In Sakurai:(adsbygoogle = window.adsbygoogle || []).push({});

The state ket for an arbitrary physical state can be expanded in terms of the |x'>

[tex]|\alpha>=\int dx'^3 |\mathbf{x'}><\mathbf{x'}|\alpha>[/tex]

(where the |x'> are the eigenkets of the position operator, [itex]\hat{x}|x'>=x'|x'>[/itex]).

(Sakurai 1.6.4, p. 42)

My question is about how this integral is mathematically defined - since the integrand is not a complex number, the Lebesgue integral as defined in analysis1 doesn't work directly. I thought about extending the defintion to kets, defining an ordering |a> <= |b> iff every <x'|a> <= <x'|b>, is this how it works? I also noticed the equation looks very much like a Fourier series (except the basis is uncountable) - it's the same idea, an expansion in a basis set... does this work formally?

Since the explanation may be very involved, a reference to a book/section would be more than sufficient. Thanks!

(secondary question - is there a better way to TeX the above equation?)

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# Q. about continuous spectra

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