Wow! Here's the short version of what's going on. My quantum II final was a bit rough because I didn't know how to decouple a pair of coupled harmonic oscillators. However, I found an excellent article on decoupling in my second favorite journal the American Journal of Physics. I wanted to capture a few notes here on the process and on one of the math operations left out of the original article, but hashing out the level of background to provide has taken awhile. I decided to go with the terser representation of the subject. It may be extremely sketchy since I'm more than a bit scattered studying for finals. Basically, these are my notes, and I hope they're helpful :) Any ideas, comments, corrections, or suggestions, as always, are more than welcome! For more detail and an excellent article on how to decouple coupled oscillators see [1].

OK, here goes.

**What does 'decoupling harmonic oscillators' mean and what is it used for?**

Pretty frequently in physics, we come across problems where you have to decouple a system of coupled harmonics oscillators. For folks who haven't had analytical mechanics yet, this can be a bit dicy. I'm one of those folks. Decoupling means taking the generic Lagrangian that describes the system of two coupled harmonic oscillators that looks like this (picture 1)

and modifying/rewriting it so that it looks like this (picture 2)

**Why does everyone say 'Just diagonalize it'?**

If you do the matrix multiplies through, you'll wind up back at the first equation above. If we actually manage to modify the equation so that the b and q coefficients go to zero, then the two square matrices will be diagonal, they'll only have non-zero entries on a diagonal going from upper left to lower right. Hence, diagonalizing is just another way of saying that we want to rearrange things so that b and q go to zero.

**Why is the next thing everyone says 'Rotate the coordinates'**

Looking back at the first equation, you might notice that it can be broken into two parts, each of which describe an ellipse when they're set to zero (picture 4).

**and, (Cool!),**it's still parallel to the coordinate axes in the way we want so that the b entries in the kinetic energy matrix are still zero. When we're done, we've rotated the potential energy ellipse as well so that the q terms are now zero!

**References:**

http://dx.doi.org/10.1119%2F1.16069

Aravind P.K. (1989). Geometrical interpretation of the simultaneous diagonalization of two quadratic forms, American Journal of Physics, 57 (4) 309. DOI: 10.1119/1.16069

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