Does every quantum system have coherent states? I am taking a course on Quantum Mechanics and last class we saw the coherent states of the harmonic oscillator, namely states $ | \lambda \rangle $ such that $$ a | \lambda \rangle = \lambda | \lambda \rangle $$ and we saw how they represent the classical situation when $ \lambda \to \infty $.
The professor also told us that there are coherent states for the hydrogen atom as well, and that these represent (in some limit I guess) states in which the electron describes classical orbits around the nucleus (circles and ellipses).
Now I am wondering: does all quantum systems give rise to some "coherent states" that can be interpreted as the classical situation? Or these two examples belong to some special type of systems that can be interpreted in classical ways?
For example I cannot think of a coherent state for a Stern-Gerlach experiment...
 A: Coherent states are eigenstates of annihilation operator. If you can reduce/transform a particular problem to a simple harmonic oscillator problem, then by construction itself there will be coherent states.
In the context of the hydrogen atom problem, hereby we consider only the angular momentum part of the problem. If we look at the Schwinger's oscillator model of angular momentum [1], we see that the Hilbert space can be characterized by using the number states of two independent harmonic oscillators. The operators $J_+$, $J_-$ and $J_z$ can be written in terms of the creation and the annihilation operators of these oscillators.
In this situation we see that coherent states are naturally arising in the problem. Since every problem cannot be mapped to a simple harmonic oscillator problem, we don't have coherent states arising in an arbitrary problem.
[1] Chapter 3, Section 3.8, Schwinger's oscillator model of angular momentum, Modern Quantum Mechanics, J.J. Sakurai.
A: Today I went back with my professor and asked this same question. Here goes my version of his answer. Apparently the answer is yes. Every quantum system has some set of coherent states that reproduce what we would wait in the classical limit. 
In the case of the harmonic oscillator these are the eigenstates of the annihilation operator. For this particular case which has a quadratic potential the coherent states are stable in time, meaning that they will be bouncing back and forth indefinitely. 
For the electron in the hydrogen atom there are "semi-classical" states in which the electron seems as orbiting the nucleus, although these states are not stable due to dispersion (in contrast to the coherent states of the harmonic oscillator). 
For a free particle we can build a gaussian profile and it is a semi-classical state representing a point like particle that moves with some speed. Again, due to dispersion, this will be true for some finite time.
For the Stern-Gerlach experiment there are also semi-classical states (or what I have previously called, maybe wrong, coherent states). A semi-classical state would be a particle with a total angular momentum $\vec{J}(=\vec{L}+\vec{S})$ consisting of a superposition (maybe gaussian, I don't know) of many different values. Thus if we repeat the experiment many times with the particle identically prepared each time, we would see a classical distribution on the screen.
A: Mathematically speaking a coherent state can be defined as a state that saturates an uncertainty relation (be it Heisenberg or Schrödinger-Robertson uncertain relation ...)
The formal development leading to the determination of coherent states doesn't need to be particularised to any system, so theoretically you can have coherent states for every system (not only with the Heisenberg algebra).
