Lebesgue Integral in QM: Tutorial & Reference

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In summary, Lebesgue integral is more general and is based on measure theory. It is used to calculate various quantities in Quantum Mechanics. Riemann integration is more rigorous and is used for more complex problems.
  • #1
teddy
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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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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 ?
 
  • #5
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
 

Related to Lebesgue Integral in QM: Tutorial & Reference

1. What is the Lebesgue integral?

The Lebesgue integral is a mathematical concept used in quantum mechanics to calculate the expected value of a physical quantity. It is a generalization of the Riemann integral and allows for more complex functions to be integrated.

2. How is the Lebesgue integral used in quantum mechanics?

The Lebesgue integral is used in quantum mechanics to calculate the expected value of physical observables, such as position or momentum. It is also used to calculate the probability of a particle being in a certain state or location.

3. Can you explain the steps to calculate a Lebesgue integral in quantum mechanics?

To calculate a Lebesgue integral in quantum mechanics, you first need to define the function you want to integrate over and the limits of integration. Then, you need to break the function into smaller intervals and calculate the Lebesgue measure for each interval. Finally, you multiply the function values by the Lebesgue measures and sum them to get the integral value.

4. What are the advantages of using the Lebesgue integral in quantum mechanics?

The Lebesgue integral allows for more complex functions to be integrated, making it more versatile than the traditional Riemann integral. It also provides a more precise calculation of expected values and probabilities, making it a valuable tool in quantum mechanics.

5. Are there any limitations to using the Lebesgue integral in quantum mechanics?

One limitation of the Lebesgue integral is that it can be more difficult to understand and calculate compared to the Riemann integral. It also requires a good understanding of measure theory and may not be suitable for simpler integration problems. Additionally, it may not always provide a unique solution for certain functions, leading to potential inaccuracies in calculations.

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