In this Appendix we derive from first principles the eigenvalues and states of muonium, in the simplest case of free muonium in vacuum with a static magnetic field applied. The polarization functions with the applied magnetic field oriented either transverse to or along the initial muon spin are calculated. Spin relaxation is not considered in this calculation.

In the situation being considered the Hamiltionian of the muonium atom,

(1) |

In order to determine the time dependence of the muon spin polarization we diagonalize this Hamiltonian and calculate the elements of the muon spin operator. In doing so we will obtain the muonium states and corresponding energies, which determine the precession frequencies observed in the muon spin polarization signal.

We start by defining a set of states of the muonium atom
in terms of the individual electron and muon states,

The spin operator has components given by the usual Pauli matrices and the muon and electron spin operators act only on their corresponding parts of the muonium wave function.

The operator in the first term of the Hamiltonian

(2) |

The hyperfine interaction term is then the sum of
Eqs. (A.5,A.6,A.7)
and we obtain

Finally, noting that

(3) |

(4) |

For the two cases it then immediately follows that
the eigenvalues are given by

(5) |

(6) |

Operating with the full Hamiltonian on the first of these, we obtain

(7) |

(8) |

Equation (A.18) can now be simplified to

(9) |

(10) |

(11) |

A similar calculation, operating on the wavefunction , yields

(12) |

The results so far are summarized in Figure A.32,
which shows the four eigenvalues as functions of the magnetic field *B*.

We shall see that the polarization signal has
components at frequencies determined by the transition
energies between these states. These frequencies are