13.1.3: Examples
- Page ID
- 5297
Define
\[G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt\, e^{i\omega t} \langle {1 \over 2}[B(0),B(t)]_+\rangle \nonumber \]
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
\[H' =-\mu \cdot \textbf{E}(t) \nonumber \]
If we take \(\textbf{E}(t)=E(t) \hat{\text{z}} \), then
\[H'=-\mu_z E(t) \nonumber \]
Identifying \( B=\mu_z \), the spectrum becomes
\[G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt\;e^{i\omega t}\langle {1 \over 2}[\mu_z(0),\mu_z(t)]_+\rangle \nonumber \]
or for a general electric field, the result becomes
\[G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt\; e^{i \omega t} \langle {1 \over 2} (\mu (0) \cdot \mu (t) + \mu (t) \cdot \mu(0) )\rangle \nonumber \]
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 \(H' \)
\[H' = -\sum_{i=1}^N S_{i,z}{\cal H}(t) \nonumber \]
The net magnetization created by the field \( {m_z}\) is given by
\[ m_z = {1 \over N}\sum_{i=1}^N S_{i,z} \nonumber \]
so that
\[ H' = -Nm_z{\cal H}(t) \nonumber \]
Identify \(B = m_z \) (the extra factor of \(N\) just expresses the fact that \(H' \) is extensive). Then the spectrum is
\[ G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt\;e^{i\omega t}\langle {1 \over 2}[m_z(0),m_z(t)]_+\rangle \nonumber \]
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 \( C_{BB} (t) \):
\[ C_{BB}(t) = \langle B^2\rangle e^{-\Gamma \vert t\vert} \nonumber \]
The spectrum corresponding to this time correlation function is
\[ G(\omega) = {1 \over 2\pi}\int_{-\infty}^{\infty}\;dt e^{i\omega t}C_{BB}(t) \nonumber \]
and doing the integral gives
\[ G(\omega) = {\langle B^2 \rangle \over \pi}{\Gamma \over \omega^2 + \Gamma^2} \nonumber \]
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:
\[C_{BB}(t) = \langle B^2 \rangle e^{-\Gamma\vert t\vert}\cos\omega_0t \nonumber \]
which describes well the behavior of a harmonic diatomic coupled to a bath. The spectrum can be shown to be
\[ G(\omega) = {\langle B^2 \rangle \Gamma \over \pi}\left[{\Gamma ^2 + \omega ^2 + \omega ^2_0 \over \left ( \Gamma ^2 + (\omega - \omega _0 )^2 \right)\left(\Gamma^2 + (\omega+\omega_0)^2\right)}\right] \nonumber \]
which contains two peaks at \( \omega = \pm \sqrt{\omega_0^2 - \Gamma^2} \) as shown in the figure below:
