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I'm currently trying to self-study quantum mechanics from Cohen-Tannoudji, and am struggling to wrap my head around the following claim. In section C-4 he says that

[representations of operators] depend on two indices and can therefore be arranged in a 'square' matrix having a countable or continuous infinity of rows and columns.

I didn't study math, but my understanding of countable vs uncountable infinities was basically that countable infinities are those which can be ordered. For example, it is well defined to talk about the integer which comes after some integer $x$, but it is not well defined to talk about the real number which comes after some real number $y$, since the real numbers are not countable.

Basically my question is, if we're dealing with an uncountable (or 'continuous') infinity, how is it possible to have a matrix representation? Doesn't a matrix representation necessarily depend on having some well-defined order which the rows and columns must adhere to? How might you construct such an ordering, for an uncountably infinite collection of rows/columns?

Qmechanic
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FrankC
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  • You may talk about matrices $\langle x|{\cal O}|y\rangle$, for x and y real. – Cosmas Zachos Sep 27 '22 at 00:16
  • Do x and y need to be real? I understand it's common to use a basis of energy eigenkets which are necessarily real since H is Hermitian, but I didn't think that was necessarily a requirement? Also accepting at face value that x and y must be real, I still have no idea what that matrix would look like for a continuous O, my question is basically asking for some intuition/explanation. – FrankC Sep 27 '22 at 00:20
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    Space coordinates such as x and y are real. Read on for context and the intuition that comes with it. The book is bursting with illustrations for this sort of thing. – Cosmas Zachos Sep 27 '22 at 00:27
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    Here a matrix is understood as a well defined map $S\times S \ni (x,y)\mapsto M_{xy}\in \mathbb{C}$. It does not matter if $S$ is ordered or not. This structure is completely enough as it stands for formal computations. – Valter Moretti Sep 27 '22 at 05:34
  • Uncountable infinities can be ordered (the real numbers are ordered!); they can in fact be well-ordered if you assume the axiom of choice, but this is all irrelevant to the question at hand. The image of an $\infty$-by-$\infty$ matrix is meant to be evocative, not rigorous, even for the case of countable infinities. It's meant to make you feel comfortable with infinite dimensional vector spaces, so that you can bring some of your intuition for finite-dimensional spaces forward to infinite-dimensional spaces. You will learn new techniques for dealing with infinite-dimensional spaces. – march Sep 27 '22 at 18:31
  • Take a look in the ADDENDUM of my answer here : Hermiticity of Momentum Operator (matrix) Represented in Position Basis. The use of Dirac $:\delta-$function for the "matrix representation" of the momentum and position operators may be help you. – Frobenius Sep 27 '22 at 18:41

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