The spinning electron

The notion introduced by Ohanian that spin is a wave property is implemented, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor components, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equations is illustrated.


Introduction
In his article "What is spin" [1], Ohanian argues that 'spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron', and that 'the spin of the electron has a close classical analog: It is an angular momentum of exactly the same kind as carried by the wave field of a circularly polarized electromagnetic wave.' Ohanian credits Belifante [2] for establishing that 'this picture of spin is valid not only for electrons but also for photons, vector mesons, and gravitons.'Dirac [3,4] regarded his four-by four matrices as 'new dynamical variables…describing some internal motion of the electron, which for most purposes may be taken to be the spin of the electron postulated in previous theories' [4].This is how the concept of spin is presented in most texts, as intrinsically relativistic, a mysterious internal angular momentum for which there is no classical analogue.For example, in his "Introduction to quantum mechanics" [5] Griffiths states '…the electron also carries another form of angular momentum, which has nothing to do with motion in space (and which is not, therefore, described by any function of the position variables , , ) but which is somewhat analogous to classical spin…'.
We shall construct, for a general relativistic or non-relativistic wavepacket, an eigenstate of the component of total angular momentum in the net direction of propagation, with eigenvalue ℏ 2 ⁄ .Such eigenstates are four-component spinors, of which two components have   azimuthal dependence.In these formulations the phenomenon of spin is incorporated into ordinary space-time: the twist is in the azimuthal dependence of two of the wavefunctions.To the question: what does a spinning electron look like?we answer, in brief, that spin in the spinor formulation, relativistic or nonrelativistic, resides in the azimuthal dependence of two of the spinor components.This contrasts with the usual spin-space formulation, and the decoupling of spin from space-time.
In Sections 2 we construct general relativistic wavepackets with spin half; these are four-component spinors.An important aspect of spin is that it is not purely a relativistic effect: Levy-Léblond [6] has proved that the Galileo group has irreducible representations with non-zero spin.A Reviewer has pointed out that Galindo and del Rio [7] show that Galilean fermions are possible, with a four-component spinor linearization of the nonrelativistic wave equation and a correct (to lowest order) g-factor.The Galindo and del Rio paper anticipates some of the work of Lévy-Leblond [6] and Gould [14].
Levy-Léblond's four-component nonrelativistic spinors are implemented in Section 3, to construct general angular momentum eigenstates with spin half.An explicit example of a non-relativistic spinning wavepacket is illustrated in Section 4.

Dirac spinors
The wavefunction Ψ(, ) of an electron wavepacket in free space is to satisfy the Dirac equation The 4 × 4 matrices ,  are written in terms of the Pauli spin matrices   ,   ,   and the unit 2 × 2 matrix  as The wave equation (2.1) thus consists of four coupled first-order partial differential equations.

Non-relativistic spinors
Lévy-Leblond [6] has shown that four coupled non-relativistic wave equations, equivalent to the Schrödinger equation, are spinors representing spin ), and consider wavepacket motion, predominantly along the  direction, but of course converging onto or diverging from the focal region, which we shall centre at the space-time origin.
Again in cylindrical polar coordinates , , and with use of (2.3), the four time-dependent freespace equations (3.1) for the spinor Ψ read When the spinor components   are independent of , solutions exist only for the   also independent of .These are the plane wave solutions   =    (−) , where the wavenumber  and the energy ℏ are constrained by ℏ = ℏ 2  2 /2.To attain localized wavepacket solutions, we need to consider azimuthal dependence.
The angular momentum operator  =  ×  does not commute with the free-particle Hamiltonian  = We shall now construct the non-relativistic spinor eigenstates of   .
Let the spinor components   have azimuthal dependence     ; the   eigenstate equations for  1 ,  2 are the same as in (2.6): The equations (3.5) and (3.6) have the same form as in the relativistic case, equations (2.5) and (2.6).
Hence as before the choice  1,3 = 0,  , with  = 0, 1 respectively, and  2 +  2 =  2 ;   are the regular Bessel functions of order .Hence spinor components of forward-propagating wavepackets have the form The amplitudes   (, ) are complex functions, subject only to the existence of the norm and of the expectation values of energy and momentum of the wave packet.A similar expression gives the wavefunctions of scalar and of electromagnetic pulses [8].
To sum up this Section: a general non-relativistic eigenstate of   with eigenvalue ℏ 2 ⁄ has been found: it is a four-component spinor, of which two components have 'twist', with   azimuthal dependence.In this formulation the spin resides in the azimuthal dependence of two of the wavefunctions, in real space-time.
Any spinor based on localized wavepacket solutions of the time-dependent Schrödinger equation, constructed as above, will be an eigenstate of   with eigenvalue ℏ 2 ⁄ .The next Section gives an explicit example.Stationary states (energy eigenstates) of the hydrogen atom are briefly discussed in Appendix A.

