# Examples

Define

$G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt e^{i\omega t} \langle {1 \over 2}[B(0),B(t)]_+\rangle$

which is just the frequency spectrum corresponding to the autocorrelation function of $$B$$. For different choices of $$B$$, $$G(\omega)$$ corresponds to different experimental measurements. Consider the example of a molecule with a transition dipole moment vector $$\mu$$. If an electric field $$\textbf{E}(t)$$ is applied, then the Hamiltonian $$H'$$ becomes

$H' =-\mu \cdot \textbf{E}(t)$

If we take $$\textbf{E}(t)=E(t) \hat{\text{z}}$$, then

$H'=-\mu_z E(t)$

Identifying $$B=\mu_z$$, the spectrum becomes

$G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt\;e^{i\omega t}\langle {1 \over 2}[\mu_z(0),\mu_z(t)]_+\rangle$

or for a general electric field, the result becomes

$G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt\; e^{i \omega t} \langle {1 \over 2} (\mu (0) \cdot \mu (t) + \mu (t) \cdot \mu(0) )\rangle$

These spectra are the infrared spectra.

As another example, consider a block of material placed in a magnetic field $${\cal H}(t)$$ in the $$z$$ direction. The spin $$S_z$$ of each particle will couple to the magnetic field giving a Hamiltonian $$H'$$

$H' = -\sum_{i=1}^N S_{i,z}{\cal H}(t)$

The net magnetization created by the field $$\underline {m_z}$$ is given by

$m_z = {1 \over N}\sum_{i=1}^N S_{i,z}$

so that

$H' = -Nm_z{\cal H}(t)$

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

$G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt\;e^{i\omega t}\langle {1 \over 2}[m_z(0),m_z(t)]_+\rangle$

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 $$C_{BB} (t)$$:

$C_{BB}(t) = \langle B^2\rangle e^{-\Gamma \vert t\vert}$

The spectrum corresponding to this time correlation function is

$G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt e^{i\omega t}C_{BB}(t)$

and doing the integral gives

$G(\omega) = {\langle B^2 \rangle \over \pi}{\Gamma \over \omega^2 + \Gamma^2}$

which is shown in the figure below:

We see that the lineshape is a Lorentzian with a width $$\Gamma$$. As a further example, suppose $$C_{BB} (t)$$ is a decaying oscillatory function:

$C_{BB}(t) = \langle B^2 \rangle e^{-\Gamma\vert t\vert}\cos\omega_0t$

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

$G(\omega) = {\langle B^2 \rangle \Gamma \over \pi}\left[{\Gamma ^2 + \omega ^2 + \omega ^2_0 \over \left ( \Gamma ^2 + (\omega - \omega _0 )^2 \right)\left(\Gamma^2 + (\omega+\omega_0)^2\right)}\right]$

which contains two peaks at $$\omega = \pm \sqrt{\omega_0^2 - \Gamma^2}$$ as shown in the figure below: