Quantum Operator Approach

We now show how to carry out those reflections formally using quantum operators. The geometry gives us a clue. The first step is reflection with respect to $|ψ_{other}⟩$. This corresponds to changing the sign of the amplitude for all states that are perpendicular to $|ψ_{other}⟩$, which is just $|w⟩$ because we have defined $|ψ_{other}⟩$ that way. So, the first step makes the amplitude of $|w⟩$ negative and produces the superposition state $|s_1⟩$.

How do we find the part of $|s_0⟩$ that lies along $|w⟩$? To find that component, we express the state vector $|s_0⟩$ as the sum of a part $|s_0⟩_∥$ parallel to $|w⟩$ and a part $|s_0⟩_⊥$ perpendicular to $|w⟩$ and, hence, parallel to $|ψ_{other}⟩$:

$$\left|s_0\right\rangle=\left|s_0\right\rangle_{\|}+\left|s_0\right\rangle_{\perp}=|w\rangle \underbrace{\left\langle w \mid s_0\right\rangle}_{\begin{array}{c} \hspace{0.1cm}\text { project } \\ \text { along }|w\rangle \end{array}}+\left|\psi_{\text {other }}\right\rangle \underbrace{\left\langle\psi_{\text {other }} \mid s_0\right\rangle}_{\begin{array}{c} \text { project} \\ \text {along}\left|\psi_{\text {other }}\right\rangle \end{array}} .$$

To understand the above equation, recall that the Dirac bracket $\left\langle w \mid s_0\right\rangle$ expresses the projection of the state vector $\left|s_0\right\rangle$ along the direction of the state vector $|w\rangle$. Then, all we need to do is change the sign of the part that is parallel to our desired state. This gives $\left|s_1\right\rangle$ :

$$\left|s_1\right\rangle=-|w\rangle\left\langle w \mid s_0\right\rangle+\left|\psi_{\text {other }}\right\rangle\left\langle\psi_{\text {other }} \mid s_0\right\rangle.$$