“You too can make the golden cut, relating the two poles of your being in perfect golden proportion, thus enabling the lower to resonate in tune with the higher, and the inner with the outer. In doing so, you will bring yourself to a point of total integration of all the separate parts of your being, and at the same time, you will bring yourself into resonance with the entire universe....
Nonetheless the universe is divided on exactly these principles as proven by literally thousands of points of circumstantial evidence, including the size, orbital distances, orbital frequencies and other characteristics of planets in our solar system, many characteristics of the sub-atomic dimension such as the fine structure constant, the forms of many plants and the golden mean proportions of the human body, to mention just a few well known examples. However the circumstantial evidence is not that on which we rely, for we have the proof in front of us in the pure mathematical principles of the golden mean.”
― Alison Charlotte Primrose, The Lamb Slain With A Golden Cut: Spiritual Enlightenment and the Golden Mean Revelation
In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom by Albert A. Michelson and Edward W. Morley in 1887 laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.
The gross structure of line spectra is the line spectra predicted by the quantum mechanics of non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy levels only depend on the principal quantum numbern. However, a more accurate model takes into account relativistic and spin effects, which break the degeneracy of the energy levels and split the spectral lines. The scale of the fine structure splitting relative to the gross structure energies is on the order of (Zα)2, where Z is the atomic number and α is the fine-structure constant, a dimensionless number equal to approximately .
The fine structure energy corrections can be obtained by using perturbation theory. To perform this calculation one must add the three corrective terms to the Hamiltonian: the leading order relativistic correction to the kinetic energy, the correction due to the spin-orbit coupling, and the Darwin term coming from the quantum fluctuating motion or zitterbewegung of the electron.
These corrections can also be obtained from the non-relativistic limit of the Dirac equation, since Dirac's theory naturally incorporates relativity and spin interactions.
Kinetic energy relativistic correction
Classically, the kinetic energy term of the Hamiltonian is
where is the momentum and is the mass of the electron.
However, when considering a more accurate theory of nature via special relativity, we must use a relativistic form of the kinetic energy,
where the first term is the total relativistic energy, and the second term is the rest energy of the electron ( is the speed of light). Expanding this in a Taylor series ( specifically a binomial series ), we find
Then, the first order correction to the Hamiltonian is
Using this as a perturbation, we can calculate the first order energy corrections due to relativistic effects.
where is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see
We can use this result to further calculate the relativistic correction:
For the hydrogen atom, , , and where is the Bohr Radius, is the principal quantum number and is the azimuthal quantum number. Therefore, the first order relativistic correction for the hydrogen atom is
where we have used:
On final calculation, the order of magnitude for the relativistic correction to the ground state is .
Main article: Spin-orbit interaction
For a hydrogen-like atom with protons, orbital momentum and electron spin , the spin-orbit term is given by:
is the electron mass, is the vacuum permittivity and is the spin g-factor. is the distance of the electron from the nucleus.
The spin-orbit correction can be understood by shifting from the standard frame of reference (where the electron orbits the nucleus) into one where the electron is stationary and the nucleus instead orbits it. In this case the orbiting nucleus functions as an effective current loop, which in turn will generate a magnetic field. However, the electron itself has a magnetic moment due to its intrinsic angular momentum. The two magnetic vectors, and couple together so that there is a certain energy cost depending on their relative orientation. This gives rise to the energy correction of the form
Notice that there is a factor of 2, called the Thomas precession, which comes from the relativistic calculation that changes back to the electron's frame from the nucleus frame.
the expectation value for the Hamiltonian is:
Thus the order of magnitude for the spin-orbital coupling is .
There is one last term in the non-relativistic expansion of the Dirac equation. It is referred to as the Darwin term, as it was first derived by Charles Galton Darwin, and is given by: