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…'. 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 1 2 particles, with g-factor 2 (see also Greiner [10]). We shall again construct a general eigenstate of with eigenvalue ℏ 2 ⁄ : it is a four- are, as before, the Pauli spin matrices defined in (2.2).
Note that the , in (3.1) have dimension differing by a speed; we could make them the same by 2) The non-relativistic limit is obtained from (3.2) by setting ( , ) = − 2 /ℏ ( , ). Then 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 (

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 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 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 ( , , ),

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 = 1 2 ( . ) 2 to show that the magnetic moment follows (correct to lowest order), just as in the Lévy-Leblond spinor formulation. There is thus an alternative formulation to the usual 'spin degree of freedom', and the total wavefunction being a product of space and spin parts, as is done in nonrelativistic quantum theory. Nevertheless, the nonrelativistic decoupling of space and spin is usually simpler, as is illustrated by the spinor version of the Hydrogen atom, Appendix A. appreciated.

Appendix A. The hydrogen atom in spinor form
The equations (3.1) become, with now an energy eigenvalue, no longer a time derivative, Considering the non-degenerate ground state, with eigenvalue ℏ 2 , 1 and 2 must satisfy the same equation. This is not possible if we choose 2 to have azimuthal dependence , as in Section 3, unless we also take 2 to be zero. The ground state spinor now consists of 1 , the hydrogenic ground state 1 , and 2 = 0, 3~1 , 4~1 . Because the Lévy-Leblond probability density is defined in terms of the first two spinor components 1 , 2 , and the probability density current can be expressed in terms of 1 , 2 , the hydrogenic ground state is, at least in the probability density and the probability density current, equivalent to the scalar ground state. The azimuthal dependence is hidden in the fourth spinor component.
For the first excited states we have a choice of 2 and 2 . The former is set up as above, the latter with 1 = 0, and 2 with ± dependence. Lévy-Leblond [15] and Mita [16] discuss the electron probability current of the 'stationary' states.

Appendix B. Probability density and flux
In the Dirac case (Section 2), Ψ + Ψ is the probability density, and = Ψ + Ψ, with is defined in 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.