Fourier transforms imply that the waves are added up with a continous range of frequencies hence wavelengths instead of discrete numbers. In QM fourier transforms are used which imply the psi function employs a continous rather than discrete range of frequencies. Why is this? Is it because psi wave is not a mechanical wave so it dosen't matter wheither the frequencies are continuous or discrete? However, I have only seen fourier transforms done on plane waves hence on free particles which can obtain a continous range of energies via E=hf. So f takes on a continous range hecne both wavenumer, k and period length lamda will all be continous. That is why the fourier transform can be done on free particles. However, if the particle is not free than fourier transforms cannot be used as the frequencies will not be continous, wavenumber and period will not be continous.