de Broglie wavelength application: molecular vibrations?

mtmentat

1. The problem statement, all variables and given/known data

In his presentation of the de Broglie wavelength and the wave/particle duality, my quantum prof. has an exercise: "Find the de Broglie wavelength of a molecular vibration. Do quantum or classical laws apply?"


2. Relevant equations

wavelength = h/(momentum)
...
E=h*c*wavenumber*100
...
momentum = Sqrt( 2*reduced mass*E)

3. The attempt at a solution

So, a typical molecular vibration can be dreamed up to be 2100cm^-1 and involve C-triple bond-O stretching (I'm a chemistry student, if you hadn't noticed). This would involve an energy of ~4E-32J. However, if I think about it as a harmonic oscillator, I think that the the center of mass doesn't change, and that the overall momentum must be zero. If I think of it as some sort of very strange particle flying through space with that energy (using the reduced mass), I get a "momentum" of 3E-29 kg*m/s, which gives a de Broglie wavelength of 2E-5 m (and thus can be treated classically). I know that classical treatment is usually used for IR vibrations, and so this result makes sense, but the way I got there definitely doesn't. My prof. mentioned quickly that you can use the uncertainty principle to get at the momentum in the problem, but I'm don't see how that works. Please let me know if this exercise makes sense or not!
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

presbyope

I'm lost. What is your criterion for classical-ness? I thought classical treatment required small de Broglie wavelengths. 2E-5 meter is huge.

mtmentat

Ha. Um... yes (and thanks), presbyope, you're quite correct about my dyslexia. I guess then that my non-method would suggest that the CO stretch would be best modeled using quantum mech.

I'm less concerned, however, with classification as with trying to figure out if there is any real way to apply de Broglie wavelength to a molecular vibration.

presbyope

Great, now I'm not lost. But you are right that IR vibrations are usually modeled with classical mechanics. So that's odd.

When we talk about light going through a slit, the wavelength is compared with the slit width (characteristic distance.) Small slits produce quantum effects. So can we treat the bond length as a "slit width" for the vibration? It should be about 1E-10 m.

When I do your calculations I get a wavelength somewhat smaller than the bond length (classical situation). I used the reduced mass of a carbon and oxygen pair. Is that what you meant by reduced mass?

mtmentat

So... I've gone back through my back-of-the envelope equations, and yes, using the reduced mass of carbon and oxygen (just for fun, 1E-26 kg) and the energy of the molecular vibration (5E-20 J), I can get a "momentum" of 3E-23 kg*m/s, leading to a de Broglie wavelength of about 20 pm. The bond length in CO is approx. 110pm (1E-10m is right on).

de Brolgie wavelength < bond length might point to classical mechanics, based on the length scale of the problem/slit width argument.

I am very unsure of the above method for getting at the de Broglie wavelength. This is the real problem, and I'm not sure if the question really makes sense at all. When do de Broglie wavelengths apply? I had thought that it mostly tied wave properties to matter, but I don't classify "molecular vibrations" as matter, leading me to be very confused.

Thanks for the comments so far!

presbyope

You're welcome. It is very puzzling. I'm still not sure how the uncertainty principle would help other than giving a upper bound for wavelength / lower bound for bond length. My notes talk about de Broglie wavelengths as describing matter waves (things with mass and kinetic energy.) The de Broglie wavelength of an object at rest is infinite.

Your CO molecule has kinetic energy: 3 modes translational and 2 modes rotational. When you expose it to IR 2 more modes appear, vibrational kinetic energy and potential energy (like a spring.) Since we're in the center of mass frame, ignore the other modes of kinetic energy and consider the molecule's mass and vibrational kinetic energy. The question was phrased oddly, I agree.