I'm reading through some lecture notes for QM in a subsection about stationary states where the definition of orthonormality involving a kronecker delta [tex] \int_{-\infty}^{\infty}\psi_n(x) \ \psi_m^*(x)dx=\delta_{m,n}[/tex]and the formula for some wavefunction that is a superposition of energy eigenstates with expanded coefficients (I think I said that all right? Maybe?) [tex]\psi(x)= \sum_{n=1}^{\infty}b_n\psi_n(x) [/tex] are related to solve for b_n, which turns out to be: [tex]b_n=\sum_{n=1}^{\infty}\psi^*_n(x)\psi(x)[/tex] which is substituted back into the second formula to give: [tex] \psi(x) = \sum_{n=1}^{\infty} \bigg( \int_{-\infty}^{\infty} \psi^*_n(x') \psi(x') \bigg) dx' \ \psi_n(x)= \int_{-\infty}^{\infty} \bigg( \sum_{n=1}^{\infty} \psi^*_n(x') \psi_n(x) \bigg) dx' \ \psi(x') [/tex]Where the order of integration and summation is switched and x' is another variable, not a derivative-It is mentioned that the above formula is of the form: [tex]f(x) = \int_{-\infty}^{\infty}K(x',x)f(x')dx[/tex] and is supposed to hold for any function x.(adsbygoogle = window.adsbygoogle || []).push({});

What I don't understand is that it says that,

"It is intuitively clear that K(x',x) must vanish for x' =/= x, for otherwise we could cook up a contradiction by choosing a peculiar function f(x)";

What exactly is even having the point of two different variables x' and x if it vanishes for all instances where they aren't equal? Isn't this basically saying, since you're multiplying the integrand by zero in all instances where x' =/= x, that the only nonzero result you have is when x' = x? What is the purpose of having the two separate variables? Is it to "preserve" one of them so one of the functions-that based on only f(x) and not at all on f(x')-is un-integrated and is only evaluated later, or something? Or is it more related to the orthonormality bit of this, so it only shows up in one exact circumstance?

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# Stat. States, Orthonorm. expansion coefficients etc qn?

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