# Operators in quantum mechanics

1. Mar 11, 2015

### Mk7492

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

2. Mar 11, 2015

### Staff: Mentor

Can you give more context?

3. Mar 11, 2015

### Mk7492

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

4. Mar 11, 2015

### Staff: Mentor

5. Mar 11, 2015

### Mk7492

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

6. Mar 11, 2015

### Staff: Mentor

Read Chapter 1 - Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/Quantum-Mechanics-Modern-Development-Edition/dp/9814578584

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/The-Theory-Distributions-Nontechnical-Introduction/dp/0521558905

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

Last edited by a moderator: May 7, 2017