What is the probability of measurement in QM, dependent on time?

Question

Consider a QM system with an observable $A$ and orthonormal eigenbasis $\{|n\rangle,n=0,1,2,\ldots\}$. Then we know that if the system is in some state $|\psi\rangle$ and we measure $A$, the probability of finding an eigenstate $|n\rangle$ is $|\langle n|\psi\rangle|^2$ and that the probability of finding a non-eigenstate $|\phi\rangle$ is zero.

How does this translate to the time-dependent setting?

In particular, what is the probability of finding a state $|\phi,t\rangle$ at time $t$ if the system evolves according to some other state $|\psi,t\rangle$? Does $|\phi,0\rangle$ have to be an eigenstate of $A$ in order for this probability to be positive? Does the time evolution of $|\phi,t\rangle$ have to be given by the Schrödinger equation? I'm really confused by this and would very much appreciate help.

Answer 1

In quantum mechanics, each state of a system is always represented as a vector (a ket) in the Hilbert space of all possible states. This space has, per definition, a scalar product and thus a geometry associated with it and consequentially there is the notion of orthogonality. This allows for the probability of measuring any state $|\phi\rangle$ when the system is in state $|\psi\rangle$ to be defined as the square of the orthogonal projection $|\langle \phi | \psi \rangle|^2$, which is the scalar product. This definition is universal and does always apply. In particular, none of the states has to be an eigenstate of any operator.

Now to answer your question explicitly, by what I just explained, the probability of finding a state $|\phi,t\rangle$ at time $t$ if the system is in the state $|\psi,t\rangle$ is $$ p = |\langle \phi,t | \psi,t \rangle |^2~, $$ and there are no special conditions which $|\phi,0\rangle$ must fulfill. Furthermore, it will be $p > 0$, if $|\psi,t\rangle$ and $|\phi,t\rangle$ are not orthogonal, by definition of orthogonality.

Remark: For the definition of the measurement probability, there are no eigenstates needed, as stated above. If one wants to conduct an actual physical measurement, though, some observable will be measured, which is represented by a self-adjoint operator, which has, of course, an eigenbasis and the only possible results of the measurement are the eigenvalues. If the system evolves over time, the eigenstates and eigenvalues as well as the operator may be constant or change themselves, depending on the system.