In quantum physics, the spinorbit interaction (also called spinorbit effect or spinorbit coupling) is any interaction of a particle's spin with its motion. The first and best known example of this is that spinorbit interaction causes shifts in an electron's atomic energy levels (detectable as a splitting of spectral lines), due to electromagnetic interaction between the electron's spin and the nucleus's electric field, through which it moves. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spinorbit effects for electrons in semiconductors and other materials are explored and put to useful work.
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In this section, we will present a relatively simple and quantitative description of the spinorbit interaction for an electron bound to an atom, using some semiclassical electrodynamics and nonrelativistic quantum mechanics, up to first order in perturbation theory. This gives results that agree well, but not perfectly, with observations. A more rigorous derivation of the same result would start with the Dirac equation, and achieving a more precise result would involve calculating small corrections from quantum electrodynamics.
The energy of a magnetic moment in a magnetic field is given by:
where μ is the magnetic moment of the particle and B is the magnetic field it experiences.
We shall deal with the magnetic field first. Although in the rest frame of the nucleus, there is no magnetic field, there is one in the rest frame of the electron. Ignoring for now that this frame is not inertial, we end up with the equation
where v is the velocity of the electron and E the electric field it travels through. Now we know that E is radial so we can rewrite . Also we know that the momentum of the electron . Substituting this in and changing the order of the cross product gives:
Next, we express the electric field as the gradient of the electric potential . Here we make the central field approximation, that is, that the electrostatic potential is spherically symmetric, so is only a function of radius. This approximation is exact for hydrogen, and indeed hydrogenlike systems. Now we can say
where U = Ve is the potential energy of the electron in the central field, and e is the elementary charge. Now we remember from classical mechanics that the angular momentum of a particle . Putting it all together we get
It is important to note at this point that B is a positive number multiplied by L, meaning that the magnetic field is parallel to the orbital angular momentum of the particle.
The magnetic moment of the electron is
where the spin angular momentum vector, μ_{B} is the Bohr magneton and is the electron spin gfactor. Here, is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum.
The interaction energy is
Let's substitute in the derived expressions.
We have not, thus far, taken into account the special relativistic correction for the electron's curved trajectory; this effect is called Thomas precession and introduces a factor of . So
[Can someone confirm: We are already taking gfactor =2 and later explaining Thomas Precession of correction factor of 1/2, but are they not the same? Edit: They are not the same. g=2.0023. See Thomas Precession and Lande' gfactor]
Thanks to all the above approximations, we can now evaluate the detailed energy shift in this model. In particular, we wish to find a basis that diagonalizes both H_{0} (the nonperturbed Hamiltonian) and ΔH. To find out what basis this is, we first define the total angular momentum operator
Taking the dot product of this with itself, we get
(since L and S commute), and therefore
It can be shown that the five operators H_{0}, J², L², S², and J_{z} all commute with each other and with ΔH. Therefore, the basis we were looking for is the simultaneous eigenbasis of these five operators (i.e., the basis where all five are diagonal). Elements of this basis have the five quantum numbers: n (the "principal quantum number") j (the "total angular momentum quantum number"), l (the "orbital angular momentum quantum number"), s (the "spin quantum number"), and j_{z} (the "zcomponent of total angular momentum").
To evaluate the energies, we note that
for hydrogenic wavefunctions (here is the Bohr radius divided by the nuclear charge Z); and
We can now say
where
For hydrogen, we can write the explicit result
For any hydrogenlike atom with Z protons
