In demonstrating the mathematics of decoherence, the simplest system we could possibly start with must entail at least two states capable of superposition. We consider a simple harmonic oscillator in the state,

(2.1) |

Generically, we define as an eigenvalue of the annhilation operator state of the oscillator,

(2.2) |

These so called coherent states represent a quantum system which is very close to being in a classical state (see appendix). Let us ignore, for now, the need for normalization (or if preferred, assume the system is normalized as is). It is well known that the Hamilton for such a system is simply given by

The unitary evolution of this wave function is,

(2.3) |

Note that

It can be written,

(2.4) |

The density matrix is

(2.5) |

To get a clear picture of the interference this superposition entails, we can take a look at this system in the coordinate representation. We have that,

We can thus obtain,

(2.6) |

Let . Then we can write,

(2.7) |

The latter follows from the fact that the two cross terms will be conjugates, and for any complex number z, . We can partition this function into real and imaginary parts as follows. We write

We note that

(2.8) |

Thus,

It is now conventional to chose a simple case in which and ,

(2.9) |

The convenient form makes calculation of the cross term easy, and it is,

(2.10) |

If Decoherence is to bring us into the classical realm, must be destroyed. Let us be more explicit here. Equation 2.11 gives us only partial information, i.e. the probabilities for any given position q'. To represent the density matrix in proper form, we need integrate over all positions, i.e. the density matrix is

(2.12) |

A classical density matrix would not have off diagonal terms. The total probability of finding the particle in one state or the other is given by

(2.13) |

Experimental demonstration of the effect of superposition has made much progress in the last decade, and we discuss one such example in the next section.