Operators in quantum mechanics

[Mk7492]

Hi,
We know the convergence of a series but what does it mean to say that "an operator converges or diverges"?

 

[DrClaude]

Can you give more context?

 

[Mk7492]

It's about the 'Propogator' of 1-D box ( -L/2 to +L/2) which is an exponential operator power series.

 

[bhobba]

It means it converges in some norm, or maybe by weak convergence:
http://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_space)

I don't know the norm used here, my suspicion is its the weak convergence of distribution theory which would be something like On|u> converges weakly to O|u> for all |u>.

However convergence in path integrals is a difficult issue requiring some pretty advanced math (eg Hida Distributions):
http://www.mathnet.or.kr/mathnet/kms_tex/99937.pdf

Thanks
Bill

 

[Mk7492]

$ U = e^{{ \frac{i \hbar t} {2m}} {\frac{d^2} {dx^2} }} $

It means it converges in some norm, or maybe by weak convergence:
http://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_space)

I don't know the norm used here, my suspicion is its the weak convergence of distribution theory which would be something like On|u> converges weakly to O|u> for all |u>.

However convergence in path integrals is a difficult issue requiring some pretty advanced math (eg Hida Distributions):
http://www.mathnet.or.kr/mathnet/kms_tex/99937.pdf

Thanks
Bill

I don't know what weak convergence is. I was reading R.Shankar's "Principle's of Quantum mechanics " & i encountered the above question. Can you suggest me the math i need to know in order to understand QM.
Thankyou for the reply.

 

[bhobba]

I don't know what weak convergence is. I was reading R.Shankar's "Principle's of Quantum mechanics " & i encountered the above question. Can you suggest me the math i need to know in order to understand QM.

Read Chapter 1 - Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

In the context of an inner product space the idea of weak convergence is fairly simple.

A sequence |an> converges to |a> if for all <b| in the space <b|an> converges to <b|a>.

Strong convergence is convergence in the norm ie |an> converges to |a> if ||an - a|| converges to zero.

Weak convergence of operators is On converges to O if On|a> converges in the weak sense to to O|a> for all |a>

For normal Hilbert spaces they are equivalent but there are spaces of great practical interest where its not the case. In particular that applies to what's called distribution theory. Its so useful it should really be part of the armoury of any physicist or applied mathematician. The best book I know to learn it from is:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

It for example makes the theory of Fourier transforms a snap. It makes use of weak convergence to ensure, for example, the Fourier transform of a convergent sequence itself always converges..

Thanks
Bill