Terms of "second order" and "fourth order"...what does this MEAN?!

[AxiomOfChoice]

I am reading the paper written by Born and Oppenheimer that explains the development of the Born-Oppenheimer approximation. The paper contains the following cryptic (to me) statement:

"The nuclear vibrations correspond to terms of second order and the rotations to fourth order in the energy, while the first and third order terms vanish."

What, EXACTLY, is a "term of second order...in the energy?" (Or fourth order, for that matter?) I'm sure this is something I should know from freshman calculus, but this vernacular gets used a lot, and my understanding of it is muddled - it just is. Should I feel bad about this?

[tiny-tim]

"The nuclear vibrations correspond to terms of second order and the rotations to fourth order in the energy, while the first and third order terms vanish."

What, EXACTLY, is a "term of second order...in the energy?" (Or fourth order, for that matter?)

Hi AxiomOfChoice! :wink:

It just means that if you expand it as ∑ anEn ,

then the nuclear vibrations are proportional to a2,

the rotations are proportional to a4,

and a1 = a3 = 0. :smile:

[maze]

Physicists take taylor series _all the time_ and don't think twice about it.

[AxiomOfChoice]

Thanks guys. On this same subject, when someone notes that "error estimates are \mathcal{O}(\alpha)" for some parameter \alpha, does this translate into English as "error estimates are order \alpha?" And, if it does, what does that mean? Does it mean there is a constant c such that the magnitude of the error is less than c|\alpha| as \alpha \to 0? And is it understood that c = 1, such that all it amounts to is that, if \Delta x is the error, we have |\Delta x| \leq |\alpha|?