_{1}

^{*}

Because magnetic moment is spatial in classical magnetostatics, we progress beyond the axiomatic concept of the point particle electron in physics. Orbital magnetic moment is well grounded in spherical harmonics in a central field. There, quantum numbers are integral. The half-integral spinor moment appears to be due to cylindrical motion in an external applied magnetic field; when this is zero , the spin states are degenerate. Consider lifting the degeneracy by diamagnetism in the cylindrical magnetic field: a uniquely derived electronic magnetic radius shares the identical value to the Compton wavelength.

There are many types of science. In mathematical quantum theory, the electron is a point particle with intrinsic spin given axiomatically. This representation is unphysical because magnetostatics provides for magnetic moment m described spatially via Ampere’s law so that m = Ia_{s}, where I is the current flowing around a loop with surface area a_{s}. Though mathematical conclusions are occasionally so unexpected that physical hypotheses are not credible without them; in the logic of physics, a hypothesis is meaningless if it is not falsifiable [^{1} since they are always “logically true” in mathematics. Here we develop a spatial representation for intrinsic spin. It will be described on the basis of dispersion dynamics [_{l} = 0) divided into two beams under a magnetic field gradient. The result demonstrated, not a continuous distribution of magnetic moment, but quantized intrinsic spin with m_{s} = ±1/2.

The orbital spin that is described in spherical wave mechanics by means of the magnetic quantum number m_{l}, is relatively unproblematic. It is described in the harmonic bases used with the Schrödinger equation. The bases describe atomic currents and associated magnetic moments that couple to magnetic intensity inside ferromagnetic and paramagnetic materials. Orbital moment is therefore a foil for exploring the less obvious intrinsic moment. Indeed, the two moments couple intimately in atomic structure, and they precess together in externally applied magnetic fields [

Perhaps the greatest significance for spin lies in the Pauli exclusion principle, because spin doubles the number of states allowed for indistinguishable Fermions having states of the same orbital quantum numbers. The spinor states are sometimes represented by Pauli spin matrices [

Dispersion dynamics [_{o} of a free body, f(E, p, m_{o}) = 0. In wave mechanics, this translates to f(ω, k, m_{o}, V) = 0, by substitution with angular frequency in Planck’s law; with wave vector in the de Broglie hypothesis; and with potential V < 0 for a bound particle. An immediate consequence is that the product of the group velocity and phase velocity in a free particle is equal to the square of the speed of light c^{2}. From the simple case of the free particle, second derivatives lead to a representation of Newton’s 2^{nd} law of motion and of forces in electromagnetic fields.

Meanwhile in special relativity, priority goes to the direction of propagation x; and this is represented in the stable wave packet by the two-dimensional space-time variable X(x, t) that will be described below. To these dimensions are added mass, and also electronic charge when electromagnetic interaction is involved. The transverse plane is minimally relativistic, in two further dimensions, Y(y, t) and Z(z, t). Probability functions in these coordinates are taken subject to the normal quantization constraints commonly understood in the Bohr model of the atom [

When we study spin, we need to progress beyond Newton’s second law of motion. One consequence of this law is that angular acceleration of a body dL/dt in a circulating frame is proportional to the torque τ applied, which is normal to the force F in the second law. Though in an ideal couple (of forces), the net force is zero; continued application of the torque causes a body to precess about an axis of circulation. This occurs in gravitational fields, as in the gyroscope, and also in electromagnetic fields, as in spin^{2}. From comparisons between orbital spin and intrinsic spin in the context of the stable wave packet, it will be possible to derive a physical model that is consistent in all three phenomena: orbital spin, intrinsic spin and precessing torques [

The most fundamental feature of modern physics is wave-particle duality^{3}. It is best expressed by the stable wave packet: self-evidently stable as the travelling wave group for a free particle or photon:

ϕ = A ⋅ exp ( X 2 2 σ 2 + X ) , with X = i ( k ¯ x − ω ¯ t ) (1)

as illustrated in

the complex exponential function exp(X). This describes the first part of the wave-particle duality. The other argument in the exponential function, exp(X^{2}/2σ^{2}), is real and describes the particle. Here, the denominator σ is particular because it depends on initial conditions, but it is stable during propagation in free space as a consequence of Newton’s first law of motion. The normalizing amplitude A depends on the coherence σ and, in free space, is therefore equally stable. The envelope depends on the square of X which is a function of four variables. Two are already considered, so we are left with the variables x and t that describe the profile. Since the other variables are all stable, this profile is also stable. Following Dirac’s opinion [

