I notice that in Quantum Mechanics when extending an infinitesimal operation to a finite one, we should end with the exponential. For example: (rf. Sakurai, Modern Quantum Mechanics)(adsbygoogle = window.adsbygoogle || []).push({});

$$

D(\boldsymbol{\hat n}, d\phi) = 1 - \frac{i}{\hbar} ( \boldsymbol {J \cdot \hat n})d\phi

$$

This is the infinitesimal version of the rotation operator, where ##\boldsymbol{\hat n}## is a unit vector and ##J## is the angular-momentum operator. Now the "finite" version of the rotation operator, that generates a rotation by a finite angle (meaning non-infinitesimal angle) is

$$

\lim_{N \to \infty}[1 - \frac{i}{\hbar} (\boldsymbol{J \cdot \hat n})\frac{\phi}{N}]^N = exp[- \frac{i}{\hbar} (\boldsymbol{J \cdot \hat n})d\phi]

$$

My question here is how does one know that the correct procedure for going to infinitesimal to finite operations is to take that limit above?

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# I Extending an infinitesimal operator

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