Operators in quantum mechanics

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Discussion Overview

The discussion revolves around the concept of convergence in the context of operators in quantum mechanics, specifically focusing on the propagator of a one-dimensional box. Participants explore the mathematical implications of operator convergence, including weak convergence and its relevance to quantum mechanics.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant asks about the meaning of an operator converging or diverging, indicating a lack of clarity on the topic.
  • Another participant provides context by mentioning the propagator of a one-dimensional box and its representation as an exponential operator power series.
  • A participant suggests that convergence may refer to some norm or weak convergence, expressing uncertainty about the specific norm used in this context.
  • There is a mention of the complexity of convergence in path integrals, which may require advanced mathematical concepts such as Hida distributions.
  • One participant shares a mathematical expression related to the propagator and reiterates the idea of weak convergence, emphasizing its importance in distribution theory.
  • A participant expresses confusion about weak convergence and seeks recommendations for mathematical resources to better understand quantum mechanics.
  • Another participant suggests reading specific chapters from texts on quantum mechanics to grasp the concept of weak convergence in inner product spaces.
  • The discussion includes explanations of weak and strong convergence, noting their equivalence in normal Hilbert spaces but highlighting exceptions in distribution theory.
  • One participant advocates for the importance of understanding weak convergence for physicists and applied mathematicians, recommending a book that covers this topic.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding weak convergence and its implications. While some provide explanations and resources, there is no consensus on the specific norms or mathematical frameworks applicable to the discussion.

Contextual Notes

The discussion touches on advanced mathematical concepts that may not be fully understood by all participants, indicating a potential gap in foundational knowledge necessary for grasping the topic of operator convergence in quantum mechanics.

Mk7492
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Hi,
We know the convergence of a series but what does it mean to say that "an operator converges or diverges"?
 
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Can you give more context?
 
It's about the 'Propogator' of 1-D box ( -L/2 to +L/2) which is an exponential operator power series.
 
$ U = e^{{ \frac{i \hbar t} {2m}} {\frac{d^2} {dx^2} }} $
bhobba said:
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.
 
Mk7492 said:
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
 
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