Self-adjoint extensions to the Dirac Coulomb Hamiltonian

Abstract

The Dirac equation is the relativistic generalization of the Schrödinger equation for spin 1/2 particles. It is written in the form -ihc -ihac OXIa+t' Omc 29 = ih-o (1.1) where V) is a four component Dirac spinor and the coefficients a and # are 4 x 4 matrices. Like the Schrödinger equation, the Dirac equation can be written as a time-independent eigenvalue equation H# = E* for a Hamiltonian operator H and energy eigenvalue E through separation of variables. The energy eigenvalues obtained by solving this equation must be real- one of the axioms of quantum mechanics is that physical observables, in this case energy, correspond to self-adjoint operators, in this case the Hamiltonian operator HI, acting on the Hilbert space 7H which describes the system in question. It can easily be shown that self-adjoint operators must have real eigenvalues. The reality of the energy eigenvalues becomes important when examining hydrogenic atoms using the Dirac equation. These atoms can be described by a Coulomb potential, V(r) = -Ze 2 /r, where Z is the number of protons in the nucleus and e is the elementary charge. When the nonrelativistic Schrodinger equation is solved for a Coulomb potential, the energy levels are given by the familiar Rydberg formula Z 2a 2mc2 1 En 2 2 (1.2) where Z is the number of protons in the atomic nucleus, a is the fine structure constant, m is the electron mass, c is the speed of light, and n a positive integer. Note that this formula assumes a stationary positive charge of infinite mass at the center of the atom, and that the energy levels for a more realistic model of an atom with a nucleus of finite mass M are given by replacing m with the reduced mass = mM/(m + M) in Eq. (1.2). When the Dirac equation in a Coulomb potential is used instead of the nonrelativistic Schrödinger equation, the energy levels are instead given by - 1/2 En, = mc2 1+ a2 (1.3) n' - j j + )2_ - 2Z2 where n' is a positive integer and j is the total angular momentum of the electron. The total angular momentum j can take on values in the range 1/2, 3/2, ..., n' - 1/2. The eigenvalues in Eq. (1.3) match those in Eq. (1.2) in the limit VZ < 1, noting that in Eq. (1.2), a free electron is considered to have an energy of 0, while in Eq. (1.3), a free electron has energy mc2 . A problem arises with Eq. (1.3) when aZ > j--. The quantity (j + 2- aZ 2 is imaginary, causing Eq. (1.3) to yield complex energy eigenvalues. Since the eigenvalues of a self-adjoint operator must all be real, this indicates that the Hamiltonian cannot be self-adjoint when aZ > j + 1. This issue raises two questions. The first is whether there is a physical explanation for the failure of Eq. (1.2) for large Z. The second is whether this problem can be addressed mathematically by defining a new, self-adjoint operator H., which is constructed from the old Hamiltonian H as a self-adjoint extension. In this thesis, I will answer both of these questions in the affirmative, relying and building upon work done by others on these questions. I will show how the failure of Eq. (1.2) can be motivated by physical considerations, and I will examine a family of self-adjoint extensions to the Dirac Coulomb Hamiltonian constructed using von Neumann's method of deficiency indices.