I've posted similar questions on a different forum previously, but since I'm feeling a bit guilty about going on at the one person there who answers, I'll post these here. Hope it's ok if I don't follow the template since they're more conceptual than a standard problem. Any help appreciated: 1. Why do we have this duality between generators of inifintesimal changes and observables in quantum mechanics? Is it a postulate in it's own right or derived from something else (is it even always the case?)? How do we know the observable from a particular generator corrisponds to the observable we attatch to it: is it just guessing based on classical mechanics? 2. Am I right in saying if you have a particle spin up in the z-direction and you measure the spin along a perpendicular axis you will get a result of + or - h(bar)/2 with a 50/50 probability of each? Thanks.