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Centre the infinite square well at zero. Its ground state wavefunction is $\psi (x)=\sqrt{2/L}\cos (\pi x/L)$ inside the well and zero outside. The momentum representation in quantum mechanics gives the wavevector amplitudeConsequently the unnormalised wavevector distribution of an infinite-well ground state isThe apparent singularities at $k=\pm \pi /L$ are removable. The normalized probability density is $4\pi L$ times the boxed expression and has limit $L/(4\pi )$ there. Translating the well changes only the Fourier transform's phase. The wavevector is continuously distributed; a box energy eigenstate is not a free momentum eigenstate.

A centred width-$L$ infinite square well has ground state wavefunction $\sqrt{2/L}\cos (\pi x/L)$ inside and zero outside. Its momentum representation in quantum mechanics follows from the Fourier transform of a truncated cosine. The probability density in the title has removable singularities at $k=\pm \pi /L$, with limit $L/(4\pi )$, and an oscillatory $k^{-4}$ tail envelope. The distribution is even, with $⟨k⟩=0$ and $⟨k^2⟩={\pi }^2/L^2$.