Examples

Define

which is just the frequency spectrum corresponding to the autocorrelation function of B. For different choices of B, corresponds to different experimental measurements.

Consider the example of a molecule with a transition dipole moment vector . If an electric field is applied, then the Hamiltonian H' becomes

If we take , then

Identifying , the spectrum becomes

or for a general electric field, the result becomes

These spectra are the infrared spectra.

As another example, consider a block of material placed in a magnetic field in the z direction. The spin of each particle will couple to the magnetic field giving a Hamiltonian H'

The net magnetization created by the field is given by

so that

Identify (the extra factor of N just expresses the fact that H' is extensive). Then the spectrum is

which is just the NMR spectrum. In general for each correlation function there is a corresponding experiment that measures its frequency spectrum.

To see what some specific lineshapes look like, consider as an ansatz a pure exponential decay for the correlation function :

The spectrum corresponding to this time correlation function is

and doing the integral gives

which is shown in the figure below:

  
Figure 2:

We see that the lineshape is a Lorentzian with a width .

As a further example, suppose is a decaying oscillatory function:

which describes well the behavior of a harmonic diatomic coupled to a bath. The spectrum can be shown to be

which contains two peaks at as shown in the figure below:

  
Figure 3: