Coherent-state overlap ($⟨\beta |\alpha ⟩=e^{-(|\beta {|}^2+|\alpha {|}^2)/2+{\beta }^{\ast }\alpha }$)

Expand both states in the Fock basis and sum the exponential series to obtain the displayed overlap. In particular $⟨\alpha |-\alpha ⟩=e^{-2|\alpha {|}^2}$. Distinct coherent states are nonorthogonal at finite separation; dropping this overlap can give incorrect normalization and small-amplitude limits.