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I'm going through problems in Quantum Field Theory for the Gifted Amateur, and have been trying to understand a problem on the forced quantum oscillator [$L = \frac{1}{2}\dot{x}(t)^2-\frac{1}{2}m\omega^2x(t)^2+f(t)x(t)$ ] and response functions.

The response function is

$$ \langle\psi(t)|\hat{x}(t)|\psi(t)\rangle = \int_{-\infty}^\infty \mathrm{d}t'\chi(t-t')f(t') $$

I want to show, using the interaction representation, that to first order in the force function $f_I(t)$

$$ |\psi_I(t)\rangle = |0\rangle + i\int_{-\infty}^t \mathrm{d}t'f_I(t')\hat{x}_I(t')|0\rangle $$

Here is what I've done so far:

I started by taylor expanding the interacting ket: $$ |\psi_I(t)\rangle = e^{i \hat{H_0}(t)t}|\psi(t)\rangle = |\psi(t)\rangle + i \hat{H_0}(t)t|\psi(t)\rangle+O(H_0^2) $$

but I am confused about how to relate the wave function to the ground state, and how to use the information I have about the response function. When you have an expression for $|psi\rangle$ there is a procedure for finding the expectation value. I don't know how to go the other way and around and extract a ket from the response function.

I also note that I can convert the response function to the interaction picture and it will have the same value, and that I can change f(t) to the interaction picture $f_I(t) = e^{i H_0 t}f(t)e^{-i H_0 t} =f(t) + O(H_0^2)$ since $H_0$ and f(t) commute.

Related: linear response for a simple harmonic oscillator

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The insight I was missing was that the $e^{i \hat{H_0}(t)} t|\psi(t)\rangle$ can be thought of more abstractly as $U(t)|psi(0)\rangle $, where $U(t) = e^{-i \hat{H}(t)t}. When the ket starts out in the ground state, doing the taylor expansion and subsitution of the interaction picture for f and x works.

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