Question from Dirac's Principles of QM

[Jimmy Snyder]

I have the fourth edition of Dirac's Principles of QM. I have a question concerning the equation in the middle of page 102 (between eqns 64 and 65 in section 25.)
$$\lim_{\delta x \rightarrow 0}(De^{i\gamma} - 1)/\delta x = \lim_{\delta x \rightarrow 0}(D - 1 + i \gamma)/\delta x$$
If you make the substitution $e^{i\gamma} \simeq 1 + i \gamma$ then it seems to me you should get
$$\lim_{\delta x \rightarrow 0}(De^{i\gamma} - 1)/\delta x = \lim_{\delta x \rightarrow 0}(D - 1 + i \gamma D)/\delta x$$
What am I missing?

 

[Dick]

You aren't missing anything. Is it a typo?

 

[Jimmy Snyder]

Is it a typo?

I doubt it. The conclusion he correctly draws from this equation is that the displacement operator is indeterminate by an arbitrary additive pure imaginary number:
$$ia_x = \lim_{\delta x \rightarrow 0}i \gamma/\delta x$$
If it were a typo, then the the right hand side would also need to be operated on by D and then would not be a number. Is there some reason that $D(i\gamma) = i \gamma$?

 

[Jimmy Snyder]

Never mind, I finally figured it out. Thanks for your help. The solution is that
$$\lim_{\delta x \rightarrow 0}D = 1$$