Yes, individual photons do have a quantized wavelength. In the quantum theory of light, electromagnetic radiation, including visible light, is quantized into discrete packets of energy called photons. Each photon carries an energy proportional to its frequency, and the frequency is inversely proportional to the wavelength according to the equation E = hf, where E is the energy, h is Planck's constant, and f is the frequency.
However, when we consider the spectrum of light, which encompasses a range of wavelengths, it appears continuous. This apparent continuity arises from the collective behavior of a large number of photons.
The spectrum of light is often described as a continuum because there are typically a vast number of photons with a range of energies and wavelengths emitted or absorbed by a light source. In many practical situations, such as thermal radiation from objects or light emitted by a glowing filament, a wide range of energy levels and corresponding wavelengths contribute to the overall emission. This collective behavior of countless photons with slightly different energies results in a continuous distribution of wavelengths and a smooth spectrum.
It's important to distinguish between the quantization of individual photons' energies and the continuous distribution of energies over a range of wavelengths when considering the spectrum of light. Although the energy of a single photon is quantized, the combined effect of a large number of photons with different energies can give rise to a continuous spectrum of light.