In this appendix we calculate the equilibrium expectation value of *X*_{k}^{2}.
In Cartesian coordinates the statistical weight of some configuration
is given by

(6.77) |

Since the transformation to normal coordinates is a linear transformation from one set of orthogonal coordinates to another, the corresponding Jacobian is simply a constant. The argument of the exponential may be written as

= | |||

= | |||

= | (6.78) |

The statistical weight therefore reads

(6.79) |

Since this is a simple product of independent Gaussians, all kinds of expectation values may easily be calculated, like for example

(6.80) |