quantum mechanics - Is it possible that we have a physical state which is a mixture of discrete eigenstates and continuous ones?

For a system has both continuous and discrete spectrum, is it possible that a physical states is something like:

$$\psi(x)=\sum_{n=n_1}^{n=n2}c_n\psi_n(x)+\int_{E_0-\Delta_E}^{E_0+\Delta_E}\psi_E(x)\mathrm{d}E$$

I am asking this because of some comments under my answer to this Phys.SE question: How is a bound state defined in quantum mechanics?

Answer

Yes, this is perfectly possible. The simplest example is the hydrogen atom, which has an infinite sequence of discrete eigenstates $|nlm\rangle$ with negative energy, and an energy continuum at $E>0$, which are known as Coulomb waves. It is perfectly possible for a state to be in a superposition of those:

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  First of all, it is required by the superposition principle. If $|\psi\rangle$ and $|\phi\rangle$ are states, the superposition $|\psi\rangle+|\phi\rangle$ must also be a state. This is the essential leap in the postulate that the state space of quantum mechanics is a Hilbert space.

  
  


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  Secondly, these superpositions get used all the time. If you ionize an atom, you will typically only remove the electron with a finite probability $p<1$, which means that you will leave population in both the bound and continuum subspaces of state space.

  
  

  I should stress that in such an ionization event you do (sometimes) need to account for the state of the system as a coherent superposition. This is particularly the case if the ionized portion of the wavefunction returns to interact with the core, as is the case in High Harmonic Generation (see e.g. this review or the original paper for bound + continuum superpositions) or Above-Threshold Ionization (as in this paper).