Definition
Probability Amplitude
Let
be a complex Hilbert space with orthonormal basis . For a normalised quantum state , The complex coefficient
is the probability amplitude of the basis vector in . Under measurement in the basis , the probability of outcome is Thus a probability amplitude is a complex coordinate of a quantum state, and the Born rule maps it to a classical probability by squared modulus.
Projection
The inner product is sesquilinear, i.e., it takes the ket as the linear side, thus:
Given that
substituting into the expression above gives
Thus the probability amplitude of a basis state