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I am currently reading "Quantum Field Theory for the Gifted Amateur". In chapter 2 Phonons are introduced as solutions (in k-space) of a coupled harmonic oscillator. In real space the oscillator is coupled, but apparently not in k-space (after doing a Fourier-Transform on the x,p Operators). During the solution the author used periodic boundary conditions but I don't see why they should accurately describe a finite crystal that is not shaped like a ring. In a different book the solution was also obtained with periodic boundary conditions.
Are more realistic boundary conditions impossible to solve? I would have guessed that we assume that the wave-function (of the phonons) is supposed to be zero outside of the lattice instead of this infinite periodic behaviour.
A fininte crystal might be very different than a infinite crystal (interpreation of the periodic boundary). They seem like two completely different systems. Why do we use periodic boundary conditions and how accurate is this?