Inner product

Inner product of states ∣u⟩\left\lvert u \right\rangle and ∣v⟩\left\lvert v \right\rangle is defined as

⟨u∣⋅∣v⟩=u1∗v1+u2∗v2+...+un∗vn\begin{gather*} \left\langle u \right\rvert\cdot \left\lvert v \right\rangle=u_1^*v_1+u_2^*v_2+...+u_n^*v_n \end{gather*}

(if the entries are real numbers, the conjugates do nothing and this reduces to the usual dot product.)

It is a Projection of ∣ψ⟩\ket{\psi} onto ∣x⟩\ket{x}

Properties

  1. Conjugate symmetric
⟨u∣v⟩∗=⟨v∣u⟩\begin{gather*} \left\langle u|v \right\rangle^*=\left\langle v|u \right\rangle \end{gather*}

Note

⟨n∣m⟩=δnm\braket{n|m}=\delta_{nm}

Where δnm\delta_{nm} is the Kronecker Delta

Example

let

∣ψ⟩=[ab]⇒⟨ψ∣=[a∗b∗]∣ϕ⟩=[cd]⇒⟨ϕ∣=[c∗d∗]\begin{gather*} \left\lvert \psi \right\rangle=\begin{bmatrix}a\\b\end{bmatrix}\quad \Rightarrow\left\langle \psi \right\rvert=\begin{bmatrix}a^* & b^*\end{bmatrix} \\ \left\lvert \phi \right\rangle=\begin{bmatrix}c\\d\end{bmatrix}\Rightarrow\left\langle \phi \right\rvert=\begin{bmatrix}c^* & d^*\end{bmatrix} \end{gather*}

so

(⟨ψ∣ϕ⟩)∗=([a∗b∗][cd])∗=(a∗c+b∗d)∗=ac∗+bd∗=c∗a+d∗b=⟨ϕ∣ψ⟩\begin{gather*} (\left\langle \psi|\phi \right\rangle)^*=\left(\begin{bmatrix}a^* & b^*\end{bmatrix}\begin{bmatrix}c\\d\end{bmatrix}\right)^*=(a^*c+b^*d)^*=ac^*+bd^*=c^*a+d^*b\\ =\left\langle \phi|\psi \right\rangle \end{gather*}

so ϕ,ψ\phi,\psi are conjugate symmetric

  1. linear
⟨u∣(a∣v⟩+b∣w⟩)=a⟨u∣v⟩+b⟨u∣w⟩\begin{gather*} \langle u|(a|v\rangle + b|w\rangle) = a\langle u|v\rangle + b\langle u|w\rangle \end{gather*}
  1. positive Definite
∀∣u⟩≠0,⟨u∣u⟩>0\begin{gather*} \forall |u\rangle \neq 0, \quad \langle u|u\rangle > 0 \end{gather*}