Projection

The projection of vv onto a vector uu is

Πu(v)=⟨v∣u⟩⟨u∣u⟩u\begin{gather*} \Pi_u(v)=\frac{\left\langle v|u \right\rangle}{\left\langle u|u \right\rangle}u \end{gather*}

Example

∣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*} Πψ(ϕ)=[c∗d∗][ab][a∗b∗][ab][ab]=c∗a+d∗ba∗a+b∗b[ab]\begin{gather*} \Pi_\psi(\phi)=\frac{\begin{bmatrix}c^* & d^*\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}}{\begin{bmatrix}a^* & b^*\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}}\begin{bmatrix}a\\b\end{bmatrix}\\ =\frac{c^*a+d^*b}{a^*a+b^*b}\begin{bmatrix}a\\b\end{bmatrix} \end{gather*}

This is similar to the dot product, but it gives a vector

ALL THESE CAN ALSO PROJECT

See Inner product See Wave function See Projector