Photons can be emitted at a wide range of wavelengths depending on the energy transitions occurring within the emitting system. However, there are certain specific wavelengths that cannot be emitted due to the nature of energy levels in atoms and molecules.
The energy levels of atoms and molecules are quantized, meaning they can only exist at specific discrete energy values. When an atom or molecule undergoes a transition from a higher energy level to a lower energy level, it emits a photon with an energy equal to the energy difference between the two levels. The energy of a photon is directly proportional to its frequency (E = hf) and inversely proportional to its wavelength (c = fλ), where h is Planck's constant and c is the speed of light.
In a given system, if there is no energy level corresponding to a particular wavelength, it means that there is no possible energy transition that can emit a photon with that specific wavelength. This restriction is a consequence of the energy quantization in the system.
For example, in atomic spectra, certain wavelengths are missing because they do not correspond to energy differences between allowed energy levels. These missing wavelengths create dark lines, known as absorption lines, in the spectrum. Similarly, in emission spectra, only certain specific wavelengths are observed as bright lines because they correspond to energy transitions that can occur in the system.
The inability to emit photons at certain wavelengths is fundamentally related to the energy level structure of the emitting system, governed by quantum mechanics. The allowed energy levels dictate the specific wavelengths at which photons can be emitted or absorbed.