# Taking Measurements of Quantities in QM

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?

You are free to choose which axis you make a measurement on and doing so will always yield and eigenvalue of the spin operator in that direction: you will always measure $\pm\hbar/2$ in whichever direction you choose to measure.
The reason is that the state vector of the particle exists in a superposition of states with various probabilites $$|\psi\rangle = a\left|\uparrow\right\rangle + b\left|\downarrow\right\rangle$$
When you make a measurement along any axis, you force the system into a specific state (say $\left|\uparrow\right\rangle$) with a projection along the axis you measure: the choice of axis is irrelevant. In other words, there is an isometry of the system that makes your choice of axis arbitrary.