In physics, an elementary (and accurate) explanation of a discrete spectrum is that it is an emission spectrum or absorption spectrum for which there is only an integer number (or countable number) of intensities. Atomic electronic absorption and emission spectrum are discrete, as contrasted with, for example, the emission spectrum of the sun, which is continuous. Discrete emission spectra of atoms are also referred to as emission line spectra, which show vertical lines at the frequencies (or wavelengths) of emission. To describe the difference between continuous and discrete spectra, we might refer to probability theory—measuring weight versus throwing dice.
Conceivably, our body-weights can be represented by any number, 50.00000 kilograms, 59.99999 kg, 39.999591 kg, the number of decimals determined by our equipment, not by any fundamental restriction on the value of a weight. By contrast, when we throw dice, the throw of a single six-sided die can yield only 1, 2, 3, 4, 5, or 6, and nothing in between; there is a fundamental restriction on the numbers we can get throwing a single die; we cannot get a 3.2, 4.5, or 1.7; we can only get whole numbers between 1 and 6 inclusive. The set of all possible weights is continuous; the set of all possible single-die tosses is discrete.
One segment of the emission spectrum of atomic hydrogen is shown below. Note that we see only four lines.
By contrast, the solar spectrum, the spectrum of electromagnetic frequencies emitted by the sun, is continuous; likewise, the Gaussian density function or the Bell Curve is continuous, showing not discrete lines but continuous curves.
When we can visualize discrete spectra and continuous spectra from experimental physics, we can better grasp, by analogy, the mathematical uses of these terms, for example, as used below.
In physics, discrete spectrum is a finite set or a countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a discrete spectrum if its spectrum consists of isolated points. If the spectrum of an operator is not discrete, we say that it is a continuous spectrum.
The position and the momentum operators (in an infinite space) have continuous spectra. But the momentum in a compact space, the angular momentum, and the Hamiltonian of various physical systems (the corresponding eigenstates are called bound state) have a discrete (quantized) spectrum. This is a major difference with the corresponding operators in classical mechanics. Quantum mechanics was therefore named in this way.
The quantum harmonic oscillator or the Hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the Hydrogen atom, it has both continuous as well as discrete part of the spectrum; the continuous part represents the ionized atom.