I have a quick question relating to Annihilation and Creation operators, and in taking observables in general.
Let's say, for instance, that I prepare a particle so that I consider the projection of Spin of a particle, say an electron, onto the 3rd axis. Now all of the eigenvlaues that I get from using operators to find expectstion values will be "with respect to" the 3rd axis, correct?
My question is, if I have an ensemble of identically-prepared particles, and am considering the 3rd axis as a point of reference, what would happen if I took a measurement of a quantity with respect to another axis? Would the wave-function just collapse, and I'd be left with a null-ket? Or would I be able to arbitrarily change the axis of projection?