Why is the momentum state of a particle in quantum mechanics given by the Fourier-Transformation of its position state? For instance, in one dimension by

$$\varphi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int dx e^{-i p x/\hbar} \psi(x)$$

Why is the momentum state of a particle in quantum mechanics given by the Fourier transform of its position state? For instance, in one dimension given by

$$\varphi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int \mathrm dx \, e^{-i p x/\hbar} \psi(x).$$