12)(cid:12) θ =0 ,π = Iω ± , (17)with the plus sign corresponding to θ = π . Combining with eq. (14) and using(5) and (11), this gives ω ± = Ω ± ω . (18)It is clear from fig. 1 that for ω ≪ Ω , the two equilibrium positions arevirtually equally stable. The difference between the two frequencies is given inthis case, according to eq. (18), by∆ ω = ω + − ω − = Ω "r (cid:16) ω Ω (cid:17) − r − (cid:16) ω Ω (cid:17) ≃ ω Ω , (19)the more energetically favourable one being slightly higher.Further, notice that the energy difference between the two stable solutionsis determined solely by the value of the applied field,∆ E = 2 µ B, (20)with independence of the values of the parameters associated with the zpf . In order to apply the above results to the specific case of the electron spin wemust assign values to the various physical parameters involved. According tothe discussion in section 2, we take ω C as the frequency of the vibrating modeof the zpf acting on the spin magnetic moment. This gives for the momentof inertia, defined as I ≃ ~ /γ for an angular momentum of the order of ~ , the(approximate) value I ≃ ~ ω C = ~ mc . (21)By writing the moment of inertia in terms of the mass and the effective radiusof the electron acquired as a result of the zitterbewegung , I ≃ mr eff , we obtainfor the latter the following value: r eff ≃ r Im ≃ ~ mc = λ C π , (22)7 .e. , the effective size of the electron turns out to be of the order of Compton’swavelength, in agreement with the standard literature (see e.g . ref. [4]) and ourprevious discussion.We need also an estimate for the magnetic field amplitude B , which isrelated to the energy density of the zpf mode of Compton’s frequency. Thisenergy density is given by the total energy of this mode, (1 / ~ ω C , divided bythe effective volume occupied by the electron, V eff = (4 / πr eff , which gives B = 32 ( 2 πλ C ) ~ ω C , (23)whence we obtain from eq. (11), with µ = | e | ~ / mc ,Ω = 38 αω C , (24)where α = e / ~ c is the fine-structure constant. With these order-of-magnitudeestimates the condition (15) for the existence of two stable equilibrium positionsreads ω < αω C , (25)or in terms of the magnitude of the applied field B , according to eq. (5), µ B < α ~ ω C = 38 αmc . (26)For a comparison with experiment it is useful to rewrite (26) as a condition onthe Larmor frequency, ω L = µ B ~ < αω C . (27)Electron paramagnetic resonance (EPR) measurements are usually carried outwith microwaves of frequencies of the order of ω L ≃ s − , whilst the Comp-ton frequency of the electron is ω C ≃ s − . Such Larmor frequenciescorrespond to magnetic field strengths of the order of 0.35 T. Magnetic fields ofextremely high intensity, of the order of 10 T, correspond to Larmor frequenciesclose to ω L ≃ s − , still eight orders of magnitude smaller than ω C . There-fore, under present experimental situations the stability condition (15) is amplysatisfied. This means that the stability of the two equilibrium positions θ = 0 , π is well guaranteed for a wide range of values of the applied magnetic field.Further, note that according to eq. (19), the relative difference between thetwo frequencies of oscillation around the points of equilibrium is of the order of∆ ω Ω ≃ ( ω Ω ) = 83 α ω L ω C , (28)which represents an insignificant deviation for usual magnetic field strengths.fig. 1 shows that the (near harmonic) potential wells are indeed very similar inwidth, and deep enough to sustain the stability of both solutions.8 Inclusion of the Larmor precession
It is possible to extend the analysis carried out above to the more general casein which the magnetic moment rotates in two dimensions. This allows us totake into account the effect of the torque exerted on the magnetic moment bythe magnetic force, τ = µ × B = d µ dt . (29)As is well known from classical electrodynamics, this torque gives rise to theLarmor precession, which is a rotation of µ about the z -axis, with angularvelocity given by ˙ ϕ = ω L = µ B/ ~ (see eq. (27)), independent of the zenithalangle θ . Since this movement of precession is orthogonal to the motion along the xz -plane described above (with θ as the only variable), and ϕ is not a rapidlyoscillating variable, the stability behaviour should be essentially the same asabove. To show that this is the case, we rewrite the complete equation (2) interms of the Larmor frequency for ˙ ϕ ,¨ θ = ω sin θ + ω L sin θ cos θ + ω B B sin θ cos γ. (30)By separating the slow motion from the fast terms in this equation, in analogywith the 1-D case, one obtains for the effective potential, using once more (11), V eff = Iω cos θ + I (cid:0) Ω − ω L (cid:1) sin θ. (31)As before, both values θ = 0 , π correspond to points of stability. From d V eff dθ = − Iω cos θ + I (cid:0) Ω − ω L (cid:1) (cid:0) cos θ − sin θ (cid:1) (32)we see that for ω (cid:18) ω L ω (cid:19) < Ω , (33)again both equilibrium positions are stable. Since, for the values of the param-eters given in section 5 (see the discussion following eq. (27)), ω L ω = µ B ~ ω C ≪ , (34)essentially the same stability condition (15) holds in the more general case thatincludes the Larmor precession. Stochastic electrodynamics shows us that an electron spin free to rotate underthe combined action of an applied field B and the high-frequency oscillatorymode B cos γt of the zpf parallel to B , has two stable equilibrium positions,parallel and antiparallel to B , just as described in quantum mechanics. Thecondition for these two spin projections, expressed in (15), is largely satisfiedfor present experimental values of the magnetic field intensity.9 eferences [1] J. J. Thomsen , Vibrations and Stability. Advanced theory, Analysis, andTools , Second ed. (Springer, 2003).[2]
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