Bra-ket notation
Bra-ket notation is Dirac’s compact language for vectors, dual vectors, inner products, and amplitudes.
The pieces
- A ket |Ψ⟩ is a state-vector.
- A bra ⟨Ψ| is the dual object that pairs with kets.
- A bracket ⟨A|B⟩ is an inner product.
- An operator can be sandwiched as ⟨A|M|B⟩.
Amplitudes
If {|j⟩} is a measurement basis, then:
α_j = ⟨j|Ψ⟩
P(j) = |α_j|²The inner product is a probability amplitude. The squared magnitude is the probability.
Operators in the middle
The expression ⟨A|M|B⟩ means: let M act on |B⟩, then take the inner product with ⟨A|. In a chosen basis these become matrix elements.
Why it is useful
Bra-ket notation keeps the physics basis-independent. A column vector is a representation after choosing a basis; the ket itself names the abstract state.
Common pitfalls
- Do not read ⟨A|B⟩ as ordinary multiplication; order matters and complex conjugation is involved.
- Do not forget that |Ψ⟩ and e^{iθ}|Ψ⟩ give the same physical probabilities when the phase is global.
- Do not confuse ⟨M⟩, an expectation value, with a guaranteed measurement outcome.