If I take a periodic wavefunction $\psi\left(\vec{r}\right)$ and then take the fourier space dispersion of the wave function as defined below $$\psi\left(\vec{k}\right)=\iiint_{-\infty}^{+\infty}\psi\left(\vec{r}\right)e^{-\vec{k}\cdot\vec{r}}\mathrm{d}^3\vec{r}$$ Is there a reason for calling $\psi\left(\vec{k}\right)$ the momentum space representation of the wavefunction? (I understand the fact that the vector space $\vec{k}$ gets quantized in accordance to the formulation, $\vec{k}\cdot\vec{R}=2\pi$, where in $\vec{R}$ is the lattice translation vector periodicity of $\psi\left(\vec{r}\right)$ in a crystal lattice), but is there some other reason for calling it momentum space?

If I take a periodic wavefunction $\psi\left(\vec{r}\right)$ and then take the Fourier space dispersion of the wave function as defined below $$ \psi(\vec{k})=\iiint_{-\infty}^{+\infty}\psi\left(\vec{r}\right)e^{-\vec{k}\cdot\vec{r}}\mathrm{d}^3\vec{r} $$

Is there a reason for calling $\psi(\vec{k})$ the momentum space representation of the wavefunction? (I understand the fact that the vector space $\vec{k}$ gets quantized in accordance to the formulation, $\vec{k}\cdot\vec{R}=2\pi$, where in $\vec{R}$ is the lattice translation vector periodicity of $\psi(\vec{r})$ in a crystal lattice), but is there some other reason for calling it momentum space?