Let us assume that we have a time dependent Hamiltonian, precisely a Hamiltonian of a harmonic oscillator with time dependent frequency. \begin{equation} \hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega(t)^2\hat{x}^2 \end{equation} I am working entirely in Schrodinger picture. I have figured the time evolution operator for this. Now, assuming at $t=-\infty$, the state of the system is the ground state i.e. $|0_{-\infty}\rangle$, I am asked to calculate the expectation value $$\langle0_{-\infty}|\hat{H}(+\infty)|0_{-\infty}\rangle$$ I want to know, what is the physical significance of this expectation value, if any. I have found this expectation value by writing $\hat{H}(+\infty)$ in terms of $\hat{H}(-\infty)$ as they are the bogoliubov trasnforms of each other $$\hat{H}(+\infty)=\hat{U}\hat{H}(-\infty)\hat{U}^\dagger$$ where $\hat{U}$ is the time evolution operator.


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