Dirac Theory

The theory of Paul Dirac represents an attempt to unify the theories of quantum mechanics and special relativity. That is, one seeks a formulation of quantum mechanics which is Lorentz invariant, and hence consistent with special relativity. For a free particle, relativity states that the energy is given by . Associating E with a Hamiltonian in quantum mechanics, one has

If H and p are associated with the same operators as in Schrödinger theory, then one expects the wave equation

This is known as the Klein-Gordan Equation. Unfortunately, attempts to utilize this equation are not successful, since that which one would wish to interpret as a probability distribution turns out to be not positive definite. To alleviate this problem, the square root may be taken to get

However, this creates a new problem. What is meant by the square root of an operator? The approach is to guess the form of the answer, and the correct guess turns out to be

With this form of the Hamiltonian, the wave equation can be written

 

In order for this to be valid, one hopes that when it is squared the Klein-Gordan equation is recovered. For this to be true, equation 63 must be interpreted as a matrix equation, where and are at least matrices and the wavefunction is a four-component column matrix.

It turns out that equation 63 describes only a particle with spin 1/2. This is fine for application to the hydrogen atom, since the electron has spin 1/2, but why should it be so? The answer is that the linearization of the Klein-Gordan equation is not unique. The particular linearization used here is the simplest one, and happens to describe a particle of spin 1/2, but other more complicated Hamiltonians may be constructed to describe particles of spin 0,1,5/2 and so on. The fact that the relativistic Dirac theory automatically includes the effects of spin leads to an interesting conclusion--spin is a relativistic effect. It can be added by hand to the non-relativistic Schödinger theory with satisfactory results, but spin is a natural consequence of treating quantum mechanics in a completely relativistic fashion.

Including the potential now in the Hamiltonian, equation 63 becomes

When the square root was taken to linearize the Klein-Gordan equation, both a positive and a negative energy solution was introduced. One can write the wavefunction

where represents the two components of associated with the positive energy solution and represents the components associated with the negative energy solution. The physical interpretation is that is the particle solution, and represents an anti-particle. Anti-particles are thus predicted by Dirac threory, and the discovery of anti-particles obviously represents a huge triumph for the theory. In hydrogen, however, the contribution of is small compared to . With enough effort, the equations for and can be decoupled to whatever order is desired. When this is done, the Hamiltonian to order can be written

where is the original Schrödinger Hamiltonian, is the relativistic correction to the kinetic energy, is the spin-orbit term, and is the previously mentioned Darwin term. The physical origin of the Darwin term is a phenomenon in Dirac theory called zitterbewegung, whereby the electron does not move smoothly but instead undergoes extremely rapid small-scale fluctuations, causing the electron to see a smeared-out Coulomb potential of the nucleus.

The Darwin term may be written

For the hydrogenic-atom potential , this is

When first-order perturbation theory is applied, the energy correction depends on . This term will only contribute for s states (l=0), since only these wavefunctions have non-zero probability for finding the electron at the origin. The energy correction for l=0 can be calculated to be

Including this term, the fine-structure splitting given by equation 58 can be reproduced for all l. All the effects that go into fine structure are thus a natural concequence of the Dirac theory.

The hydrogen atom can be solved exactly in Dirac theory, where the states found are simultaneous eigenstates of H, , and , since these operators can be shown to mutually commute. The exact energy levels in Dirac theory are

This can be expanded in powers of , yielding

This includes an amount due to the relativistic energy associated with the rest mass of the electron, along with the principle energy levels and fine structure, in exact agreement to order with what was previously calculated. However, even this exact solution in Dirac theory is not a complete description of the hydrogen atom, and so the the next section describes further effects not yet discussed.