B Question about this equation for the expectation value

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Hi all,

I found this notation of expectation values in a NMR text.

In class, I learned that expectation values are given by
$$<\hat{X}>=\int_{-\infty}^\infty\psi^*x\psi dx$$

why does this textbook divide by the integral of probability density ##\int \psi^*\psi dx##?

Screen Shot 2021-03-16 at 11.06.08 PM.png
 
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docnet said:
a NMR text.

Which textbook?

docnet said:
why does this textbook divide by the integral of probability density ##\int \psi^*\psi dx##?

Because it is not assuming that ##\psi## is normalized. The formulas you are used to seeing are based on the assumption that ##\psi## is normalized, which is a typical assumption in QM textbooks; but that just means the integral in the denominator is ##1## so the textbooks often leave it out. A strictly correct formula keeps it in.
 
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PeterDonis said:
Which textbook?

University of Cambridge published lectures of NMR under Introduction to NMR on this website

Link: http://www-keeler.ch.cam.ac.uk/lectures/

PeterDonis said:
Because it is not assuming that ##\psi## is normalized. The formulas you are used to seeing are based on the assumption that ##\psi## is normalized, which is a typical assumption in QM textbooks; but that just means the integral in the denominator is ##1## so the textbooks often leave it out. A strictly correct formula keeps it in.

Ah, that makes so much sense. Thank you I am truly glad for to have your knowledge as a resource for studying. :bow:

edited for grammar
 
docnet said:
Thank you I am truly glad for to have your knowledge as a resource for studying. :bow:

You're welcome! Glad I could help.
 
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