= Solution
With $y=x-x_0$ and <oscillator length> $a_0=\sqrt{\hbar/(m\omega)}$, the first two <harmonic oscillator eigenfunctions> and <energies> are
$$
\psi_0=\frac{e^{-y^2/(2a_0^2)}}{\pi^{1/4}\sqrt{a_0}},\qquad
\psi_1=\sqrt2\frac{y}{a_0}\psi_0,\qquad
\boxed{E_0=\frac{\hbar\omega}{2},\quad E_1=\frac{3\hbar\omega}{2}.}
$$
The <ground state> is even and nodeless; the first excited state is odd with one <node of a wavefunction>. Their rms widths are $a_0/\sqrt2$ and $\sqrt{3/2}\,a_0$. Also, $m\omega^2a_0^2\sim\hbar\omega$ gives the characteristic extent directly.
For the <shaken harmonic trap>, expanding about the unshifted <Hamiltonian> gives
$$
H'(t)=-m\omega^2\varepsilon x\cos\Omega t+\frac12m\omega^2\varepsilon^2\cos^2\Omega t\,I
\quad(0\le t\le T).
$$
The scalar term gives only a common phase and no transitions. In <time-dependent perturbation theory>, initially $c_0=1$, $c_{n\ne0}=0$ and the oscillatory factor for $0\to1$ is $e^{i(E_1-E_0)t/\hbar}=e^{i\omega t}$. Thus
$$
c_1(T)=-\frac{\varepsilon m\omega^2}{i\hbar}\langle1|x|0\rangle\int_0^T e^{i\omega t}\cos\Omega t\,dt.
$$
The odd operator $x$ makes transitions to even $n>0$ vanish by <parity>. In fact the <first-order selection rule for a linearly driven oscillator> is stronger: since $x=a_0(a+a^\dagger)/\sqrt2$,
$$
\langle n|x|0\rangle=\frac{a_0}{\sqrt2}\delta_{n1}.
$$
\b[All excited $n\ne1$ amplitudes vanish at first order], including odd $n>1$. This assertion does not apply to the surviving $c_0$ or to higher perturbative orders.
Put $\delta=\omega-\Omega$, $\Sigma=\omega+\Omega$. The exact first-order time integral is
$$
I(T)=e^{i\delta T/2}\frac{\sin(\delta T/2)}{\delta}
+e^{i\Sigma T/2}\frac{\sin(\Sigma T/2)}{\Sigma}.
$$
Near resonance, the second term oscillates rapidly at approximately $2\omega$ and is bounded by $1/\Sigma$, whereas the first grows as $T/2$ on resonance. Thus $\omega T\gg1$ justifies the <rotating-wave approximation>, $\cos\Omega t\mapsto e^{-i\Omega t}/2$. At zeros of the resonant term, the omitted small contribution can dominate the residual; the approximation has no uniform relative-error bound there.
Using the requested order-one approximation $\langle1|x|0\rangle\simeq a_0$ and $m\omega^2a_0/\hbar=\omega/a_0$ gives the <square-pulse resonance profile>
$$
\boxed{p_1\simeq\left(\frac{\varepsilon\omega}{a_0}\right)^2\frac{\sin^2(\delta T/2)}{\delta^2},\qquad
p_1(\omega)\simeq\left(\frac{\varepsilon\omega T}{2a_0}\right)^2.}
$$
The exact <harmonic oscillator> matrix element instead gives \b[half this probability]. The finite central maximum is at $\Omega=\omega$; first zeros are at $|\Omega-\omega|=2\pi/T$, with smaller side lobes and a $1/\delta^2$ envelope. Increasing $T$ narrows the peak as $1/T$ and increases its height as $T^2$, as long as <weak resonant driving> keeps $p_1\ll1$. Small displacement alone does not ensure this for arbitrarily long $T$; near resonance one also needs $\varepsilon\omega T/a_0\ll1$.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2013/paper-1-aqp2013-oscillator-resonance.png]
{title=Even and odd oscillator wavefunctions and the finite-duration sinc-squared excitation profile}
{source=paper-1-aqp2013-oscillator-resonance.py}
Back to article page
