How to prepare a desired quantum state? Given a quantum state function, we can Fourier expand it in terms of stationary states of the Hamiltonian. So if we want to build that same quantum state approximately all we need to do is to superpose stationary states with proper amplitudes. Assuming that we can prepare such stationary states individually, how is their superposition done experimentally? 
As a specific example, how to prepare experimentally, an ensemble described by the quantum state that is a superposition of 4 stationary states of the particle in a box (or the hydrogen atom) that corresponds to n = 1, 2, 3, and 4 say ?  
 A: For particles in a beam, making a superposition of spin is easy; you just split the beam and recombine it after making appropriate modifications to the split beams. In creating a superposition, this works even if the beam has only a single particle in it.
A particle in a box is a tougher situation as it is difficult to split the states. The problem is that the usual methods of measuring a particle's position or momentum (in a box) are destructive. For example, after you measure the position of an electron in a potential well you destroy all information about its momentum and its previous positions. I'll take a whack at it...
To get what you're asking for, you would need to have a measurement that (a) is non destructive, and (b) allows you to make some modification to the electron(s). Suppose that the electron begins in its lowest energy state. You want a measurement that will split that energy state into two different states and then you make a modification to one of the states.
Suppose that you begin with an electron with spin-up and lowest energy in your potential well. One thing you could do is to apply a magnetic field in a direction other than up or down. This will split the electron into appropriate superpositions of spin along that new direction. If you make the new magnetic field in some horizontal direction u (so you're perpendicular to spin-up), you'll have a superposition: $(|+u\rangle+|-u\rangle)/\sqrt{2}$. Those two superpositions have two different energies.
When an electron absorbs a single photon, its spin flips. So you can convert a $|-u\rangle$ electron to the $|+u\rangle$ state by arranging for it to absorb a photon oriented in the +u direction. The $|+u\rangle$ electron can't absorb such a photon because of conservation of angular momentum. Therefore, you've just added energy to one half of your superposition.
Now turn off the magnetic field. You've created a state that is a superposition of two energy states. (Maybe more, depending how copacetic the excited energy state with the magnetic field on are with the excited energy states with no magnetic field.) In fact, the magnetic field really wasn't necessary. I put it in there to make you think of the electron in terms of the +-u basis for its spin. All you really have to do is to arrange for the photon to have spin in the +u direction.

By the way, a photon with spin in the +u direction is "circularly polarized". Also, the turning on and off of the magnetic field needs to be done slowly. To arrange for particular relative phases in the superposition between the + and - spins you can use the technique of "quantum phase", or "Berry-Pancharatnam phase". This is the phase acquired when a system is slowly sent through a sequence of states (but which also applies to sudden state changes with the same topology). One can induce a phase by a slow change to the spin axis. The phase one obtains is equal to half the spherical area cut out by the spin axis in the Bloch sphere (i.e. the set of possible directions for a spin axis). To get a relative phase you'd want to "park" one of the spin states by, say, arranging for it to have an energy that prevents its spin state from being modified, and then send the other state through a sequence.
