Born Rule

If a Quantum System is in state ∣ψ⟩\left\lvert \psi \right\rangle and you measure observable A^\hat{A} — the probability of getting eigenvalue λα\lambda_\alpha (which corresponds to eigenstate ∣α⟩\left\lvert \alpha \right\rangle) is

Pα=∣⟨α∣ψ⟩∣2\begin{gather*} P_\alpha=|\left\langle \alpha|\psi \right\rangle|^2 \end{gather*}

This only holds true if eigenstate ∣ϕn⟩\left\lvert \phi_n \right\rangle is non-degenerate and normalized. Usually, the convention is that we pick the representative state ⟨ϕn∣ϕn⟩=1\left\langle \phi_n|\phi_n \right\rangle=1 (i.e., it is normalized).

Born’s rule is a postulate, meaning that it is based on experimental observations and cannot be derived from first principles.

Note that ∣⟨α∣ψ⟩∣2|\braket{\alpha|\psi}|^2 is a probability amplitude