- #1

bhobba

Mentor

- 10,196

- 3,374

In discussing stuff in another thread I used the standard Dirac notion expanding a state in position eigenvectors namely |u> = ∫f(x) |x>. By definition f(x) is the wave-function. I omitted the dx which is my bad but the following question was posed which I think deserved a complete answer. It was also off at a tangent to the main threads topic so really required a separate thread.

Its the Dirac bra-ket notation.

You are not the only one to be confused by it even though its in standard use in QM - so was the great Von-Neumann. He was scathing about it in his famous book on QM which with your mathematical background may be the presentation of QM you are most familiar with. Nowadays it can be made rigorous using the Rigged Hilbert Space (RHS) formalism:

https://en.wikipedia.org/wiki/Rigged_Hilbert_space

|x> is the eigenvector of the position operator. It does not exist in Hilbert Space - but in a RHS.

Heuristically here is what's happening. Suppose you have a very fine grid of positions with eigenvectors |xi> where each lies in some range Δx. We assume that positions lie in a large but finite range. These can be found in a finite dimensional Hilbert space and any element can be represented by ∑f(xi) |xi>. We then divide f(xi) and |xi> by √Δx so we have the new equation as ∑f(xi) |xi> Δx using these new f(xi) and |xi>. Then we let the range of positions go to infinity at the same time as Δx goes to zero. Then, heuristically |xi> goes to |x> where x is an exact position. f(xi) goes to a function f(x) and the sum goes to an integral ∫f(x) |x> dx Then we have this new continuous basis |x> - each is of infinite length and do not belong to a Hilbert space, but instead belong to this strange beast a RHS.

This is all simply heuristics. I have attached a document giving the full rigor - but its no easy read nor short.

Interestingly with your background in probability it also finds application in White Noise Theory (it called by its other name there Generalized Functionals and Schwartz Space - but it just is an example of a RHS):

http://www.asiapacific-mathnews.com/04/0404/0010_0013.pdf

Also interestingly since Von-Neumans scathing attack on it mathematicians were not lying down - it took the efforts of 3 great mathematicians - Gelfland, Schwartz and Grothendieck to sort it out.

Thanks

Bill

I'm confused. What is the state |x> supposed to be? Does ∫ f(x) |x> = ∫ f(x)dx•|x> or ∫ f(x) |x>dx ?

Its the Dirac bra-ket notation.

You are not the only one to be confused by it even though its in standard use in QM - so was the great Von-Neumann. He was scathing about it in his famous book on QM which with your mathematical background may be the presentation of QM you are most familiar with. Nowadays it can be made rigorous using the Rigged Hilbert Space (RHS) formalism:

https://en.wikipedia.org/wiki/Rigged_Hilbert_space

|x> is the eigenvector of the position operator. It does not exist in Hilbert Space - but in a RHS.

Heuristically here is what's happening. Suppose you have a very fine grid of positions with eigenvectors |xi> where each lies in some range Δx. We assume that positions lie in a large but finite range. These can be found in a finite dimensional Hilbert space and any element can be represented by ∑f(xi) |xi>. We then divide f(xi) and |xi> by √Δx so we have the new equation as ∑f(xi) |xi> Δx using these new f(xi) and |xi>. Then we let the range of positions go to infinity at the same time as Δx goes to zero. Then, heuristically |xi> goes to |x> where x is an exact position. f(xi) goes to a function f(x) and the sum goes to an integral ∫f(x) |x> dx Then we have this new continuous basis |x> - each is of infinite length and do not belong to a Hilbert space, but instead belong to this strange beast a RHS.

This is all simply heuristics. I have attached a document giving the full rigor - but its no easy read nor short.

Interestingly with your background in probability it also finds application in White Noise Theory (it called by its other name there Generalized Functionals and Schwartz Space - but it just is an example of a RHS):

http://www.asiapacific-mathnews.com/04/0404/0010_0013.pdf

Also interestingly since Von-Neumans scathing attack on it mathematicians were not lying down - it took the efforts of 3 great mathematicians - Gelfland, Schwartz and Grothendieck to sort it out.

Thanks

Bill

#### Attachments

Last edited: