Question

I was studying canonical quantization for the electromagnetic fields. I know that we can expand our fields in other normal modes which one of them is the Laguerre–Gaussian (LG) wave set but in most books the procedure is not written.

I wanted to ask that if there is some good reference for this?

edit : i wanted this refrences because i wanted to see why the orbital angular momentum of these beams have well defined values and their eigenstates which probably are these normal egein modes (LG beams)

Answer 1

Quantization of angular momentum is a principle where angular momentum of particles such as electrons in atomic orbits can only have discrete values. Laguerre-Gaussian beams are electromagnetic modes with quantized orbital angular momentum, a property arising from their wave-like characteristics. For a detailed examination of these concepts, the literature on quantum mechanics and quantum optics by credible authors is recommended.

Step-by-step explanation:

Quantization of Angular Momentum

The concept of the quantization of angular momentum is fundamental in quantum mechanics and arises from the wave-like properties of particles. According to Niels Bohr's model, electrons in atoms can only occupy orbits where their wave nature allows for constructive interference, leading to discrete energy levels and angular momentum values. When describing orbits in an atom, the quantized angular momentum L can be calculated using L = mvr, where m is the electron's mass, v is the velocity, and r is the orbit radius. The allowed quantized values of this angular momentum are integer multiples of Planck's constant h divided by 2π (h-bar), i.e., ℓ/2π, 2ℓ/2π, 3ℓ/2π, and so on.

Laguerre-Gaussian (LG) Beams

Laguerre-Gaussian (LG) beams are a set of solutions to the paraxial wave equation and represent modes of electromagnetic fields with well-defined orbital angular momentum (OAM). The distinct helical phase fronts and ring-shaped intensity distributions characterize LG beams. Their OAM values are quantized since these beams carry an angular momentum of ℓl per photon, where l is the azimuthal index representing the number of intertwined helices. This characteristic is attributed to the eigenstates of the quantized OAM.

For a comprehensive understanding of canonical quantization in electromagnetic fields and LG beams, references such as "Quantum Mechanics" by Cohen-Tannoudji, "Quantum Optics" by Scully and Zubairy, and related scientific papers on the subject are valuable resources that delve into quantization and the properties of LG beams.