From equation (1) are also derived Planck’s law, E = ħω, the de Broglie hypothesis p = ħk, and several conservation rules. Solve f(ω, k, m_{0}, V) = 0, first for the free particle with rest mass m_{o} in potential V = 0. The relativistic Klein-Gordon equation, ( □ 2 − m 0 2 ) ϕ ( x ) = 0 , operating on Equation (1) yields, for free particles, an algebraic equation in second order:

ℏ 2 ω 2 = ℏ 2 k 2 c 2 + m 0 2 c 4 (2)

ħ being the reduced Planck constant and c the speed of light. This is the same equation as is obtained from Einstein’s relativistic formula, E 2 = p 2 c 2 + m 0 2 c 4 , by substituting for energy using Planck’s law and for momentum using the de Broglie hypothesis. The equation can be simplified with appropriate units c = 1 = ħ. Differentiation then gives a new result in relativity, for the product of group velocity dω/dk [

d ω d k ⋅ ω k = v g ⋅ v p = 1 (=c^{2} in generalized units) (3)

The result is plotted in the positive quadrant of _{o} = 1. The group velocity is well behaved: it tends to zero at low k and to c at large k exactly as in the special theory of relativity^{4}. The phase velocity is faster than the speed of light c and is singular when k → 0: within this rest frame, time is Newtonian within the coherence σ. This has significance in the reduction of the wave packet during a quantum transition [

Progressing to second derivatives, we can represent Newton’s second law of motion in terms of dispersion dynamics. An important application is the Hall effect which shows negative coefficients in Cu, n-type semiconductors, and high

temperature superconductors; but positive coefficients in Al, p-type semiconductors and low temperature superconductors, all consistent with dispersion dynamics [

d 2 ω d k 2 = d v g d k = d v g d p = ( 1 m ′ − k 2 m ′ 3 ) = 1 m e f f = a F (4)

where m' is the relativistic mass m o / ( 1 − v g 2 / c 2 ) 1 / 2 ; effective mass m_{eff} is as defined in the brackets; a is acceleration in Newton’s second law of motion corresponding to applied force F, such as the Lorentz force in magnetism. Notice that a negative second derivative, or curvature, causes negative effective mass and negative acceleration due to an applied Lorentz force. This is why the Hall coefficient is positive in p-type semiconductors, Al metal and high temperature superconductors, where the charge carriers in these cases can only be electrons [

Then Quantum physics becomes a consequence of wave motion, combined with spatio-temporal, self-interference constraints on bound states: as in spectral emissions and absorptions between quantized atomic terms. For example, without the quantization provided in the Bohr atomic model and by the Schrödinger equation, wave functions would destructively self-interfere. The energy is quantized in the harmonic basis vectors.

Two important facts are: the expected energy or relativistic mass, integrated in time over the packet in Equation (1), is equal to ℏ ω ¯ ; while the expected momentum, integrated over space, is equal to ℏ k ¯ . This description is a physical, non-axiomatic description of quantization. These conditions are set by initial and final states. We will see how the model applies to intrinsic spin. Notice that in the calculation of spectral lines, such as the Lyman α for the hydrogen atom, rest mass energy cancels between the ground and excited terms.

As another physical description, the Uncertainty Principle can be derived from Equation (1) by Fourier transforms [

Dirac’s calculation for the speed of the electron found it equal to c [

Two extreme regimes are commonly identified: relativistic when p ≫ m o c , and non-relativistic when p ≪ m o c .

Relativistically, at high k ≫ m o , (simplified units) both the group velocity and phase velocity tend to the speed of light: v_{g}, v_{p} → c, as in the massless photon travelling in free space. Then d ω / d k = ω / k = ν ′ λ , the product of oscillational frequency ν' with wavelength. Conductance depends on the group velocity.

