Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before.  How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try?

If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious.  Why should this be the case, or at least, why might we expect this to be the case? 

Then there is the issue of measurement which is even more difficult to motivate.  In the usual formulation of quantum mechanics, we assume that, given a state $|\psi \rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\langle a|\psi \rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$).  How would you convince your students that this had to be the case?

I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be.  So, what do you guys think?  Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results?

Note that, at some point, you will have to make reference to experiment.  After all, this is the reason why we needed to develop quantum mechanics.  In principle, we could just say "The Born Rule is true because its experimentally verified.", but I find this particularly unsatisfying.  I think we can do better.  Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory.  You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment.

Edit:  To clarify, I will try to make use of a much simpler example.  In an ideal gas, if you fix the volume, then the temperature is proportional to pressure.  So we may ask "Why?".  You could say "Well, because experiment.", or alternatively you could say "It is a trivial corollary of the ideal gas law.".  If you choose the latter, you can then ask why that is true.  Once again, you can just say "Because experiment." or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example).  The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment.  And in this particular case, the objective is to do this with quantum mechanics.

I am late to this party here, but I can maybe advertize something pretty close to a derivation of quantum mechanics from pairing classical mechanics with its natural mathematical context, namely with Lie theory. I haven't had a chance yet to try the following on first-year students, but I am pretty confident that with just a tad more pedagogical guidance thrown in as need be, the following should make for a rather satisfactory motivation for any student with a little bit of mathematical/theoretical physics inclination.
For more along the following lines see at nLab:quantization.

