Consider a normed quantum state $|\Psi(0)\rangle$ and let $\Delta_\Psi E\equiv \langle (\hat{H}-\langle\hat{H}\rangle_\Psi)^2\rangle_\Psi^{1/2}$ be the energy uncertainty.
a. Compute the probability that the system is still in state $|\Psi(0)\rangle$ after a time $\delta t$. (Meaning $|\langle\Psi(\delta t)|\Psi(0)\rangle|^2$.
b. Determine the energy uncertainty of a one-dimensional harmonic oscillator in the state $|n\rangle$, whereas $\hat{H}=\hbar \omega(\hat{N}+1/2)$ and $\hat{N}|n\rangle=n|n\rangle$.
c. Let $|\Psi(0)\rangle =1/\sqrt{2}(i|1\rangle-|3\rangle)$. What's the energy uncertainty of that state at time t=0 and when is the probability of it being still in the initial state after $\delta t$ less than 0.9?
Hello everyone,
I've been doing exercises from my workbook and I'm kinda stuck at this one (and I tried looking for a exercises manual of that book, but apparently there is none).
As far as a. goes I have no idea how to start there. I'm guessing this is a different expression for the Born Rule, but I never really understood how to apply on a particular problem (but I get the theory of the rule a bit, but apparently not enough).
As for b: I just thought of taking the expressions for H and N and plug it into the term for the energy uncertainty (I will leave out the hats):
$\Delta_\Psi E=\langle(\hbar\omega(N+1/2)-\langle(\hbar\omega(N+1/2))^2\rangle_\Psi\rangle_\Psi^{1/2}=\langle(\hbar\omega(N+1/2)-\hbar\omega(n+1/2))^2\rangle_\Psi^{1/2}=\langle(\hbar\omega N-\hbar\omega n\rangle_\Psi^{1/2}=\langle\hbar^2 \omega^2N^2-2\hbar^2\omega^2Nn+\hbar^2\omega^2n^2\rangle_\Psi^{1/2}=0.$
Is that right?
And I'm also lost on c. there.