I Does there exist momentum-shift operator?

[PRB147]

As is well known there is translation operator in position space, such that.,
$$\exp(i\hat{p}a)x\exp(-i\hat{p}a)=x+a.$$
While in momentum space, can we have analog of the above mentioned translation operator? i.e., momentum shift operator?
$$\exp(-i\hat{x}q)p\exp(i\hat{x}q)=p+q.$$
If so, why many many quantum mechanics books never mention it?

 

[PeterDonis]

As is well known there is translation operator in position space

Yes, and this operator is the momentum operator.

While in momentum space, can we have analog of the above mentioned translation operator?

Yes, the translation operator in momentum space is the position operator.

If so, why many many quantum mechanics books never mention it?

I don't know what QM books you've read, but there are plenty that do mention the above.

 

[PeterDonis]

Btw, @PRB147, there is no need to use both double dollar signs and tex tags; just one will do. I have used magic mentor powers to fix your OP to remove the unnecessary tags.

 

[PRB147]

Yes, and this operator is the momentum operator.Yes, the translation operator in momentum space is the position operator.I don't know what QM books you've read, but there are plenty that do mention the above.

Thank you very much for your reply.