Quantization of course was and is motivated by experiment, hence by observation of the observable universe: it just so happens that quantum mechanics and quantum field theory correctly account for experimental observations, where classical mechanics and classical field theory gives no answer or incorrect answers. A historically important example is the phenomenon called the “ultraviolet catastrophe”, a paradox predicted by classical statistical mechanics which is not observed in nature, and which is corrected by quantum mechanics.
But one may also ask, independently of experimental input, if there are good formal mathematical reasons and motivations to pass from classical mechanics to quantum mechanics. Could one have been led to quantum mechanics by just pondering the mathematical formalism of classical mechanics? (Hence more precisely: is there a natural Synthetic Quantum Field Theory?)
The following spells out an argument to this extent. It will work for readers with a background in modern mathematics, notably in Lie theory, and with an understanding of the formalization of classical/prequantum mechanics in terms of symplectic geometry.
So to briefly recall, a system of classical mechanics/prequantum mechanics is a phase space, formalized as a symplectic manifold $(X,ω)$. A symplectic manifold is in particular a Poisson manifold, which means that the algebra of functions on phase space $X$, hence the algebra of classical observables, is canonically equipped with a compatible Lie bracket: the Poisson bracket. This Lie bracket is what controls dynamics in classical mechanics. For instance if $H\in C^{∞}(X)$ is the function on phase space which is interpreted as assigning to each configuration of the system its energy – the Hamiltonian function – then the Poisson bracket with $H$ yields the infinitesimal time evolution of the system: the differential equation famous as Hamilton's equations.
To take notice of here is the infinitesimal nature of the Poisson bracket. Generally, whenever one has a Lie algebra $\mathfrak{g}$, then it is to be regarded as the infinitesimal approximation to a globally defined object, the corresponding Lie group (or generally smooth group) $G$. One also says that $G$ is a Lie integration of $\mathfrak{g}$ and that $\mathfrak{g}$ is the Lie differentiation of $G$.
Therefore a natural question to ask is: Since the observables in classical mechanics form a Lie algebra under Poisson bracket, what then is the corresponding Lie group?
The answer to this is of course “well known” in the literature, in the sense that there are relevant monographs which state the answer. But, maybe surprisingly, the answer to this question is not (at time of this writing) a widely advertized fact that would have found its way into the basic educational textbooks. The answer is that this Lie group which integrates the Poisson bracket is the “quantomorphism group”, an object that seamlessly leads over to the quantum mechanics of the system.
Before we say this in more detail, we need a brief technical aside: of course Lie integration is not quite unique. There may be different global Lie group objects with the same Lie algebra.
The simplest example of this is already the one of central importance for the issue of quantization, namely the Lie integration of the abelian line Lie algebra $\mathbb{R}$. This has essentially two different Lie groups associated with it: the simply connected translation group, which is just $\mathbb{R}$ itself again, equipped with its canonical additive abelian group structure, and the discrete quotient of this by the group of integers, which is the circle group
$$
  U(1) = \mathbb{R}/\mathbb{Z}
  \,.
$$
Notice that it is the discrete and hence “quantized” nature of the integers that makes the real line become a circle here. This is not entirely a coincidence of terminology, but can be traced back to be at the heart of what is “quantized” about quantum mechanics.
Namely one finds that the Poisson bracket Lie algebra $\mathfrak{poiss}(X,ω)$ of the classical observables on phase space is (for X a connected manifold) a Lie algebra extension of the Lie algebra $\mathfrak{ham}(X)$ of Hamiltonian vector fields on $X$ by the line Lie algebra:
$$
  \mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X)
  \,.
$$
This means that under Lie integration the Poisson bracket turns into a central extension of the group of Hamiltonian symplectomorphisms of $(X,ω)$. And either it is the fairly trivial non-compact extension by $\mathbb{R}$, or it is the interesting central extension by the circle group $U(1)$. For this non-trivial Lie integration to exist, $(X,ω)$ needs to satisfy a quantization condition which says that it admits a prequantum line bundle. If so, then this $U(1)$-central extension of the group $Ham(X,\omega)$ of Hamiltonian symplectomorphisms exists and is called… the quantomorphism group $QuantMorph(X,\omega)$:
$$
  U(1) \longrightarrow QuantMorph(X,\omega) \longrightarrow Ham(X,\omega)
  \,.
$$
While important, for some reason this group is not very well known. Which is striking, because there is a small subgroup of it which is famous in quantum mechanics: the Heisenberg group.
More exactly, whenever $(X,\omega)$ itself has a compatible group structure, notably if $(X,\omega)$ is just a symplectic vector space (regarded as a group under addition of vectors), then we may ask for the subgroup of the quantomorphism group which covers the (left) action of phase space $(X,\omega)$ on itself. This is the corresponding Heisenberg group $Heis(X,\omega)$, which in turn is a $U(1)$-central extension of the group $X$ itself:
$$
  U(1) \longrightarrow Heis(X,\omega) \longrightarrow X
  \,.
$$
At this point it is worthwhile to pause for a second and note how the hallmark of quantum mechanics has appeared as if out of nowhere from just applying Lie integration to the Lie algebraic structures in classical mechanics:
if we think of Lie integrating $\mathbb{R}$ to the interesting circle group $U(1)$ instead of to the uninteresting translation group $\mathbb{R}$, then the name of its canonical basis element 1∈ℝ is canonically ”i”, the imaginary unit. Therefore one often writes the above central extension instead as follows:
$$
  i \mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X,\omega)
$$
in order to amplify this. But now consider the simple special case where $(X,\omega)=(\mathbb{R}^{2},dp∧dq)$ is the 2-dimensional symplectic vector space which is for instance the phase space of the particle propagating on the line. Then a canonical set of generators for the corresponding Poisson bracket Lie algebra consists of the linear functions p and q of classical mechanics textbook fame, together with the constant function. Under the above Lie theoretic identification, this constant function is the canonical basis element of $i\mathbb{R}$, hence purely Lie theoretically it is to be called ”i”.
With this notation then the Poisson bracket, written in the form that makes its Lie integration manifest, indeed reads
$$
  [q,p] = i
  \,.
$$
Since the choice of basis element of $i\mathbb{R}$ is arbitrary, we may rescale here the i by any non-vanishing real number without changing this statement. If we write ”ℏ” for this element, then the Poisson bracket instead reads
$$
  [q,p] = i \hbar
  \,.
$$
This is of course the hallmark equation for quantum physics, if we interpret ℏ here indeed as Planck's constant. We see it arise here by nothing but considering the non-trivial (the interesting, the non-simply connected) Lie integration of the Poisson bracket.
This is only the beginning of the story of quantization, naturally understood and indeed “derived” from applying Lie theory to classical mechanics. From here the story continues. It is called the story of geometric quantization. We close this motivation section here by some brief outlook.
The quantomorphism group which is the non-trivial Lie integration of the Poisson bracket is naturally constructed as follows: given the symplectic form $ω$, it is natural to ask if it is the curvature 2-form of a $U(1)$-principal connection $∇$ on complex line bundle $L$ over $X$ (this is directly analogous to Dirac charge quantization when instead of a symplectic form on phase space we consider the the field strength 2-form of electromagnetism on spacetime). If so, such a connection $(L,∇)$ is called a prequantum line bundle of the phase space $(X,ω)$. The quantomorphism group is simply the automorphism group of the prequantum line bundle, covering diffeomorphisms of the phase space (the Hamiltonian symplectomorphisms mentioned above).
As such, the quantomorphism group naturally acts on the space of sections of $L$. Such a section is like a wavefunction, instead that it depends on all of phase space, instead of just on the “canonical coordinates”. For purely abstract mathematical reasons (which we won’t discuss here, but see at motivic quantization for more) it is indeed natural to choose a “polarization” of phase space into canonical coordinates and canonical momenta and consider only those sections of the prequantum line bundle which depend on just the former. These are the actual wavefunctions of quantum mechanics, hence the quantum states. And the subgroup of the quantomorphism group which preserves these polarized sections is the group of exponentiated quantum observables. For instance in the simple case mentioned before where $(X,ω)$ is the 2-dimensional symplectic vector space, this is the Heisenberg group with its famous action by multiplication and differentiation operators on the space of complex-valued functions on the real line.

For more along these lines see at nLab:quantization.
