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The difference between a continuous and a line spectrum is well understood in that a clear cut boundary is present in the case of a line spectrum. When will continuous spectrum be obtained and when a line spectrum for example in hydrogen? What method can be used to observe either or both these spectra?

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I will quote part of this answer:

Quantum mechanics dictates that the energy levels of atomic hydrogen are discrete--just from the way the math works out. The reason why hydrogen's peaks are so spaced out is that it has so few degrees of freedom (just the one electron). Xenon, in contrast, has many closely-spaced peaks because it has 54 electrons (so it has many more accessible energy levels than hydrogen).

The implication is that to a certain resolution, line spectra appear to become continuous once the energy level spacing and the excitation energies become close together. If you look closely enough, they are still discrete.

However, there comes a point where the electron that's being excited is no longer bound to the atom; that would be the first ionization potential (IP) or ionization energy. In the case of the hydrogen atom, that is the Rydberg constant. See this answer for how this arises mathematically.

For more details on the hydrogen atom, see here:

The infinity level represents the highest possible energy an electron can have as a part of a hydrogen atom. If the electron exceeds that energy, it is no longer a part of the atom. The infinity level represents the point at which ionization of the atom occurs to form a positively charged ion.

At this point it is important to distinguish between hydrogen's absorption/excitation spectrum and its emission/deexcitation spectrum (taken from here):

enter image description here

Since energy is inversely proportional to wavelength, the higher-energy excitations are toward the left of the spectra. This is visual confirmation that as energy level spacing decreases, lines in spectra get closer together. Past that point, what do the spectra look like? Certainly there will be peaks at the ionization energy, which is at a wavelength of about 91 nm. Applying more energy will also strip the electron from the proton, but the excitation from $n = 1$ to $n = \infty$ will still occur at the same energy, and likewise for the absorption from $n = \infty$ to $n = 1$. So there is only one peak in the hydrogen emission and exitation spectra directly resulting from the interaction between the electron and the proton (related). Any potential continuous band structure at higher energies due almost entirely to the free electron in the continuum, or you could think of any point in that spectral region being composed of two contributions, one being the fixed electron-proton interaction energy and the other being related to the kinetic energy of the free electron.

Note that this is for a single atom; for a collection of atoms approaching a bulk phase, you may have continuous spectra under other situations. Any time there are free electrons, energy levels may be continuous. There is also above threshold ionization, but without doing any calculations I am not sure how intense these effects are in terms of directly contributing to UV/vis spectra; if the $x$-axis is electron energy in eV and the $y$-axis is related to excitation probability, the effect is most likely negligible in practice.

enter image description here

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I apologize if I misunderstand, but it seems like you are asking two questions.

  1. Why does the hydrogen spectrum have so few, widely spaced peaks?

  2. Why do spectral peaks have width?

Here are answers:

  1. Quantum mechanics dictates that the energy levels of atomic hydrogen are discrete--just from the way the math works out. The reason why hydrogen's peaks are so spaced out is that it has so few degrees of freedom (just the one electron). Xenon, in contrast, has many closely-spaced peaks because it has 54 electrons (so it has many more accessible energy levels than hydrogen).

  2. Spectral lines are broadened for a number of reasons (both internal to the system--perhaps states are mixing, and external--the pressure on the system has some effect on line widths). However one type of broadening that is inherent to all systems (including simple, little hydrogen) is something called "lifetime broadening," and it is a result of the uncertainty principle. The principle puts a lower bound on the product of the uncertainty of the energy and the uncertainty of the lifetime of any quantum state, like the first excited state of hydrogen (2p), for example. Hydrogen's 2p has a lifetime of ~1.6 ns which gives it a peak width (an uncertainty in the energy of the transition) of ~0.000004 eV.

(see: http://farside.ph.utexas.edu/teaching/qmech/Quantum/node122.html )

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