Eigen functions of momentum operator

esornep

Hello,
I am going through the book Introduction to QM by D.Griffiths. In the third chapter the book says the eigen functions of the momentum operator do not belong to the Hilbert space ...... But the only condition that a vector belongs to the Hilbert vector space is
that the integral gives a value between intervals ...... and the function we get is also giving a finite value ...... i mean the constant remains ...... please help and correct if i am wrong ...... thanks

genericusrnme

It's because in quantum mechanics we require that the functions be square integrable, the integral of e^ipx over all space does not exist so it is not a member of the space of square integrable functions.

I'd advise dropping Griffiths' book though, it's a very bad introduction to qunatum mechanics imo.
Try Landau and Lifshitz' book or even Zettili's book.

esornep

the solution to the eigen function of the momentum operator is A*exp(ihx/p) where p is the eigen value which is satisfying the condition for the L2(a,b) space ... i.e., integral of the modulus of the complex number square which turns out to be A*2 .... which is less than infinity ....... so y does it not fall in the L2(a,b) group .... ???

fzero

I think you mean ##A \exp i px/\hbar##. In any case, this is square-integrable if the domain is compact. It is not square-integrable if the domain is ##\mathbb{R}##.