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We discuss the interaction picture with respect to a Hamiltonian $H=H_0+V$ in this problem.

a) Prove that for any given operator $M$, the interaction picture operator $M_I(t)\equiv e^{iH_0 t} M e^{-iH_0 t}$ fulfills

$$\frac{d}{dt}M_I(t) = i[H_0,M_I(t)],$$ (6.3)

where $[A,B]\equiv AB-BA$ is the commutator of two operators.

b) Prove the rule $[AB,C] = A[B,C]+ [A,C]B$ for any three operators $A$, $B$, $C$.

c) Now consider the harmonic oscillator $H_0 = \omega a^{\dagger} a$. Use b) and the fundamental relation $[a,a^{\dagger}]=1$ to find the interaction picture operator $a_I(t)$ and $a_I^{\dagger}(t)$.

d) Now consider the time-dependent Hamiltonian of a harmonic oscillator in a damped, oscillating radiation field,

$$H(t) = H_0 + V(t),\quad H_0 = \omega a^{\dagger} a, \quad V(t)= V_0e^{-t/\tau} \left[e^{-i\omega_0t} a^{\dagger} + e^{i\omega_0t} a\right].$$ (6.4)

i) Use c) to calculate the transition probability from the ground state $\vert\rangle$ at $t=0$ to the first excited state $\vert 1\rangle$ after time $t\to \infty$,

$$P_{0\to 1}(t\to \infty) = \left\vert\int_{0}^{\infty} dt' \langle 1\vert V_I(t')\vert\rangle\right\vert^2.$$ (6.5)

Hint: Use $a\vert \rangle =0$ and $a^{\dagger}\vert \rangle =\vert 1\rangle$.

ii) Sketch $P_{0\to 1}(t\to \infty)$ as a function of the radiation frequency $\omega_0$.