Bra-Ket Notation

quantum

Definition

Bra-Ket Notation

Let $\mathcal{H}$ be a complex Hilbert space with topological dual ${\mathcal{H}}^{\ast }$. The bra–ket notation defines:

$$\begin{array}{rlrl}|\psi ⟩& \in \mathcal{H}& & \text{(ket, column vector)}\\ ⟨\psi |& ={|\psi ⟩}^{†}\in {\mathcal{H}}^{\ast }& & \text{(bra, row vector)}\end{array}$$

The inner product $⟨\cdot |\cdot ⟩:{\mathcal{H}}^{\ast }\times \mathcal{H}\to \mathbb{C}$ satisfies:

$$⟨\varphi |\psi ⟩={⟨\psi |\varphi ⟩}^{\ast },⟨\psi |\psi ⟩\ge 0$$

For orthonormal basis $\{|i⟩\}$ with $⟨i|j⟩={\delta }_{ij}$:

• Resolution of identity: $\hat{I}={\sum }_i|i⟩⟨i|$
• State expansion: $|\psi ⟩={\sum }_i⟨i|\psi ⟩|i⟩$
• Operator matrix elements: $\hat{A}={\sum }_{i,j}⟨i|\hat{A}|j⟩|i⟩⟨j|$