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I study Bethe ansatz, although my background is mathematics not physics. Can somebody explain to me what is plane waves? I have seen in many papers this expression that "The idea of the Bethe ansatz is that eigenfunctions are a linear combination of plane waves", or superposition of plane waves.

My understanding is for each particle a complex number which is called plane wave has been defined! then an eigenfunction is a combination of these complex numbers over all possible permutations. But why is such a complex number for each particle defined, and why is it called a plane wave?

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This depends a bit on the context, but in essence you have a position variable $x$ (which is typically real but it can be discrete), and any function of the form $$ f(x) = e^{ikx}, $$ where $k$ is a real number, is called a plane wave.

This isn't just "some complex number": it's a complex-valued function of position. Moreover, each particle has some wavefunction $f:S\to\mathbb C$, which is a complex-valued function of your position space $S$ (which could be $\mathbb R$, or could be discrete), but not all such functions are plane waves: you could have $f(x) = 1/(x+i)$, say, or $f(x) = e^{ix^2}$, neither of which are plane waves.

The term plane wave comes from the fact that if you give them a time dependence, and you look at it in three dimensions, $$f(x,y,z,t) = e^{ikx}e^{-i\omega t} = e^{i(kx-\omega t)},$$ then $f$ oscillates harmonically with position, and its wavefronts (the surfaces of constant phase) form planes which move with time. For further details, consult a solid optics or advanced-EM textbook.

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