Definition
Observable
An observable is a Hermitian operator
whose eigenvalues are the possible numerical outcomes of a quantum measurement. For a projective measurement, an observable has a spectral decomposition
where
are the possible measurement values and are orthogonal projections onto the corresponding eigenspaces. If the system is in state
, then outcome occurs with probability After observing outcome
, the state becomes The expectation value of observation in state
is Intuitively, the projectors
define the measurement branches, while the eigenvalues are the classical numbers reported as outcomes.