= Solution
{c}
For one electromagnetic <normal mode>, $a$ and $a^\dagger$ are the <annihilation operator> and <creation operator>, with <commutator> $[a,a^\dagger]=I$. Their action on normalized <number states> is
$$
a|n\rangle=\sqrt n\,|n-1\rangle,\qquad
a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle.
$$
They remove or add one <photon>, whose <energy> is $\hbar\omega$. The <vacuum state> $|0\rangle$ is normalized and obeys $a|0\rangle=0$. It has no <photons>, but retains <zero-point energy> $\hbar\omega/2$ for the <quantum harmonic oscillator> <Hamiltonian> $H=\hbar\omega(a^\dagger a+1/2)$. The <coherent state> $|\alpha\rangle=D(\alpha)|0\rangle$ is a displaced <vacuum state>.
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