**5.1. Low Lying States for Particles in Uniform Magnetic Fields**

Following Refs.
[216,
217] in this section
we will
consider questions related to the mass shifts and decay of
bound states of quarks. Mass shifts occur both due to the effect
that magnetic fields have on the strong binding forces, and due to
the direct interactions of charged spinning particles with
external fields. The modifications of the strong forces are such
as to close the gap between the proton and neutron masses and
ultimately make the proton heavier. A delicate interplay between
the anomalous magnetic moments of the proton and neutron drives
the mass shifts due to the direct interactions in the same
direction. For *B* > 1.5 × 10^{18} G the neutron
becomes stable
and as the field is increased past 2.7 × 10^{18} G the
proton becomes unstable to decay into a neutron, positron and
neutrino.

The quantum mechanics of a Dirac particle with no anomalous magnetic moment in a uniform external magnetic field is straightforward. We shall present the results for the case where particles do have such anomalous moments. In reality, in fields so strong that the mass shifts induced by such fields are of the order of the mass itself one cannot define a magnetic moment as the energies are no longer linear in the external field. Schwinger [37] calculated the self energy of an electron in an external field and we shall use here his results. We cannot follow this procedure for the proton or neutron as we do not have a good field theory calculation of the magnetic moments of these particles, even for small magnetic fields; all we have at hand is a phenomenological anomalous magnetic moment. However, for fields that change the energies of these particles by only a few percent, we will consider these as point particles with the given anomalous moments. In the Sec. 5.1.7 we will discuss possible limitations of this approach.

**5.1.1 Protons in an External Field**

The Dirac Hamiltonian for a
proton with a uniform external magnetic field **B** is

(5.1) |

The vector potential **A**(**r**) is related to the magnetic field
by **A**(r) = 1/2 **r** × **B** and
*g*_{p} = 5.58 is the proton's
Landé g factor. We first solve this equation for the case
where the momentum along the magnetic field direction is zero and
then boost along that direction till we obtain the desired
momentum. For **B** along the **z** direction and
*p*_{z} = 0 the energy levels are
[38]

(5.2) |

In the above, *n* denotes the Landau level, *m* the orbital
angular
momentum about the magnetic field direction and *s* = ± 1
indicates
whether the spin is along or opposed to that direction; the levels are
degenerate in *m*. *n* = 0 and *s* = +1 yield the lowest
energy

(5.3) |

As we shall be interested in these states only we will drop the *n*
and *s* quantum numbers. The Dirac wave function for this
state is

(5.4) |

_{m}'s are the
standard wave functions of the lowest Landau level:

(5.5) |

Boosting to a finite value of *p*_{z} is
straightforward; we obtain

(5.6) |

with a wave function

(5.7) |

where 2, the rapidity,
is obtained from tanh 2
= *p*_{z} / *E*_{m}(*p*_{z}).

In the non-relativistic limit the energy becomes

(5.8) |

and the wave function reduces to

(5.9) |