# Operator relations

## Homework Statement

Consider the operator ##F_a(\hat{X}) =e^{ia \hat{p} / \hbar} \cdot F(\hat{X}) e^{-ia \hat{p} / \hbar}## where a is real.

Show that ##\frac{d}{d_a} F_a(\hat{X}) \cdot \psi = F'(x) \psi## evaluated at a=0.

And what is the interpretation of the operator e^{i \hat{p_a} / \hbar}?

## The Attempt at a Solution

By just starting taking the derivative I find that,
## \frac{d}{d_a} F_a(\hat{X}) = (\frac{i \hat{p}}{ \hbar}) e^{ia \hat{p} / \hbar} \cdot F(\hat{X}) e^{-ia \hat{p} / \hbar} - e^{ia \hat{p} / \hbar} \cdot F(\hat{X}) (\frac{i \hat{p}}{\hbar}) e^{-ia \hat{p} / \hbar} ##

Plugging a=0 gives,

## \frac{i}{h}[ \hat{p}, \hat{F}]##. But how do I take it from here to get the required form?

And finally, what is the interpretation of ##e^{i \hat{p_a} / \hbar}##?

Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
But how do I take it from here to get the required form?
Are you perhaps familiar with some commutation relation involving the argument of ##\hat F##?

And finally, what is the interpretation of ei^pa/ℏeipa^/ℏe^{i \hat{p_a} / \hbar}?

Are you perhaps familiar with some commutation relation involving the argument of ^FF^\hat F?
If the operators are hermitian I know they commute, but other than that it dosen't ring a bell.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
If the operators are hermitian I know they commute, but other than that it dosen't ring a bell.
This is not correct. Hermitian operators need not commute. Just look at, just off the top of my head for no apparent reason whatsoever, say ##\hat x## and ##\hat p## ...

This is not correct. Hermitian operators need not commute. Just look at, just off the top of my head for no apparent reason whatsoever, say ##\hat x## and ##\hat p## ...
Okey. Anyway I'm pretty stuck at the problem. I only get as far as in my attempt of solution.

Orodruin
Staff Emeritus
$$\frac{i}{\hbar}[\hat p, F(\hat x)]\psi,$$