Svnthesis Problem 9. Time (or lifetime) and spectral bandwidth are related to each other by a Fourier transform IK width The width of a vibrational band in an IR spectrum can be determined, for example, by the lifetime of the excited vibrational level. Suppose we have a vibration with a frequency 2-00 and suppose that the vibrational state decays exponentially in time leading to an expression: E(t) cos(o,t)e . The width of the absorption band is then determined by the E(O Fourier transform g()-Re | | E(t)e"dt |-| E(t) cos(or) d. 0 0 t (ps) (a) Let o-40 radians per ps, (1 ps- 1012 s). k-1 ps1. These are typical values for low frequency vibrations. What is Vo? (b) Calculate the Fourier transform and use Matlab to plot your function g(o) to determine the width at half the peak height in the spectrum. When this expression is correct, the width of a vibrational band is lifetime limited. The width can be greater due to other effects. Svnthesis Problem 9. Time (or lifetime) and spectral bandwidth are related to each other by a Fourier transform IK width The width of a vibrational band in an IR spectrum can be determined, for example, by the lifetime of the excited vibrational level. Suppose we have a vibration with a frequency 2-00 and suppose that the vibrational state decays exponentially in time leading to an expression: E(t) cos(o,t)e . The width of the absorption band is then determined by the E(O Fourier transform g()-Re | | E(t)e"dt |-| E(t) cos(or) d. 0 0 t (ps) (a) Let o-40 radians per ps, (1 ps- 1012 s). k-1 ps1. These are typical values for low frequency vibrations. What is Vo? (b) Calculate the Fourier transform and use Matlab to plot your function g(o) to determine the width at half the peak height in the spectrum. When this expression is correct, the width of a vibrational band is lifetime limited. The width can be greater due to other effects