Suppose I have two identical systems, such as two copies of a mass attached to a spring (anchored to an immovable wall). Each system by itself has some characteristic response, like a frequency of motion, and those responses are identical because the independent systems are identical. Now consider coupling the two systems together, such as linking the two masses by another (weak) spring, and ask what the total coupled system response looks like. With classical oscillators like our example, we would say that we find the "new normal modes" of the coupled system - instead of writing separate equations to describe Newton's laws for each mass separately, we can do some kind of change of variables and consider new coordinates that combine the motions of the two masses. When we do this, we end up again with two characteristic frequencies (basically two effectively independent oscillators), but now the frequencies differ a bit, one being higher and one being lower than the original independent oscillator frequency. You can generalize this to \(N\) oscillators and find \(N\) new normal modes with a

*band* of frequencies, with the bandwidth determined by the strength of the couplings.

This coupling+splitting the modes crops up again and again. You can use this to make an electronic band pass filter by capacitively coupling together a bunch of \(LC\) circuits. You can make a mechanical band pass filter by elastically coupling together a bunch of mechanical oscillators. You can make a miniband of electronic states by coupling together a bunch of quantum wells that each individually would have identical bound states. You can make a band of electronic states by coupling together the single-particle atomic states of a periodic array of atoms. The stronger the coupling, the wider the resulting bandwidth. This basic idea (taking noninteracting systems, coupling them, and re-solving for new effectively noninteracting modes) is so powerful that when it doesn't work sometimes, it can be jarring.

## 2 comments:

Thank you for putting this (two-post so far!) series together: I'm familiar with most of the examples you discuss, but don't think about the connecting threads often enough.

What's the formal (mathematical) property that is shared by all of the systems exhibiting hybridization? Is it the linearity of the underlying equations (SHO, LC circuit, Schroedinger equation)? Can one make a simple rigorous argument that shows how hybridization arises in a generic linear system? Or am I on the wrong track here?

Hi Ted - Thanks for the kind words. More coming, though real life is intrusive.

In answer to your question, I think (but don't know with mathematical rigor) that you're on the right track. Linearity and second/fourth order differential equations seem to be key ingredients. (I include fourth order in space PDEs here because that's what is relevant in the elastic response of clamped beams or a drumhead.)

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