Spectral projection
= Spectral projection
{title2=$E_A(S)=\mathbf1_S(A)$}
For a <self-adjoint operator> $A$, its projection-valued <spectral measure of a normal operator> assigns each Borel set $S\subseteq\mathbb R$ an <orthogonal projection> $E_A(S)$. For a normalized <quantum state> $\psi$, the probability of a measurement result in $S$ is $\langle\psi,E_A(S)\psi\rangle$. In finite dimensions, $E_A(S)$ is the sum of the orthogonal projections onto eigenspaces whose eigenvalues lie in $S$. This generalizes an individual eigenvector probability to degenerate eigenvalues and continuous spectra. The functional-calculus notation $\mathbf1_S(A)$ uses the indicator function of the set.