When textbooks in QM give example of finite dimensional Hilbert spaces they give examples of photon polarizations or of 2-states systems and sometimes they mention how one can achieve superposition in such cases experimentally.
On the other hand when they talk about simple potentials like particle in an infinite potential well, and talk about superposition of the stationary states of this problem they never mention how such superposition can be achieved experimentally (despite the fact that we are in the era of nanotechnology and scientists can "engineer" effectively such potentials and many others). Typical example that may appear in textbooks can be as simple as $\Psi(x)=\frac{4}{5}\phi_1(x)+\frac{3}{5}\phi_5(x)$, where {$\phi_n(x)$} are the normalized eigenfunctions of the particle in a box Hamiltonian. Other more general superpositions could be between infinite numbers of stationary states via $\Psi=\sum a_n\phi_n(x)$.
That makes me wonder, is there a fundamental reason that prevents us from engineering such superposition in case of particle in a box and the like, or that we just do not know how to do it yet? why is it possible with spin and seem to be hard with particle in a box? or is it something related to the energy eignestates in the position representation?
It is kind of frustrating to study for long hours/read/solve problems/HW on all kinds of potentials and on superpositions without knowing how/if they can be realized in experiment.
If someone knows references in which this issue is discussed it would be greatly appreciated.
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