Emuberable and Demumerable sets

1. Oct 18, 2014

sweet springs

Hi. I have a question about numbers of basis in quamutum mechanics space.

Hamiltonian of harmonic oscillator is observable and have countably infinite sets |En>s

Together with position or momentum basis identity equation is,

$$|state>=\int|x><x|state>dx=\int|p><p|state>dp=\Sigma_n\ |E_n><E_n|state>$$

The same state is expressed as both enumerable and denumerable infinite sets.

Is it OK? Denumerable sets should be interpreted correctle as enumerable or vice versa?

2. Oct 18, 2014

bhobba

Sure - that's fine.

To really understand it though you need to investigate Rigged Hilbert Spaces, but that requires considerable background in analysis:
http://physics.lamar.edu/rafa/webdis.pdf [Broken]

At he beginning level simply accept you can have both.

Thanks
Bill

Last edited by a moderator: May 7, 2017
3. Oct 19, 2014

mathman

Enumerable is essentially a mathermatics term meaning that the set is ordered. Denumerable means countbly infinite, but not necessarily ordered.

4. Oct 19, 2014

bhobba

You are of course correct.

But reading between the lines I am pretty sure he is asking about the existence of continuous basis in separable spaces which is a bit strange until you are used to it.

Thanks
Bill

5. Oct 20, 2014

dextercioby

But of course, continuous bases do not exist in separable spaces.

6. Oct 20, 2014

aleazk

7. Oct 20, 2014

sweet springs

Thanks ALL for your advise. I wil learn it.

8. Oct 20, 2014

bhobba

Again of course. The RHS formalism is sneaky in making it look like it does by allowing a continuous spectrum.

To the OP its tied up with the so called Nuclear Spectral Theorem which is very difficult to prove - in fact I haven't even seen a proof - the one in the standard text by Gelfland that I studied ages ago is in fact incorrect - don't you hate that sort of thing

However the following details an approach useful in QM based on so called Hilbert-Schmidt Riggings:
http://mathserver.neu.edu/~king_chris/GenEf.pdf

Thanks
Bill

Last edited: Oct 20, 2014
9. Oct 21, 2014

dextercioby

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