# Sakurai's approach to $\hat p$ as the generator of translations

Often times one simply postulates that the momentum operator is the generator of translations or follows the reasoning here, for example. I am working through Sakurai and Napolitano's Modern Quantum Mechanics 3rd Ed, and below is the part I want clarified (copied from 2nd Ed.) When he says, "We are led to speculate," I do not see why. The two expressions

$$\vec x\cdot\vec P,~~\quad\text{and}\qquad 1-i\hat{\vec K}\cdot d\vec x ~~,$$

seem quite different to me. I can look at these two expressions and, without the suggestion in the book, I am not drawn to speculate in this manner. What connection suggests this speculation?

• Dec 27, 2020 at 20:45
• Thanks. I take it as a good sign when my question about a passage in book has been asked before. Dec 27, 2020 at 20:52

I suspect what is confusing you is the apparent typo in (1.6.29) where the p is supposed to be P to yield the first of the equations (1.6.28): it is a type 2 generating function, after all, F(x,P), with $${\mathbf X}=\partial F/\partial {\mathbf P}$$, in our case $$= {\mathbf x} +d {\mathbf x}$$. So, as he reminds you, F is a machine consisting of the identity plus a piece that ads a small increment $$d {\mathbf x}$$ to $${\mathbf x}$$ and nobody else, as $${\mathbf P}=\partial F/\partial {\mathbf x}={\mathbf p}$$, so, basically, a gradient w.r.t. x.
This is meant to evoke the standard linear QM shift operator of Lagrange, a mere rewriting of Taylor's series around the constant dx, $$e^{id{\mathbf x}\cdot {\mathbf K}}f({\mathbf x})= f({\mathbf x}+d{\mathbf x})\\ \approx ( I +id{\mathbf x}\cdot {\mathbf K} )f({\mathbf x}) = f({\mathbf x})+ d{\mathbf x}\cdot \partial f/\partial {\mathbf x},$$ the increment term also involving a gradient.
The evocation and the name are not that necessary, really; if you were a Martian, the commutation relations of K with x dictating representation thereof with a gradient $$-i\nabla$$ would tell you everything you need. But the two S's remind you that you should expect to observe this shadow-dance between QM and classical mechanics, and K should strongly evoke the classical phase-space variable p, so it's good mental hygiene to call it momentum.
The minus sign you have is because S defines his translation on kets, which transform with a minus sign w.r.t. bras, and functions are basically bras, $$f(x) =\langle x | f\rangle$$. I could have done the whole thing with the minus sign and the kets, without reminding you that you are really discussing a bland Lagrange shift, in pompous language, after all.