I have an inner product ## \langle \alpha|f| \beta \rangle## where ##f## is an operator that is a function of position ##x## operator (1D). According to the book I read (and I'm sure in any other book as well), that inner product can be written in position representation as ## \int u^*_{\alpha}(x) f(x) u_{\beta}(x) dx ##. This book doesn't discuss how to get this integral expression but I can guess how it's done, namely sandwiching the operator ##f## by two completeness relations of position eigenkets. In doing so we get ##f## sandwiched by a ket and a bra of ##x## and since ##f## is a function of ##x## operator, it can be(adsbygoogle = window.adsbygoogle || []).push({}); expanded into power seriesof ##x##. This way we will have a dirac delta ##\delta (x'-x") ## which will simplify to the integral above. But my question is what if ##f## is, for example, ##1/(1+x) ## whose power series only converges between x=-1 and 1? What will happen with the integral of all space in the expectation value of ##f(x)##? Or is my argument expanding f into power series wrong?

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# Expectation value of 1/(1+x)

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