So, we apply Equation 3.28 (above) to our initial vector state, following the equation below, to get $|\psi(t)\rangle$.
What I obtained was the basically the same equation, except now we have a $|up\rangle$ vector on the $\cos$ term (as the unity matrix returns the same vector) and "-" the $\sin$ term with $|down\rangle$, as Pauli-$X$ will "flip" the spin state.
Now for the following part, the equation is as follows:
But I feel I need some clarification. Obviously here our vector state is different (the one calculated above), but we have this superposition in the term (South-west pointing). Just before this exercise is asked, we are given the following relationships:
For Pauli-$X$ and Pauli-$Y$ respectively. Because in the calculated vector term, we have $|up\rangle - |down\rangle$, does this mean we are to use the superposition in the top right $|North-East\rangle$, and this would be the syntax used?
- Main Question:
- What is the $|\psi(t)\rangle$ term, just to verify I am calculating properly. In the equation for $P_+$, we are shown the result is a $1/2$[1+term]. This obviously makes sense, as when calculating $P_-$, and adding the two terms, you should result in a "$1$". In my $|\psi(t)\rangle$ I am not getting this $1+$ term, I have a feeling there is something to do with an Euler identity here?.