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While the wavefunctions are vectors by virtue of being functions, it still remains to be seen that the set of "physically possible" wavefunctions is also a vector space. For example, if the physics were such that we were equating the local kinetic energy of a string to, I don't know, it's position or something, we would end up with an equation looking like (D_t f(x,t))^2 - f(x,t) = 0. It still can be written as an operator (not all operators are linear), but the addition thing wouldn't work anymore. It's a remarkable pattern in physics that those squared terms tend not to appear in reality.

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