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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 tex2html_wrap_inline1610 . Associating E with a Hamiltonian in quantum mechanics, one has

equation439

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

equation442

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

equation447

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

equation450

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

  equation454

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 tex2html_wrap_inline1618 and tex2html_wrap_inline1620 are at least tex2html_wrap_inline1622 matrices and the wavefunction tex2html_wrap_inline1624 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

equation466

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

equation474

where tex2html_wrap_inline1634 represents the two components of tex2html_wrap_inline1624 associated with the positive energy solution and tex2html_wrap_inline1638 represents the components associated with the negative energy solution. The physical interpretation is that tex2html_wrap_inline1634 is the particle solution, and tex2html_wrap_inline1638 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 tex2html_wrap_inline1638 is small compared to tex2html_wrap_inline1634 . With enough effort, the equations for tex2html_wrap_inline1634 and tex2html_wrap_inline1638 can be decoupled to whatever order is desired. When this is donegif, the Hamiltonian to order tex2html_wrap_inline1652 can be written

equation490

where tex2html_wrap_inline1654 is the original Schrödinger Hamiltonian, tex2html_wrap_inline1656 is the relativistic correction to the kinetic energy, tex2html_wrap_inline1658 is the spin-orbit term, and tex2html_wrap_inline1660 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

equation500

For the hydrogenic-atom potential tex2html_wrap_inline1662 , this is

equation505

When first-order perturbation theory is applied, the energy correction depends on tex2html_wrap_inline1664 . 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

equation511

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, tex2html_wrap_inline1490 , and tex2html_wrap_inline1492 , since these operators can be shown to mutually commute. The exact energy levels in Dirac theory are

equation517

This can be expanded in powers of tex2html_wrap_inline1680 , yielding

equation526

This includes an amount tex2html_wrap_inline1682 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 tex2html_wrap_inline1578 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.


next up previous
Next: Smaller Effects Up: Everything You Always Wanted Previous: Kinetic Energy Correction

Randy Telfer
Mon Oct 26 15:21:07 EST 1998