Solution To Triatomic Molecule Problem

 Problem:

For each of the frequency solutions (h 1  etc. )applied to the linear tri-atomic molecule, give the relation between the normalized amplitudes: A1, A2 and A3 and the angular frequency.

Solution:

The three solutions for the angular frequency were already given in the Aug. 18 blog post, from the secular determinant:

(k - w ² m    …-k…   . .0)

(-k …..2 k - w ² m  ..  -k)

(0…. ….-k….. k - w ² m)

I.e.

w1  =  0,     w2  = Ö(k/ m),   w3  = Ö{ k/ m (1 + 2m/ M)}

The amplitudes from these solutions are then obtained (as shown in the post) from the Lagrange tensor:

h 1  =   A1 exp (iw t)

 h 2  =   A2 exp (iw t)

h 3  =   A3 exp (iw t)