D.17 In­ner prod­uct for the ex­pec­ta­tion value

To see that $\left\langle\vphantom{A}\Psi\hspace{-\nulldelimiterspace}\hspace{.03em}\right.\!\left\vert\vphantom{\Psi}A\right\rangle$ works for get­ting the ex­pec­ta­tion value, just write $\Psi$ out in terms of the eigen­func­tions $\alpha_n$ of $A$:

$$\langle c_1 \alpha_1 + c_2 \alpha_2 + c_3 \alpha_3 + \ldots... ...rt c_1 \alpha_1 + c_2 \alpha_2 + c_3 \alpha_3 + \ldots\rangle$$

Now by the de­f­i­n­i­tion of eigen­func­tions $A\alpha_n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $a_n\alpha_n$ for every $n$, so you get

$$\langle c_1 \alpha_1 + c_2 \alpha_2 + c_3 \alpha_3 + \ldots... ...\alpha_1 + c_2 a_2 \alpha_2 + c_3 a_3 \alpha_3 + \ldots\rangle$$

Since eigen­func­tions are or­tho­nor­mal:

$$\langle\alpha_1\vert\alpha_1\rangle = 1 \quad \langle\alph... ... 1\quad \langle\alpha_3\vert\alpha_3\rangle = 1 \quad \ldots$$

$$\langle\alpha_1\vert\alpha_2\rangle =\langle\alpha_2\vert\a... ...ha_3\rangle =\langle\alpha_3\vert\alpha_2\rangle = \ldots = 0$$

So, mul­ti­ply­ing out pro­duces the de­sired re­sult:

$$\langle \Psi\vert A \Psi\rangle = \vert c_1\vert^2 a_1 + \... ... c_3\vert^2 a_3 + \ldots \equiv \left\langle{A}\right\rangle$$