If the emission of a photon from an excited state to the ground state occurs with a precise energy, then why is it that wavelengths that differ from lambda max have values on a fluorescence vs. wavelength graph?
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1$\begingroup$ In a low pressure gaseous state atomic transitions would be a very particular wavelengths. However in liquids and solids there are all sorts of vibrational, rotational, and intermolecular attractions that broaden line spectra in "bands", very broad peaks. $\endgroup$– MaxWApr 22, 2019 at 17:47
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There are two main reasons for what you're describing.
The first is the fundamental properties of the molecules involved, as mentioned in the comments. Even for isolated molecules, the transition is not always between the same vibrational states of the excited and ground state. Since the vibrational states are quantized, we would expect a series of very closely spaced emission lines, each representing a different combination of vibrational states.
If the molecules are in a solution or in liquid or solid phase, then the interactions of the emitting molecule with the molecules around it can also cause small changes in the energy of the emission as the electronic structure of the molecule is perturbed.
The general term for these phenomena is "line broadening," and most reference books about spectroscopy will have a discussion of the various mechanisms that lead to it.
The second reason for it is the error in measurements. Most spectrometers cannot filter wavelengths perfectly, so, for example, light of wavelength 590 nm will contribute to a signal at 589.5 and 591.5 nm, less as you get farther away. Sometimes the slit width is an adjustable parameter, but even at the narrowest setting, most instruments will not give extremely high resolution. As a result, a series of closely spaced lines will appear instead as a smooth curve.
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2$\begingroup$ There is a third reason involving Heisenberg uncertainty and lifetime of the excited state ... $\endgroup$– FeodoranApr 22, 2019 at 19:39
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2$\begingroup$ You make good points about why "line spectra" wouldn't have monochromatic peaks. // In UV-Vis range one thinks of "band spectra" as opposed to line spectra. // In x-ray spectra it is rare to observe "line spectra" broadening due to anything other than instrumental errors. $\endgroup$– MaxWApr 22, 2019 at 20:54
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$\begingroup$ @Fedoran, I would not add anything about the uncertainty principle as up today its widely accepted interpretation is debatable. $\endgroup$ Apr 24, 2019 at 13:13
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$\begingroup$ @user1420303 it's a fundamental part of quantum mechanics. the Interpretation of it is not the point here. There is debate about interpretation of whole of quantum mechanics, but that does mean it is not applied. $\endgroup$– FeodoranApr 25, 2019 at 6:13