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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 $\langle \xi'x | \xi'y \rangle$

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I might think that this way to make this integral finite would be the Dirac's delta.

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.

Dirac's explanation of the idea is of course pretty loose and hand-wavy, and has been made mathematically precise only more recently.

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  • $\begingroup$ 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? $\endgroup$ – Luca Mattioni Mar 1 at 13:04
  • $\begingroup$ @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. $\endgroup$ – J. Murray Mar 1 at 13:07
  • $\begingroup$ you are perfectly right, my fault XD. Thank you for your time! $\endgroup$ – Luca Mattioni Mar 1 at 14:21

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