Lebesgue Integral in QM: Tutorial & Reference

[teddy]

hi
can anyone give me a short tutorial on lebesgue integral in QM.
I am doing my first course in QM and got stuck in the mathematical formalism of Hilbert space .Even a good reference on the web will do.
my math back ground is only upto Reimann(spelling?) integral.
thanks

bye.

 

[mathman]

The main difference between Lebesgue and Riemann from a user perspective is that Lebesgue integration is more general, based on measure theory. For a physicist, I don't believe there is enough of a difference as far as understanding QM.

If you want to get some quick background try using google and search for "Lebesgue". You will get a lot of good hits.

 

[arivero]

As for Hilbert space L^2 stands, the whole point is that two functions that are different in a set of null measure are to be considered the same function.

This does not apply to distributions (ie delta functions and pure waves), which do not live in Hilbert space, although they are used as a "rigged" structure over it.

I believe that for most aplications the imaginery of Riemannian integration is enough, even if the rigour asks for Lebesgue.

 

[teddy]

thanks.

On going thru your replies and the web i got the following points :

1) in Lebesgue integ we divide the y-axis into small intervals instead of x-axis for integration.

2)the lebesgue integration of wave-fn for two physically equivalent systems is always same while riemann integration may differ due to 1)

3)For distribution functions we use Dirac measure instead of lebesgue.

am i missing something ?
is it more to it,speaking in physical terms ?

 

[Hurkyl]

Actually, whenever a function is Riemann integrable, the Lesbegue integral gives the same result. The reason the Lesbegue integral is "better" is because the Lesbegue integral works for a much vaster set of functions than the Riemann integral.

Hurkyl