# 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

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