Quantum experiments in the pre-industrial era Could an 18th century or earlier scientist have come across phenomena which require quantum theories to explain them, given the apparatus available at the time?
I'll choose 1805 as the cut-off date, because that's when Maudslay's micrometer revolutionised precision in instruments.
 A: *

*Christian Huygens discovered in 1690 polarized light - this is the first quantum effect ever observed. The transformation behavior of rays of completely polarized light was ﬁrst described by Etienne-Louis Malus 1809 (who coined the name ”polarization”), and that of partially polarized light by George Stokes 1852. 
In modern terminology, the behavior described by Malus (resp. Stokes) is identical to that of a qubit in a pure (resp. mixed) state. Stokes 1852 paper contains all modern quantum phenomena for a single qubit, discussed in classical terms.
(For details, see my lecture http://arnold-neumaier.at/ms/optslides.pdf)

The transverse nature of polarization was discovered by Augustin Fresnel 1866, and the description in terms of (what is now called) the Bloch sphere by Henri Poincare 1892. In modern terminology, polarization is a manifestation of the massless spin 1 nature of the unitary representation of the Poincare group deﬁning photons.


*The second oldest observed quantum effect are spectral lines, apparently first discussed in 1802 by William Hyde Wollaston. (For the history of spectroscopy, see http://www.spectroscopyonline.com/spectroscopy/article/articleDetail.jsp?id=381944)
Both phenomena require quantum physics for their explanation (though polarization can also be explained by a statistical version of classical electrodynamics).

But, of course, before 1900 nobody considered these to be quantum effects.
Spectral lines were first described as a quantum effect in 1913 by Niels Bohr.
Polarization was first described as a quantum effect in 1930 by Norbert Wiener.
A: Very little in the field of chemistry makes any sense in detail without quantum mechanics. If you want "phenomena which require quantum theories to explain them", just look around ... why is wood brown and leaves green and iodine yellow? What chemicals are stable versus unstable, why do different elements react different ways, why do salt crystals form cubes while ice forms hexagons?
None of these questions can be answered in a correct and consistent way except in the framework of quantum chemistry (and its consequences like orbital hybridization, delocalized electrons, resonance stabilization, Pauli exclusion principle, electron orbitals, the relation of light absorption and emission to electronic structure, etc. etc.)
These sorts of things are not usually discussed as motivation for quantum mechanics because it is a long and difficult road from the basic principles of quantum mechanics to explaining facts in chemistry like why ice is hexagonal. One might imagine that there is an alternative explanation of all the facts of chemistry that does not require quantum mechanics ... well, there isn't, but there is no particularly easy and pedagogical way to convince people of that. By comparison, there is a relatively simple path from basic principles of quantum mechanics to the two-slit experiment. 
For the exact same reason, the conceptual breakthroughs in quantum mechanics did not historically arise from the attempt to explain why iodine is yellow. The path is too indirect. It required years of work AFTER quantum mechanics was established to understand how quantum mechanics was the only sensible explanation of almost everything in chemistry.
A: using an analogy between optics and PHYSICS ROWAN HAMILTON, could have discovered quantum mechanics by means of the Eikonal Equation $ (\nabla S)^{2}=N $
however there were no empirical evidences of Hamilton's discovery by this era so Hamilton rejected the idea of a 'wave mechanics' it was XIX century
it is a pity, science could have advance more than 50 years if Hamilton and othere had taken their ideas seriosly :(
http://quantum-history.mpiwg-berlin.mpg.de/eLibrary/fileserverPub/Joas-Lehner_2009_Optical-mechanical.pdf/V1_Joas-Lehner_2009_Optical-mechanical.pdf
for more
