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## 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}##?

Thanks in advance!