Why are most QM Operator defined as Identity minus Generator? I am currently rehearsing my lectures in quantum mechanics for the exam. I recognized that there is a pattern for different types of operators such as: Rotation operator, Time evolution operator and so on. The way we got it presented in our course is that they all look the following way:
$R(d\Phi k) = I-\frac{i}{\hbar}d\Phi J_z$
And if you do this $N$ times for $N \rightarrow \infty $ we can write as
$R(\Phi k) = e^{-iJ_z\Phi/\hbar}$.
So far I get that we take the Original and subtract from it depending on the angle. But I can't figure out how to get to the primary equation.
 A: The identity operator is the same as doing nothing. If you want to construct an operator that is 'small' it better be close to the identity. The goal is to construct a 'big' (read: finite) operator by composing (infinitely) many 'small' operators. We want to do this because these small operators are easy to study and we can deduce many properties of the big group just by looking at the small group.
You can't derive the first equation because it is a definition. When you insert an infinitessimal parameter in an operator$^\dagger$ you will get the identity operator + another infinitessimal $\times$ a matrix. That matrix is defined as the generator. In mathematics this is simply the definition but in physics we like to slap a factor $\frac i\hbar$ in front. This way the generators become Hermitian and we can interpret them as physical observables.
$\dagger$ this assumes that your operator becomes the identity when the parameter it depends on is zero, i.e. $R(\theta):\ R(0)=I$. This is a fundamental assumption in Lie groups.
A: This is described well enough in textbooks such as Modern Quantum Mechanics by J.J. Sakurai. Essentially, you choose this form of the infinitesimal operators such that the transformation is unitary, additive, reversible in time and reduced to identity in the limit where the parameter by which you are changing the system goes to 0.
You can find details for this procedure in regards to translation within section 1.6, and for rotations in section 3.1. Time evolution is discussed in section 2.1.
