How to find the probability of a state from non-degenerate spectral decomposition

Question

I have just begun learning the topics of time evolution in quantum mechanics, and I'm having trouble understanding how to calculate the probabilities of certain eigenvalues of an operator at a later time.

In my textbook, there is this example: For a quantum system with a Hamiltonian $H$ and operator $A$ with spectral decomposition given by $$A = \sum_n a_n \lvert \psi_n \rangle \langle \psi_n \rvert \, .$$ The system is initially in the eigenstate $|\psi\rangle$ of $A$, with eigenvalue $a_n$. $H$ and $A$ do not commute. After time θ, the probability of obtaining the eigenvalue $a_n$ is given by $$w_{nn}(\theta) \approx 1 - (\Delta E)_n^2 \, \theta^2 \, .$$ I am completely clueless as to how this equation is derived. My attempt at deriving this would be to begin with $|\langle\psi_n|\psi_n\rangle|^2$ as this gives the probability of the system being at the same initial state, but I'm unsure if this approach is correct for time-evolving systems. I also noticed the θ in the equation, which leads me to think the time evolution operator $U$ is involved $$U = \exp(-i H t / \hbar) \, .$$ May someone provide an explanation or clue as to how measurements are calculated in this case?

Answer 1

Well, how we would find the probability of being in the same initial state (say $|\psi_n\rangle$) is as follows: $$P(|\psi_n\rangle, t) = |\langle\psi_n| \Psi \rangle|^2$$ where $|\Psi\rangle = e^{-i H t}|\psi_n \rangle$. Physically, how you should think about this expression is that I am finding the projection of my quantum state at time $t$ on my initial state which yields the probability of measuring my initial state. Now, we can do a simple calculation: $$P(|\psi_n\rangle, t) = |\langle\psi_n|e^{-i H t}|\psi_n \rangle|^2 \approx |\langle\psi_n| 1- i H t - \frac{1}{2} H^2 t^2|\psi_n \rangle|^2 \approx | 1 - i \langle E \rangle - \frac{1}{2} \langle E^2\rangle t^2|^2 $$ $$\approx 1 - \langle E^2 \rangle t^2 - \langle E\rangle ^2 t^2 \approx 1 - \langle \Delta E^2\rangle t^2 $$ Note that this is a short time approximation where "short" is set by the characteristic energy scale.