Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Bra-Ket Notation

Bra-ket notation, \langle \Psi| and |\Psi\rangle, provides a concise and elegant way to represent wavefunctions and operators in matrix mechanics.
📜

The statement of the theorem

Let H\mathcal{H} be a separable Hilbert space over C\mathbb{C}. Define the state vector ∣Ψ⟩∈H|\Psi\rangle \in \mathcal{H} and its corresponding dual element ⟨Ψ∣\langle\Psi| as the linear functional on H\mathcal{H} such that ⟨Ψ∣Φ⟩=⟨Φ∣Ψ⟩∗\langle\Psi|\Phi\rangle = \langle\Phi|\Psi\rangle^*. The inner product ⟨Ψ1∣Ψ2⟩\langle\Psi_1|\Psi_2\rangle is defined by the Hermitian inner product ⟨Ψ1,Ψ2⟩H\langle\Psi_1, \Psi_2\rangle_{\mathcal{H}}. The bra-ket notation is formalized by the identity operator I=∑k=1N∣k⟩⟨k∣\mathbf{I} = \sum_{k=1}^{N} |k\rangle\langle k| (or I=∫∣k⟩⟨k∣dμ(k)\mathbf{I} = \int |k\rangle\langle k| d\mu(k) in continuous bases), which ensures the completeness relation: ⟨Ψ∣Φ⟩=⟨Ψ,Φ⟩H\langle\Psi|\Phi\rangle = \langle\Psi, \Phi\rangle_{\mathcal{H}}. Furthermore, the expectation value of a self-adjoint operator A^\hat{A} for state ∣Ψ⟩|\Psi\rangle is given by ⟨A^⟩=⟨Ψ∣A^∣Ψ⟩=⟨Ψ,A^Ψ⟩H\langle\hat{A}\rangle = \langle\Psi|\hat{A}|\Psi\rangle = \langle\Psi, \hat{A}\Psi\rangle_{\mathcal{H}}. This structure defines the action of operators A^:H→H\hat{A}: \mathcal{H} \to \mathcal{H} via matrix multiplication in a chosen basis.
Source: Wikipedia