Can somebody please explain to me how 1) there is the exponential term for the e
ID: 1668643 • Letter: C
Question
Can somebody please explain to me how1) there is the exponential term for the equation for themotion the ath mode? 2) how you use that to get the equation below it as well asthe determinant?
http://bilder.buecher.de/zusatz/23/23603/23603855_lese_1.pdf (pg. 195)
My teacher wants us to use that to calculate the harmonicfrequency for a given diatomic molecule, but I don't see how.(undergraduate course in statistical mechanics)
1) there is the exponential term for the equation for themotion the ath mode? 2) how you use that to get the equation below it as well asthe determinant?
http://bilder.buecher.de/zusatz/23/23603/23603855_lese_1.pdf (pg. 195)
My teacher wants us to use that to calculate the harmonicfrequency for a given diatomic molecule, but I don't see how.(undergraduate course in statistical mechanics)
Explanation / Answer
I haven't taken a course onstatistical mechanics, but it looks like that what your professoris doing is breaking Hamilton's equations of motion into a Fourierseries. If you recall, any continuous, periodic function can bedescribed as the summation of sines and cosines or exponentials.This would explain the idea of an "ath mode." The sine, cosine, orexponential terms in a Fourier series are defined by an integera, or more frequently n, which ranges from 1 toinfinite. These modes are essentially individual components thatcan be summed to produce the function. The next equation is the eigenvalue problem. If you plug in theexponential equation into Hamilton's equations (you're taking thederivative twice so that's where the w2 comes from), youget eigenvalue problem. From linear algebra, we know that there isa non-zero (non-trivial solution) if the determinant is 0, which iswhy we calculate the determinant. Hope that helps!