#### George Jones

Staff Emeritus

Science Advisor

Gold Member

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My project is simply to answer the very general request by the tutor:

"The set up of the Hamiltonian requires a proper set up (i.e. Hilbert space etc)". Can you imagine now, why I am am confused?

Well, if ##x## lives in the set of real numbers, then ##e^{ipx}## does not live in the standard Hilbert space of (equivalence classes of) square-integrable functions. Restricting ##x## (i.e., the domain of the functions) to the bounded interval ##\left[a , b\right]## seems to help, but, in this case, the momentum operator is unbounded (see page 569). Then, by page 525, a self-adjoint moment momentum operator can't have the Hilbert space all the square-integrable functions on ##\left[a , b\right]## as its domain. Done "properly", this stuff is quite subtle.How does one prove that an operator can transform a function, say f= e^ipx in a Hilbert space, even though it is evident from doing a simple operation with it on f?