Quantum Chemistry 5.2 - Classical Harmonic Oscillator 1: Trajectory

[@holleyhester471]

Hi! Your videos are wonderful. I was wondering if it would be possible for you to make a video about applying ladder operators to harmonic oscillators. Thanks from University of Florida!

[@TMPChem]

Thanks Holley. Give my regards to your fellow gators. I've considered including ladder operators before, and the primary limitation is that I haven't found a clear and efficient way to include them without it taking up half a chapter's worth of videos to go through all the introductory algebra before you arrive at the final energy expressions. At the undergraduate level they typically only show up in a pchem course if the professor has a super physics / math / theory etc heavy background, though UF certainly does have a few titans in that field, so it's entirely possible you're currently experiencing such a situation. Though for a broader, more general audience at first introduction I find them to be somewhat more trouble than they're worth, and merely give the quantum H.O. solutions as given and move forward in the result. That doesn't preclude their future inclusion, but it's not a top priority at the moment. Cheers.

[@diepimmelkuh1341]

Thank you for the video. But how do we get omega?

[@TMPChem]

The equation for omega is defined in purple on the left side of the slide. Omega is equal to the square root of the spring constant k divided by the mass m. Once you have specified values for k and m, the value for omega is completely determined.

[@diepimmelkuh1341]

First of all, thank you very much for the fast answer, that is just awesome. The problem I have is the connection from the omega equation to the step before (blue) and the step afterwards (red)

[@TMPChem]

As far as going from d2x/dt2 = -k/m x to x(t) = Acos(wt) + Bsin(wt)? This is from the general form to a solution to any differential equation where the second derivative of a function equals itself times a negative constant. In such a case, the general solution is a sum of complex exponentials [k1 exp(iwt) + k2 exp(-iwt)], which we typically use Euler's formula to rearrange into sines and cosines of the same exponent. I skipped that step here and went straight to sines and cosines, as I often do in this playlist. If you haven't studied differential equations yet this probably doesn't make sense, but you can still verify for yourself that the given solution satisfies the original differential equation if you take its second derivative.

[@tag_of_frank]

V above is potential energy V below is the velocity; that was confusing because both V look the same. You can't tell from the writing on the board, so if you are just trying to read the board you get confused.

[@samitannir6830]

No it's not, the v for velocity is a small v while the V for potential energy is a capital V

[@tag_of_frank]

@@samitannir6830 Prove it.

[@samitannir6830]

@@tag_of_frank wym prove it, can't you see one v is smaller than the bracket, and the other v is larger? either increase the resolution or get some glasses

[@tag_of_frank]

​@@samitannir6830 You said it is not confusing so I said prove it. Are you claiming less than 5 percent of viewers will find find it confusing? For over a year I have represented 1/3 = 33% of the main comments on this video. Thus 33% of the comments are about how confusing the V v notation is.