ANSWER & QUESTION

Answer & Question

Answer & Question

Question 1

This answer will present a relatively simple and quantitative description of the spin-orbit interaction for an electron bound to an atom, up to first order in perturbation theory, using some semiclassical electrodynamics and non-relativistic quantum mechanics. This gives results that agree reasonably well 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 (Levine, 2008).

Energy of a magnetic moment

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.

Magnetic field

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, in SI units 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 hydrogen-like systems. Now we can say

where  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 spin factor affects primarily the degeneracy of the energy levels associated with the hydrogen and helium atoms (Levine, 2008). To a good approximation, the spin factors do not affect the energy levels of such atoms. The hydrogen atom, however, has three electrons. An anti-symmetric spin wave function of three electrons could in principle be written as the Slater determinant,

(1)

Such Slater determinant, however, is equal to zero because two of the columns are equal to each other. This fact rules out the possibility of having a zero order wave function that is the Fock product of three hydrogen like functions:

(2)

Only if the construction of an anti-symmetric spin wave function was possible, we could proceed in analogy to the Helium atom and compute the perturbation due to repulsive coupling terms as follows,

(3)

Where  is the product of hydrogen like functions of Eq above.

Having ruled out such possibility, we construct the zeroth order ground-state wave function for lithium in terms of a determinant similar to Eq. 3, but where each element is a spin-orbital (i.e., a product of a one electron spatial orbital and one-electron spin function),

(4)

Where the third column includes the spatial orbital, instead of the orbital, because the Pauli exclusion principle rules out the possibility of having two electrons in the same ...