Question 33.8: Show that eqn 33.103 holds for the problem considered in Exa......
Show that eqn 33.103 holds for the problem considered in Example 33.5.
\boxed{\widetilde{C} _{xx} (ω)=2 k_{B} T\frac{\widetilde{χ} ^{\prime \prime }(ω)}{ ω },} (33.103)
Recall from Example 33.7 that
\widetilde{C} _{xx} (ω)=\int e^{−iωt} \left\langle x(0)x(t)\right\rangle dt = \left\langle |\widetilde{x} (ω)^{2} \right\rangle = A|χ(ω)|^{2} , (33.104)
and hence using \widetilde{χ}(ω) from eqn 33.76 and A from eqn 33.94, we have that
\widetilde{χ} (ω) =\frac{\widetilde{x} (ω) }{\widetilde{f} (ω)} =\frac{1}{m}\left[\frac{1}{ω^{2}_{0} −ω^{2} −iωγ} \right]. (33.76)
A =2αk_{B} T. (33.94)
\widetilde{C} _{xx} (ω) = \frac{2γk_{B} T}{m} \left[\frac{1}{(ω^{2} − ω^{2} _{0} )^{2} +(ωγ)^{2} } \right]. (33.105)
Equation 33.77 shows that
\widetilde{χ}^{\prime \prime } (ω) = \frac{1}{m} \left[\frac{ωγ}{(ω^{2} − ω^{2} _{0} )^{2} +(ωγ)^{2} } \right], (33.77)
2k_{B} T\frac{\widetilde{χ} ^{\prime \prime } (ω) }{ω} =\frac{2γk_{B} T}{m} \left[\frac{1}{(ω^{2} −ω^{2}_{0} )^{2} +(ωγ)^{2} } \right], (33.106)
and hence eqn 33.103 holds.