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3

Your complex algebra is messed up: $$|e^{2\pi i n x/L}|^2= e^{2\pi i n x/L} e^{-2\pi i n x/L}=1.$$

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In fact, these are just two different bases for the same space. Any function on a compact interval $[0,a]$ can be written as a sum of sines, cosines, or both. In the language of Fourier series, what we’re doing is taking the function defined on $[0,a]$ and extending it to a periodic function on the entire real line in three different ways. We can extend the ...

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You are right; the wave function does not need to be sinusoidal for the particle in a box. We can have other functions that satisfy the boundary conditions. The reason so much focus is given to the sine waves is because those functions are wave functions for particles with definite energy. Furthermore, these functions serve as a basis to describe any other ...

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One needs to impose appropriate$^1$ boundary or fall-off conditions at both temporal and spatial infinity in field theory. For starters, to ensure that the action $S$ has a mathematically well-defined variational/functional derivative $\delta S/ \delta\phi$. -- $^1$ To assume that the field $\phi$ vanishes might be too strong to describe relevant physical ...

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