|
After quantization of the electromagnetic field, the EM (electromagnetic) field consists of discrete energy parcels, photons. Photons are massless particles of definite energy, definite momentum, and definite spin. In order to explain the photoelectric effect, Einstein assumed heuristically in 1905 that an electromagnetic field consists of parcels of energy ''h''ν, where ''h'' is Planck's constant. In 1927 Paul A. M. Dirac was able to weave the photon concept into the fabrics of the new quantum mechanics and to describe the interaction of photons with matter.〔P. A. M. Dirac, ''The Quantum Theory of the Emission and Absorption of Radiation'', Proc. Royal Soc. (London) A114, pp. 243–265, (1927) (Online ) (pdf)〕 He applied a technique which is now generally called second quantization,〔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.〕 although this term is somewhat of a misnomer for EM fields, because they are, after all, 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 Planck's constant enters the expressions. In his original work, Dirac took the phases of the different EM 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 will be done below. A quantum mechanical photon state |k,μ⟩ belonging to mode (k,μ) will be introduced. It will be shown that it has the following properties : These equations say respectively: a photon has zero rest mass; the photon energy is ''h''ν=''hc''|k| (k is the wave vector, ''c'' is speed of light); its electromagnetic momentum is ℏk (); the polarization μ=±1 is the eigenvalue of the ''z''-component of the photon spin. ==Second quantization== Second quantization starts with an expansion of a scalar or vector field (or wave functions) in a basis consisting of a complete set of functions. These expansion functions depend on the coordinates of a single particle. The coefficients multiplying the basis functions are interpreted as operators and (anti)commutation relations between these new operators are imposed, commutation relations for bosons and anticommutation relations for fermions (nothing happens to the basis functions themselves). By doing this, the expanded field is converted into a fermion or boson operator field. The expansion coefficients have been promoted from ordinary numbers to operators, creation and annihilation operators. A creation operator creates a particle in the corresponding basis function and an annihilation operator annihilates a particle in this function. In the case of EM fields the required expansion of the field is the Fourier expansion. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quantization of the electromagnetic field」の詳細全文を読む スポンサード リンク
|