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[tex]

\lim_{R \to \infty} \int_{-R}^{R} dk\,\lvert \tilde{\psi}(k)\lvert^2\, \hbar k

[/tex]

instead of the improper integral

[tex]

\int_{-\infty}^{\infty} dk\,\lvert \tilde{\psi}(k)\lvert^2\, \hbar k = \lim_{R_1, R_2 \to \infty} \int_{-R_1}^{R_2} dk\,\lvert \tilde{\psi(k)}\rvert^2\,\hbar k

[/tex]

I ask because a solution from the 8.04 Quantum Physics course on MIT OCW only makes sense taking the principal value of the improper integral for the momentum operator's expectation, and not taking the mathematical definition of an improper integral. Is it assumed integrals over the real line are principal values in quantum physics? Fourier transforms are usually interpreted as principal values, correct? Or no?