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I've learned that the wave function corresponding to a free particle has this form:

Psi(x,0)=1/sqrt(2*Pi)*Integral[g(k)*E^(ikx)dx] (i can't write it in Latex, sorry)

Is it just for the free particle, or any quantum state of a system can be represented in this form, where g(k) is the fourier transf. of Psi?

Can any function of x, represent the state of the free particle?