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Section 21.9 Fourier Uncertainties
UNDER DEVELOPMENT
In quantum mechanics, the probability interpretation requires that the quantum state is normalized. In the position representation, the normalization condition for the wave function \(\psi(x)\) becomes
\begin{align}
1\amp =\int_{-\infty}^{\infty}
\left(\Psi(x)\right)^*
\left(\Psi(x)\right)\, dx\tag{21.9.1}
\end{align}
Parceval’s theorem
Section 21.4 says that this normalization condition extends to the Fourier transform
\(\tilde\Psi(k)\) and therefore, by a simple change of variables (
\(p=\hbar k\) ), to the momentum space representation of the state,
\(\Phi(p)\text{.}\)
\begin{align}
1\amp =\int_{-\infty}^{\infty}
\left(\tilde\psi(k)\right)^*
\left(\tilde\psi(k)\right)\, dk\tag{21.9.2}\\
\hbar\amp =\int_{-\infty}^{\infty}
\left(\Phi(p)\right)^*
\left(\Phi(p)\right)\, dp\tag{21.9.3}
\end{align}