Spinning Gaussian wavepackets
A free-particle wavepacket solution of Schrödinger's time-dependent equation dates back to the early days of quantum mechanics (Kennard [11], Darwin [12]).This is the Gaussian wavepacket.
It is a compact exact solution, but with a physical flaw, to be discussed below.For propagation along the  axis, and with cylindrical symmetry, it has the form The Gaussian wavepacket (4.1) is normalized so that  *  = 1 at the space-time origin.In (4.1) the spatial origin  = 0,  = 0 is the position of maximal || a time  = 0,  is the dominant  component wavenumber,  is the mass of the particle,  = ℏ  ⁄ is the group speed, and  = ℏ 2 ⁄ is the spreading speed.The length  gives the spread of the wavepacket at  = 0. Earlier and later the longitudinal and lateral spread of the packet is greater, proportional to . Thus  = 0,  = 0 can be thought of as the centre of the focal region of the wavepacket, occupied at  = 0.As  increases towards zero the wavepacket converges to its most compact form, reaches it at  = 0, and then expands as it continues to propagate in the positive  direction.The packet used by Ohanian [1] is equivalent to (4.1) evaluated at  = 0 (zero momentum expectation value) and  = 0.
For the Gaussian wavepacket  the momentum operator has the expectation values (see for example [13]) The wavepacket  is neither an energy nor a momentum eigenstate, but it is an eigenstate of the orbital angular momentum operator   =   −   = −ℏ(  −   ) = −ℏ  .The orbital angular momentum eigenvalue is zero, because  is independent of the azimuthal angle .
Eigenstates of the  component of orbital angular momentum, with eigenvalues which are integer multiples of ℏ, may be generated from any such  by differentiation, as shown in [13].
The probability density of the scalar wavepacket is  * : the probability that the particle described by g(, ) is within the volume element  3  is  3   * .The norm  = ∫  3   *  (integration over all of space) is independent of time.The probability density flux, or the probability current density vector , satisfies the conservation law What are the corresponding relations for spinors?The conservation law is now (Lévy-Leblond [6], Section IIIe, and Appendix B) The first term in the second expression for  corresponds to the Schrödinger current in (4.3), the second is a spin current.Ohanian [1] derived the relativistic analogue of last term in (4.5).He showed that it leads, in the nonrelativistic limit, to an azimuthal current.In his words, "such a circulating flow of energy will give rise to an angular momentum.This angular momentum is the spin of the electron." We shall calculate the radial, azimuthal, and longitudinal components of the probability current density,   ,   ,   in the simplest case, in which the spinor components are  1 =  1 (, , ),  2 = 0,  A problem with the Gaussian solution is apparent in   : for positive  and negative  (or vice versa) the longitudinal component is negative if the magnitude of  exceeds that of 2 3 .The probability current then flows backward.Far from the focal region (here centred on the space-time origin) there should be no backward flow for free-space propagation.Note that the Gaussian wavepacket cannot be put in the purely forward-propagating form (3.10).
Nevertheless, the Gaussian packets demonstrate the azimuthal current component which arises in the spinor formulation.Figures 1 and 2 show the current components in the focal plane, and at a transverse plane cutting through the wavepacket center at a later time.The azimuthal part   gives the electron wavepacket its spin.

Summary
The spinning electron may be described by a four-component spinor, depending on space and time coordinates, in both relativistic and non-relativistic quantum theory.The non-relativistic quantum theory and its azimuthal dependence is similar to the relativistic Dirac spinor formulation of Section 2. In both cases the spin is contained in the azimuthal dependence of wavefunctions in ordinary space-time.Gould [14] used the Hamiltonian  = The first term in this expression for  corresponds to the Schrödinger current in (3.3), the second is a spin current, which gives the correct  factor at leading order [6].The spin term is the curl of a vector, and so does not contribute to the conservation law (B.1).See also Landau and Lifshitz [17] Section 114, and Mita [16] for the spin current term.
We shall calculate the radial, azimuthal, and longitudinal components of the probability Let  1 ,  2 be solutions of (3.8) and (3.9), respectively, and  1 =

Figure 1 .
Figure 1.Focal plane section through a Gaussian spinor wavepacket, at  = 0.The contours give the probability density, the arrows the transverse current density (the longitudinal current is not shown).The direction of motion is out of the page.The transverse current density is purely azimuthal at this instant.

Figure 2 .
Figure 2. Gaussian spinor wavepacket, at  =  = 2.The transverse current density now has radial and azimuthal components.The group speed is , so the section is through the centre of the wavepacket.The longitudinal current density is not shown.
[10]articles, with g-factor 2 (see also Greiner[10]).We shall again construct a general eigenstate of   with eigenvalue ℏ 2 ⁄ : it is a fourcomponent spinor.It is based on localized wavepacket solutions of the time-dependent Schrödinger equation, with no restriction on the wavepacket parameters.In Section 4 we shall explore some properties of exact Gaussian solutions of the equations satisfied by the spinor components.
3 ~  1 ,  4 ~    1 .From Appendix B, the components of the probability current The components   ,   are the same for the scalar wavepacket, the azimuthal component   is zero in the scalar case based on .The conservation law (4.4) is satisfied.