Dirac’s Equation

1Before we attempt to follow a general outline of Dirac’s mathematical logic, which leads to the somewhat abstract-looking equation embedded in the diagram, it might be worth recapping why Dirac was motivated to look for a solution in the form of the equation that bears his name. Leading up to this period, the Bohr model of the atom had been established in 1913, but had been subject to some considerable modification in order to try to account for a number of additional observations associated with the fine structure spectra of hydrogen. Bohr's initial model had suggested that that there could only be one electron per orbital, although subsequent calculations by Pauli had indicated that there could be two. An initial resolution to this conflict had been proposed by Goudsmit and Uhlenbeck, in 1925, in the form of an electron spinning as it moved around its orbit, although we might now reflect on the physical nature of the ‘spin’ and ‘orbit’ associated with the quantum model. However, there were also problems with the implied speed of rotation, within this model, in order that the electron produce the required magnetic field. One calculation suggested that the rotational speed would have to be 137 times the speed of light, but at this speed the electron would have immediately torn the atom apart. There was also a major discrepancy in what is called the ‘geomagnetic ratio [g]’ that defines the ratio of a charged particle's magnetism to its angular velocity of rotation. Classical calculations suggested a value of [g=1], while Goudsmit and Uhlenbeck postulated [g=2]. By 1928, this situation had been compounded in terms of there now appearing to be two seemingly different formulations underpinning the concept of quantum theory, i.e. Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics. As previously outlined, the derivation of the time-dependent Schrodinger wave equation is predicated on a non-relativistic assumption, which is linear in time, but not in space. In order to obtain a relativistic wave equation, further consideration of the relativistic relationship between energy, momentum and mass of a particle is necessary based on the following relationship:

[1a]    1

At first glance, the form of [1a] seems to bear no relationship to the form of the Dirac equation embedded in the diagram above. However, it is possible to show that [1a] is still at the heart of the Dirac equation by dissembling some of the mathematical abstraction as follows:

[1b]    2

While we might now see some basic similarity in the form of the final expression in [1b] with [1a], there is clearly some discrepancy between the two, because we appear to have lost the square on each component, as shown in [1a]. However, this aspect of [1a] is still contained in [1b], but is now in the form of a series of matrix elements [γ], which will be explained in more detail as we proceed. However, this outline is only intended to represents some of the historical backdrop against which Paul Dirac was working, when he started to develop his relativistic quantum wave equation. As it turned out, not only did this equation address some of the basic requirements of relativity, it also provided a description of elementary ½-spin particles, such as electrons, that was consistent with both the principles of quantum mechanics and the theory of special relativity. While it was the first theory to account for relativity, within the context of quantum mechanics, it also helped explain some of the fine structural details of the hydrogen spectrum in a more rigorous way. However, what was more surprising, in the context of the timeline itself, was the implication that postulated the existence of a new form of matter, i.e. antimatter, which had previously been both unsuspected and unobserved. Unfortunately, the full telling of this story involves a lot of abstracted mathematics, which is way beyond the scope of this website. This said, because Dirac's equation is possibly central to much of the subsequent development of quantum theory, some attempt will be made, in the following sub-pages, to reconcile the mathematics to some form of physical description.