A.18 Inner product for the expectation value

To see that $\langle \Psi\vert A \vert\Psi\rangle$ works for getting the expectation value, just write $\Psi$ out in terms of the eigenfunctions $\alpha_n$ of $A$:

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

Now by the definition of eigenfunctions $A\alpha_n = a_n\alpha_n$ for every $n$, so you get

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

Since eigenfunctions are orthonormal:

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

$$\langle\alpha_1\vert\alpha_2\rangle =\langle\alpha_2\vert\... ..._3\rangle =\langle\alpha_3\vert\alpha_2\rangle = \ldots = 0$$

So, multiplying out produces the desired result:

$$\langle \Psi\vert A \Psi\rangle = \vert c_1\vert^2 a_1 +... ...a_2 + \vert c_3\vert^2 a_3 + \ldots \equiv \langle A\rangle$$