1. The problem statement, all variables and given/known data
Chlorine has two naturally occurring isotopes, Cl-35 and Cl-37. Show that the vibrational spectrum of HCl should consist of closely spaced doublets, with a splitting given by ##\Delta \nu = 7.51x10^{-4}\nu ## where ##nu## is the frequency of the emitted photon. Hint: think of it as an harmonic oscillator, with ##\omega = \sqrt{k/\mu} ## where ##mu## is the reduced mass, and k is the same for both isotopes.
2. Relevant equations
To put it into to context this comes after an explanation of how a two-particle quantum system can be reduced to a central-force problem.
3. The attempt at a solution
I am at a loss here. I'm not even sure what "vibrational spectrum" or "splitting" mean. Are they the range of the frequency and the space between consecutive values?
My guess is that I should compute the energies for the 35 and 37 excited states of an harmonic oscillator; their difference should be the energy of a photon emitted for that transition. However I can't understand when is the photon emitted. Does the chlorine spontaneously change from Cl-53 to Cl-37?
Chlorine has two naturally occurring isotopes, Cl-35 and Cl-37. Show that the vibrational spectrum of HCl should consist of closely spaced doublets, with a splitting given by ##\Delta \nu = 7.51x10^{-4}\nu ## where ##nu## is the frequency of the emitted photon. Hint: think of it as an harmonic oscillator, with ##\omega = \sqrt{k/\mu} ## where ##mu## is the reduced mass, and k is the same for both isotopes.
2. Relevant equations
To put it into to context this comes after an explanation of how a two-particle quantum system can be reduced to a central-force problem.
3. The attempt at a solution
I am at a loss here. I'm not even sure what "vibrational spectrum" or "splitting" mean. Are they the range of the frequency and the space between consecutive values?
My guess is that I should compute the energies for the 35 and 37 excited states of an harmonic oscillator; their difference should be the energy of a photon emitted for that transition. However I can't understand when is the photon emitted. Does the chlorine spontaneously change from Cl-53 to Cl-37?
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