Valid Intuition? - Why observables are represented by eigenstates/eigenvalues So I've been frustrated with the usual presentation of the operator formalism being presented as an axiom, and have been after a more intuitive explanation. Would the following intuition be considered valid?
Suppose we know nothing besides the fact that states of a physical system are represented by complex state vectors. And suppose we know of a particular continuous physical degree of freedom, x.
We can then ask a question, "are there any quantum states which are fundamentally unaffected by translations in x?". We find that infinitesimal translations may be represented by the operator:
$T(dx) = 1 - dx\cdot i\hbar\frac{d}{dx}$
So we find those quantum states which are unaffected by translations are precisely those which only get multiplied by a constant when operated on by $i\hbar\frac{d}{dx}$, i.e. an eigenstate of  $i\hbar\frac{d}{dx}$. I.e. states of definite momentum.
So then, this line of reasoning seems to makes clear why we'd be interested in eigenstates/eigenvalues. We're looking for states which have a continuous symmetry, and therefore are definite states of the conjugate momentum, which is conserved by Noether's theorem.
Question: Does this line of reasoning suffice to derive the operator formalism or have I misunderstood something?
 A: I can just offer my personal view on how the operator formalism was discovered. The most important point (which I don't remember if it was the first step) is the analogy to mechanical vibration systems. Any bound (stable) conserved mechanical system can be described by a stiffness matrix and a mass matrix, which together form a frequency squared matrix, that is symmetric positive definite (or can be chosen so). That is, if you are not confined by prior knowledge about the configuration variables, one can always choose those variables so as to guarantee a symmetric positive definite operator. Any such operator always has a complete set of eigenstates that allow the representation of an arbitrary initial state of the system by superposition. Since initial condition is somehow synonymous to measurement, it was probably quite intuitive to consider a symmetric positive definite operator as an observable, even at the times of the first QM theorists. Mathematically, it can then be proven, that it is possible to enrich the system (frequency) matrix by other spd. operators, so as to form a complete set, i.e. resolve ambiguity (degeneracy, if any) in the frequency eigenstates.
The experimental physicists had discovered that atoms and molecules can vibrate and radiate. But it couldn't be the vibrating states themselves that radiate, because that quickly leads to charges collapsing into the nuclei. Moreover the radiated frequencies made the most sense, when one assumed that they were related to frequency differences between vibrating states rather than their absolute levels (especially the hydrogen line series). Hence, there must be the possibility that "pure" vibrating states exist, that do not radiate. Since any real vibrational state evolution involves zero-crossings, it is incompatible with electromagnetism, that real vibrating states do not radiate EM waves. Therefore, the necessity arises that the states are complex, and therefore, the concept of symmetric positive definite operators must be extended to hermitian operators, if we accept that only transitions/interferences between states can cause radiation.
The concept you propose, i.e. invariance under infinitesimally small translations, is only applicable in case of spatial homogeneity, which is nothing but representing empty space. You can even extend that to finite translations, which are given by the exponential map
$$\hat T(\delta x_i)=\exp(\delta x_i \frac{d}{dx_i})$$
and which allow to see the nature of translation invariance more clearly. Beware that what you have defined is not a translation, because the imaginary unit does not belong there. My $\hat T$ is nothing but a clever way of writing Taylor's theorem, and the imaginary unit is not part of Taylor's.
But the wave function of an electron in an atom is not translation invariant (because it is bound by the nucleus). The only thing you can relate to translation invariance is the quantum behavior of matter waves in empty space. And that has been discovered by de-Broglie as the relations $p=h/\lambda$ and $E=h\nu$. Again, what might have guided the way to hermitian operators (besides the sheer will to satisfy the de-Broglie relations) was the awareness, that an electron beam can exist, which, although behaving like a wave in some contexts, does not possess locations with zero density/zero crossings, but which is completely homogeneous. Again this leads to assuming a complex wave function, and forces the attention to hermitian operators instead of symmetric positive definite operators as the observables.
But in the end, I am pretty sure, that the first quantum guys did a lot of creative guessing, which is naturally difficult to reconstruct in hindsight, because there were also so many erratic ideas involved, which were later discarded (think of Bohr's atomic model). What I don't believe is that the first quantum physicists followed a strictly axiomatic path to find something yet unknown about nature.
A: Generally we say observables are represented by Hermetian operator - this is a mathematical condition and not directly a physical one. Their eigenvalues represent the concievable values that this observable can take. Or one could say, the observable values.
One good question to ask is why do we require observables to be represented by Hermetian operators? The standard textbooks are silent on this. It is because physical laws preserve probability. Mathematically, again, this means they are unitary. Now, observables are the generators of change. This means in particular that we need to look at the Lie algebra. And the Lie algebra of the unitary group is Hermetian (roughly speaking - there is a mathematical convention here which is at odds with the physical convention) and this is the basic reason why we represent observables by Hermetian operators.
