Quantization of the electromagnetic field

The quantization of the electromagnetic field means that an electromagnetic field consists of discrete energy parcels called photons. Photons are massless particles of definite energy, definite momentum, and definite spin.

To explain the photoelectric effect, Albert Einstein assumed heuristically in 1905 that an electromagnetic field consists of particles of energy of amount hν, where h is the Planck constant and ν is the wave frequency. In 1927 Paul A. M. Dirac was able to weave the photon concept into the fabric of the new quantum mechanics and to describe the interaction of photons with matter.^[1] He applied a technique which is now generally called second quantization,^[2] although this term is somewhat of a misnomer for electromagnetic fields, because they are solutions of the classical Maxwell equations. In Dirac's theory the fields are quantized for the first time and it is also the first time that the Planck constant enters the expressions. In his original work, Dirac took the phases of the different electromagnetic modes (Fourier components of the field) and the mode energies as dynamic variables to quantize (i.e., he reinterpreted them as operators and postulated commutation relations between them). At present it is more common to quantize the Fourier components of the vector potential. This is what is done below.

A quantum mechanical photon state $|\mathbf {k} ,\mu \rangle$ belonging to mode $(\mathbf {k} ,\mu )$ is introduced below, and it is shown that it has the following properties:

{\begin{aligned}m_{\textrm {photon}}&=0\\H|\mathbf {k} ,\mu \rangle &=h\nu |\mathbf {k} ,\mu \rangle &&{\hbox{with}}\quad \nu =c|\mathbf {k} |\\P_{\textrm {EM}}|\mathbf {k} ,\mu \rangle &=\hbar \mathbf {k} |\mathbf {k} ,\mu \rangle \\S_{z}|\mathbf {k} ,\mu \rangle &=\mu |\mathbf {k} ,\mu \rangle &&\mu =\pm 1.\end{aligned}}

These equations say respectively: a photon has zero rest mass; the photon energy is hν = hc|k| (k is the wave vector, c is speed of light); its electromagnetic momentum is ħk [ħ = h/(2π)]; the polarization μ = ±1 is the eigenvalue of the z-component of the photon spin.

^ P. A. M. Dirac, The Quantum Theory of the Emission and Absorption of Radiation, Proc. Royal Soc. Lond. A 114, pp. 243–265, (1927) Online (pdf)
^ The name derives from the second quantization of quantum mechanical wave functions. Such a wave function is a scalar field (the "Schrödinger field") and can be quantized in the very same way as electromagnetic fields. Since a wave function is derived from a "first" quantized Hamiltonian, the quantization of the Schrödinger field is the second time quantization is performed, hence the name.

[1] P. A. M. Dirac, The Quantum Theory of the Emission and Absorption of Radiation, Proc. Royal Soc. Lond. A 114, pp. 243–265, (1927) Online (pdf)

[2] The name derives from the second quantization of quantum mechanical wave functions. Such a wave function is a scalar field (the "Schrödinger field") and can be quantized in the very same way as electromagnetic fields. Since a wave function is derived from a "first" quantized Hamiltonian, the quantization of the Schrödinger field is the second time quantization is performed, hence the name.

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