Talk:Translation operator (quantum mechanics)

Editing this page!!

Hi all!
I wish to edit this wiki article. I went through this page, and found that some portions have scope for improvement. Moreover I was thinking of adding one or two subtopics to make this article more complete.
Here is a brief outline of my plan of editing.
1. The definition of Translation operator can be improved.
2. Next, the introduction part: I think this portion can be made more clear, more rigorous and more generalised. I noticed, that here the 'shift' or the 'translation' λ is treated as a scalar, right from the beginning, considering the problem as a one dimensional one. I want to show that even if we start from a vector, we can make use of Baker Campbell Hausdorff formula to factorise the exponential into pieces containing only the components of the vector since the higher order terms in the Baker Campbell Hausdorff formula become zero as the components of momentum commute among themselves.
3. Translation operators form an Abelian group. I want to add some materials about this fact under a heading "Translation Group". I think that would be worth mentioning in this article.
4. I have planned to improve the materials under Translational invariant Hamiltonian. Actually I have planned to write it in a somewhat different way. I want to state that in general the Hamiltonian of a system does not commute with the translation operator. i.e. in general Hamiltonians do not have translational invariance. However in some spacial cases, like in free particle it might have translation invariance. And there is actually another special case, where the Hamiltonian of the system will commute with the translation operator, when the potential is periodic. Unlike the previous case here the Translation will no longer be continuous, rather it becomes discrete.
5. Here it comes, another new subheading, Discrete Translation. I want to add some materials here stating what happens whenever the potential is periodic. I have also planned to show how Bloch theorem follows from here.
This is how I have planned to proceed so far.
Suggestions, discussions and questions on my plan are always welcome. I will try my best to explain them.Sumeruhazra (talk) 18:27, 16 October 2014 (UTC)https://wiki.riteme.site/wiki/User:Sumeruhazra[reply]

Hello! You've created a lot of really great content here! Here is a suggestion. (Or I'm happy to try editing it myself if you prefer.)

I think the approach currently used to define the translation operators is not the best way to do it, especially to provide an entry point for people who are just beginning and don't really understand how momentum works in quantum mechanics. My preferred approach is from these lecture notes (starting section 3).

Step 1: A translation operator is defined as "an operator which shifts particles and fields by a certain amount in a certain direction"
Step 2: Momentum is defined in terms of infinitesimal translation operators.

The article currently uses the reverse approach: Translation operators are defined as an exponential of momentum operators. But I think that "an operator which shifts particles and fields by a certain amount in a certain direction" is something which is very intuitive for beginners, whereas the quantum mechanical momentum operator is not. For example, I had already taken 4 quantum mechanics courses before [x,p]=iħ made intuitive sense to me, and I know that this is typical.

You're welcome to agree or disagree of course :-D --Steve (talk) 17:55, 10 November 2014 (UTC)[reply]

Suggestion: Get rid of the inline LaTeX (convert it to HTML). It (sadly) doesn't display well on many computers. YohanN7 (talk) 10:34, 11 November 2014 (UTC)[reply]

Momentum and Infinitesimal Translations

Hi Steve!
Thanks for your valuable suggestions and the reference lecture notes.
Actually under the sub-heading definition, I wanted to write the compact mathematical form of translation operator, which I have derived in the previous section, Momentum as Generator of Translation. In this derivation I started with looking for a form of infinitesimally translated state, and eventually came up with $\nabla$ operator. Then I used the common expression for the momentum operator in position basis and replaced $\nabla$ operator with momentum operator. Then I claimed that the momentum operator is thus the generator of translation.
I could have, and I think I should have gone the other way round. As you nicely pointed out, I could go on with $\nabla$ operator and finally, since it is a generator of translation, in analogy with classical mechanics (and with proper units and dimensions) I could have defined (derived the expression for) the momentum operator in terms of translation.
I will make those edits as soon as possible. But once we have shown all these, we can still keep the compact form of the mathematical definition, right? Sumeruhazra (talk) 11:29, 13 November 2014 (UTC)Sumeruhazra[reply]

Obviously the article should have the equation

{\hat {T}}(\mathbf {x} )=\exp \left(-{\frac {i\mathbf {x} \cdot \mathbf {\hat {p}} }{\hbar }}\right)

but I'm suggesting that we call this equation the "definition of p" and not the "definition of T".

A lot of readers know that p=-iħ∇ only because of rote memorization. It would be nice if the article could answer the question "Why is p equal to -iħ∇?"

Just my opinion :-D --Steve (talk) 12:51, 13 November 2014 (UTC)[reply]

Yes, I agree! It will be a nice intuitive way of defining ${\hat {p}}$ . Thanks again, for your suggestion. Sumeruhazra (talk) 18:44, 13 November 2014 (UTC)Sumeruhazra[reply]

OK, I did some rewriting and reorganization along those lines a few days ago. I just want to say: I tried to avoid deleting good content and good references, but maybe I did it anyway by accident. Sorry! It wasn't intentional. You can put things back if you disagree with a change.

And thanks for already cleaning up my math formatting errors. :-D --Steve (talk) 16:43, 21 November 2014 (UTC)[reply]