Importance of bound states

Question

While solving a potential well problem we get scattering states and bound states (if exist). Number of the bound states we get depends on the potential profile. What I want to ask is, what is the importance of number of bound states we are getting? For example for Dirac delta potential well we have only one bound state. What more information we can derive by this knowledge?

Answer 1

Modern electronic devices like quantum well lasers, resonant tunneling diodes, quantum cascade lasers and detectors heavily rely on the spatial and energetic position of such bound states. This defines their transport and optical properties.

On a separate notice: any well, no matter how shallow or narrow, has at least one bound state.

Answer 2

Generally speaking, bound states in a system give rise to resonant behaviour. The number of states defines the number of different resonances that can be observed.

For example, consider the absorption of photons by an electron trapped in a quantum well:

If the well only contains one bound state then it could absorb any photon with any energy greater than the binding energy of the state (i.e., a transition from state $|1\rangle$ to anywhere in the continuum), so a broad spectrum would be observed.

If the well contains two bound states with energies $E_1$ and $E_2$, then an incident photon with energy $\hbar\omega_{21} = E_2 - E_1$ would resonate with the $|2\rangle\to|1\rangle$ transition, and this would give rise to an additional strong, and narrow peak in the absorption spectrum.

If there were three bound states, then there could be three resonant peaks in the spectrum corresponding to $\hbar\omega_{32} = E_3 - E_2$, $\hbar\omega_{31}=E_3 - E_1$ and $\hbar\omega_{21} = E_2 - E_1$.

So, the number of bound states determines the number of possible resonant interactions. Note, though, that not all of these transitions are necessarily permitted... symmetry rules can often forbid light from interacting with certain pairs of states.