Scalar product between two kets using eigenkets (QM)

I'm reading Dirac's "Principles of QM" and I reached the point (page 39), when he introduces the concept of expressing the scalar product of two kets using the complete set of eigenkets of an observable $$\xi$$. So, as I understood, he take into exam the simpler scenario where a ket can be expressed only using a complete and continuos set of eigenkets of $$\xi$$, excluding the hypothesis of some discrete eigenkets outside this interval. Therefore he writes two arbitrary kets as:

$$| X \rangle = \int |\xi'x \rangle d\xi'$$ $$| Y \rangle = \int |\xi''y \rangle d\xi''$$

He then says that it is possible to calculate the scalar product $$\langle X | Y \rangle$$ by taking the conjugate imaginary (his words) of the first:

$$\langle X | Y \rangle = \iint \langle \xi'x | \xi''y \rangle d\xi' d\xi''$$

He says that taking only $$\int \langle \xi'x | \xi''y \rangle d\xi''$$, the result vanishes for every eigenket except for the one associated with $$\xi' = \xi ''$$ for the orthogonality theorem. He concludes that for this reason also $$\langle X | Y \rangle$$ vanishes since the above conclusion holds for every $$\xi'$$. And then it comes the part I was struggling about: Dirac says that normally $$\langle X | Y \rangle$$ doesn't vanish, "so in general $$\langle \xi'x | \xi''y \rangle$$ must be infinitely great in such a way as to make $$\int \langle \xi'x | \xi''y \rangle d\xi' d\xi''$$ non-vanishing and finite" (page 39 in my edition). I really couldn't understand this part: I might think that this way to make this integral finite would be the Dirac's delta. But I don't understand how making $$\langle \xi'x | \xi''y \rangle$$ infinitely great could help fixing the integral...

The same situation occurs after, when Dirac's discuss the case of $$|X \rangle$$ and $$| Y \rangle$$ are equal: he, in fact, says that in general the result is $$\langle \xi'x | \xi'y \rangle$$

That's precisely what Dirac is saying here. Since $$\langle \xi' x|\xi'' y\rangle=0$$ for all $$\xi' \neq \xi''$$ (that is, almost everywhere), if $$\langle \xi' x|\xi'' y\rangle$$ is finite then $$\int \langle \xi' x|\xi''y\rangle \mathrm d\xi' \mathrm d\xi''=0$$. Therefore, if you want that integral not to vanish, you need $$\langle \xi' x| \xi'' y\rangle$$ to be infinite for $$\xi'= \xi''$$. In other words, it is zero for $$\xi'\neq\xi''$$ and infinite for $$\xi'=\xi''$$. This should sound a lot like the delta function $$\delta(\xi'-\xi'')$$ with which you are already familiar; the reason Dirac doesn't use that language is because he invented the idea, and it didn't have a name before he thought of it.
• but the passage I don't understand that if $\langle \xi'x | \xi''y \rangle$ is finite, then the integral too is finite beacause to a fixed quantitity you are adding other infinite zeros quantities, so why it is possible to equal it to zero? Mar 1, 2021 at 13:04
• @LucaMattioni It's zero except at a single point. If a function is zero except at a single point, then its integral is equal to zero. Loosely speaking, the area underneath a single point is zero. In the language of Riemann sums, as $\Delta x\rightarrow 0$ the Riemann sum converges to zero because only one interval has a nonzero value in it. In the language of Lebesgue integrals, the integrand is zero almost everywhere and so the integral vanishes. Mar 1, 2021 at 13:07