At low k ≪ m o , non-relativistic values approximate:

E = m o ( 1 + p 2 c 2 / ( m o 2 ) ) 1 / 2 ≈ m o + p 2 / 2 m o (5a)

In classical mechanics, the mass energy is ignored as constant in mechanical or chemical changes, as it is in Schrödinger’s equation which is likewise non-relativistic. Moving on from the free particle, when the potential V < 0 is included, the Schrödinger eigenvalue ε corresponds with the result of the virial theorem, so that the expectation value for 〈 | V | 〉 ≈ − 2 〈 | p 2 / 2 m 0 | 〉 (

ε ≈ 〈 | ℏ 2 k 2 / 2 m o | 〉 (5b)

using the simplified units previously described. The kinetic energy for the free particle is positive; while the eigenvalue in a potential V < 0 is similar, but negative.

Whether free or bound, the group velocity is given by:

v g = E − m o 2 p = ω − m o 2 k (6)

or ε/2p in the system of Schrödinger. This is the velocity that is proportional to the Lorentz force of magnetism for a charged particle moving in a magnetic field, as in Hall effect measurements mentioned earlier.

Intrinsic spin is a physical quantity. Our new starting point is the stable wave packet of Equation (1) which is 2-dimensional in time and space. To these are added relativistic mass and electromagnetic charge. It is well known that the transverse plane is weakly relativistic, for example after application of a transverse force, the transverse component of the energy:

E y = p y 2 2 p x (7)

where momentum in the propagation direction replaces rest mass in the classical expression for kinetic energy. This fact shows that the transverse wave motion is dependent on the propagation. We assume therefore that the transverse components and propagation components of the 4-dimensional wave must be related in phase. The force transforms the propagation direction:

p ′ x → p x i ^ + p y j ^ (8)

as indicated by the Cartesian unit vectors. Components of angular momentum are illustrated in _{z} = 0, i.e. for a plane wave in free space. The spin states are degenerate. When B z e x t > 0 , presume that ω_{z} increases by induction, as in normal diamagnetism and in the Meissner-Ochsenfeld effect in superconductors. The degeneracy is then lifted and spin up states split from spin down. (The spin current does not decay; it is resistanceless. Under the diamagnetic hypothesis, spin in a Fermion is superconductive at room temperature, lacking a critical temperature.)

In the case of the Ag atoms observed in the Stern-Gerlach experiment, the origin for the magnetic quantization is due, not to the central atomic potential, but to diamagnetic Bohr orbits conceived, not at the atomic scale, but in an electron scale in the diamagnetic response to B_{z}. Without quantization the orbits would self-interfere and self-distruct, so the fact that m_{s} = ±1/2 implies that the diamagnetic Bohr orbit diameters are restricted.

In principle, a free beta ray is restricted in the same way. An experimental observation of splitting would indicate localization within the wave group because the splitting might not be uniquely quantized if it depends on the vagary in the

volume of the wave packet extended by its coherence σ. Notice however that, whereas a beta ray is less restricted than the atomic orbit in Ag, Stokes’ theorem applies in an evaluation of the moment in a travelling wave group, and that its electronic charge is normalized through the amplitude A in Equation (1). We will return to this after finding the magnetic radius.

More generally, there are further features that delineate the nature of intrinsic spin. Firstly, the spin and is independent of wavevector k, but depends on mass (ω in simplified units) when in the non-relativistic regime. Secondly, the induced currents in atomic beams imply a magnetic moment μ with electron magnetic radius r_{m}:

μ = I a s = I π r m 2 (9)

Meanwhile in the transverse plane, suppose that any angular frequencies are in phase with the frequency in the direction of propagation x, so that ω = ω_{x} = ω_{y} = ω_{z}. These angular frequencies are coherent because of intersecting overlaps on the two-dimensional planes that are normal to them. (The frequencies are independent of mass components (Equation (7)) in angular momentum.)

These frequencies determine the current, so that:

μ = e ω 2 π π r m 2 = μ B g s m j (10)

where e is the electron charge; μ_{B} = eħ/2m_{e} is the Bohr magneton with electron mass m_{e} ~ m_{o}; and g_{s} ≈ 2 is the gyromagnetic ratio which makes the moment of the spinor approximately integral in μ_{B} when the spinor is half integral. The magnetic electron radius is defined when:

μ μ B = 1 = e ω 2 π ⋅ π r m 2 ⋅ 2 ω e c 2 (11)

by substituting ħω = m_{e}c^{2}, while I = eω/2π. Then the magnetic electron radius,

r m = c / ω = 3.86159323 ( 35 ) × 10 − 13 (12)

A value that is identical to the electron Compton wavelength λ_{e} [_{o} where α is the fine structure constant. The value is also α^{−1} times the classical electron radius r_{e} that is calculated by supposing the mass of the electron is all electrostatic energy. The magnetic radius provides a limiting size for magnetic moment due to intrinsic spin that owes to a cylindrical applied field intensity.

