Why do molecules absorb "exact" frequencies when given a "continuous" beam of light? This question is in the context of performing simple rotational spectroscopy on molecules, but could be extended for more general cases.
In rotational (microwave or IR) spectroscopy of molecules, the energy levels are discrete and given by
$$E_J=BJ(J+1)$$
where $B$ is the rotational constant, and $J$ is the angular momentum quantum number for the molecule.
Given the selection rule is $\Delta J=\pm 1$, one can show that we get allowed transitions between levels, with energy spacings at
$$\Delta E=2B,\; 4B,\; 6B,\; ...$$
and so on, which correspond to peaks in the spectrum.
My confusion is that due to the discrete nature of this formalism, the molecule needs to absorb a photon of energy exactly $2nB$, where $n$ is a positive integer.  However if we feed in a continuous spectrum of light, mathematically speaking one gets rather worried when we are required to select a discrete value!  Yet we see peaks, so light is definitely being absorbed!
My thoughts so far on explanations include the fact that the peaks aren't infinitely thin - i.e. they have some thickness to them.  Is this from inherent quantum mechanical uncertainty in energy of the photons that need to be absorbed (maybe from the Heisenberg relation for $\Delta E$ and $\Delta t$)?  Are there Doppler broadening effects occurring?  I don't quite understand why these "errors" or "uncertainties" would be present if the molecule has the above "exact" spacings however.
 A: 
My thoughts so far on explanations include the fact that the peaks aren't infinitely thin - i.e. they have some thickness to them.  Is this from inherent quantum mechanical uncertainty in energy of the photons that need to be absorbed (maybe from the Heisenberg relation for $\Delta E$ and $\Delta t$)?

Yes.

Are there Doppler broadening effects occurring?

Potentially, also yes.

Spectral lines are never exact delta-function spikes of zero width $-$ they always have a nonzero spectral bandwidth.
The sources of this bandwidth are generally divided into two different classes:

*

*Inhomogeneous broadening occurs when you have an ensemble of molecules absorbing at different frequencies, such as the Doppler effects you mentioned (so, there will be a thermal distribution of velocities, and each will produce a different Doppler shift), as well as e.g. different Zeeman shifts from inhomogeneities in the magnetic field, and so on.


*On the other hand, even if all the molecules are in identical conditions, there are also homogeneous-broadening effects. The main culprit here is, as you point out, the $\Delta E$-$\Delta t$ uncertainty relationship, where $\Delta t$ is given by the radiative decay lifetime of the excited state: frequencies any closer together than $1/\Delta t$ require times at least as long as $\Delta t$ to distinguish, and in that time any initial absorption has time to decay by re-emission and start absorbing again.
A: Indeed the absorption line is not infinitely thin. If it were, the absorption would take infinitely long!
There are several sources of line broadening. Lines are broadened by the effect of lifetime. The transition takes a finite time and therefore a finite number of wavelengths. This results in broadening. Also the molecule feels effects from its environment. In a gas it will collide, which this will affect the resonance and the lifetime, and doppler broadening due to thermal or other motion. If the molecule is in a solid it may experience variation in its environment. In a liquid it may experience motional narrowing.
See https://en.wikipedia.org/wiki/Spectral_line_shape
