I've been reading Thomas Jordan's Linear Operators for Quantum Mechanics, and I am stalled out at the bottom of page 40. He has just defined the projection operator E(x) by E(x)(f(y)) = {f(y) if y≤x, or 0 if y>x.} Then he defines dE(x) as E(x)-E(x-ε) for ε>0 but smaller than the gap between eigenvalues. Okay, so long as the eigenvalue spectrum is discrete...but then he just announces that ∫dE= E (limits -∞ to ∞) when it is supposed to be a discrete sum. I can sort of almost accept the handwave there, but then in the next line, he states ∫xdE = A, and that just doesn't make any sense to me. Sure, in a vague handwaving sense, maybe, but how does one define this so that it rigorously makes sense for operators with continuous eigenvalue spectrum?
Is there a particular reference you recommend? Please, not Von Neumann--I spent months on that book and found over 130 errata; it was the worst way imaginable to learn this math.
Relatedly, I've looked at several texts on the theory of integration (I keep running into "Stiejles" integrals references and Lesbegue integration, but haven't found a good book yet for learning the details that didn't make it more boring than doing taxes.) Any suggestions there? Thanks.
Is there a particular reference you recommend? Please, not Von Neumann--I spent months on that book and found over 130 errata; it was the worst way imaginable to learn this math.
Relatedly, I've looked at several texts on the theory of integration (I keep running into "Stiejles" integrals references and Lesbegue integration, but haven't found a good book yet for learning the details that didn't make it more boring than doing taxes.) Any suggestions there? Thanks.
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