The derivation applies to excited states of Ag where the angular momentum l > 0, as it does to atoms having multiple electrons in Russell-Saunders, L-S, spin-orbit coupling,; or in J-J coupling etc. Returning to Equation (12), the magnetic radius is a limiting value when the current is given by eω/2π. A smaller current that is spread over a larger radius might yield the same moment consistent with Stokes’ theorem. However, such variation in radius would break the requirement for quantization in both wave phenomena and in magnetization: the wave function, where it repeats, must not self-destruct; and concomitantly, the magnetization is itself quantized as demonstrated by Stern and Gerlach. It therefore appears that the magnetic radius is fixed by quantization and describes the quantized size of the electron, i.e. less than two orders of magnitude shorter than the Bohr radius for the atom, λ_{e} = αa_{o}. Atomic spin-orbit coupling then becomes the interaction of a localized electron with an atomic wave function. The explanation represents logically-pure, quantum physics.

An extension of the method (Equations (10)-(12)) shows, within an order of magnitude, that:

μ μ B ~ v p r m v g a 0 (13)

the numerator being due to phase harmonics in the cylindrical symmetry of B about the localized electron; the denominator to real charge current about the spherical atomic field (Equation (9)). The former is on particle scales; the latter on atomic scales. These scales physicalize the mathematical short-cut implicit in its idea of a point particle.

Notice that, in terms of wave-particle duality, ω is a wave property dependent on energy and equal to relativistic mass (using simplified units); k is a particle property dependent on momentum, tending to zero in the rest frame where it is opposed by vibrational uncertainty. Quantized energy depends on k in the circumference for Bohr model for the atom, or in the time independent Schrödinger equation; Equation (1) describes a probability amplitude for finding an electron with finite size and magnetic radius c/w.

The derivation described here for intrinsic spin takes evidence from dominant data obtained from atomic and chemical physics, but it extends to isospin in atomic nuclei and magnetic moments in elementary particles. The same transverse motion exists there with similar outcomes on different scales. Likewise, uncharged mesons and hadrons contain charged quarks and have magnetic moments [^{k} + e^{−k}), do the same.

Intrinsic spin is often described by a supposedly physical axiom in a mathematical construct of possible worlds. Orbital magnetic moment from wave functions in a central field is comparatively unproblematic. By contrast, intrinsic spin states are degenerate in zero field, so that lifting of degeneracy by a cylindrical magnetic field implies diamagnetism with a magnetic electronic radius. We use the stable wave packet and dispersion dynamics to derive, independently, a value that turns out identical to the Compton wavelength.

The author declares no conflicts of interest regarding the publication of this paper.

Bourdillon, A.J. (2018) Dispersion Dynamical Magnetic Radius in Intrinsic Spin Equals the Compton Wavelength. Journal of Modern Physics, 9, 2295-2307. https://doi.org/10.4236/jmp.2018.913145

B z e x t z-component of magnetic field

m magnetic moment

I current flowing around a loop

a_{s} surface area vector normal

l azimuthal quantum number

m_{l} orbital magnetic quantum number

m_{s}_{ }spin magnetic quantum number

E energy

p momentum

m_{o} rest mass

ω angular frequency, ω ¯ mean

k wave vector, k ¯ mean

V potential

c speed of light

X space-time variable (propagation direction)

x coordinate in direction of propagation

t time coordinate

Y,y,Z,z corresponding transverse coordinates

dL/dt angular acceleration

τ torque

F force

f wave function

σ coherence

ħ reduced Planck constant

v_{g} group velocity = dω/dk

v_{p} phase velocity = ω/k

m_{eff} effective mass

a acceleration

m’ relativistic mass

v’ oscillational frequency = ω/2π

ε eigenvalue

p_{x},p_{y},p_{x} components of momentum

ω_{x},ω,ω_{z}, components of angular frequency

r_{m} electron magnetic radius

μ magnetic moment

e electronic charge

μ_{B} Bohr magneton

g_{s} gyromagnetic ratio

λ_{e} Compton wavelength

a_{o} Bohr radius

α fine structure constant

L total orbital quantum number

S total spin quantum number

J total magnetic quantum number