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Quantizzazione del campo elettromagnetico 電磁場の量子化 Quantization of the electromagnetic field
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The quantization of the electromagnetic field, means that an electromagnetic field consists of discrete energy parcels, photons. Photons are massless particles of definite energy, definite momentum, and definite spin. A quantum mechanical photon state belonging to mode is introduced below, and it is shown that it has the following properties: In fisica, la quantizzazione del campo elettromagnetico è la descrizione del campo elettromagnetico, responsabile delle onde elettromagnetiche, nel formalismo della meccanica quantistica. La trattazione è, sotto alcuni aspetti, simile a quella dell'oscillatore armonico, anche se più complicata data la maggiore complessità delle equazioni che descrivono il campo. 量子電磁力学では電磁場の量子化（でんじばのりょうしか）により、粒子の運動量は演算子に置き換わる。量子化によって電磁場は光子の集まりであることがわかる。つまり、光子の状態を表す電磁ポテンシャルの時間微分が電場、空間微分が磁場である。 電磁場の量子化には2通り考えられる。1つ目の方法は、場の量子論の知識によって古典的な電磁場を量子化して、量子化された電磁場を得る方法である。 2つ目の方法は、古典電磁気学と解析力学によって「古典的な電磁場は、無限個の古典的な調和振動子の集まりと等価である」ことを示し、その調和振動子を量子力学の知識によって量子化する。すると無限個の量子的な調和振動子を得られるが、それを量子化された電磁場と考える。以下ではこちらの方法について述べる。
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量子電磁力学では電磁場の量子化（でんじばのりょうしか）により、粒子の運動量は演算子に置き換わる。量子化によって電磁場は光子の集まりであることがわかる。つまり、光子の状態を表す電磁ポテンシャルの時間微分が電場、空間微分が磁場である。 電磁場の量子化には2通り考えられる。1つ目の方法は、場の量子論の知識によって古典的な電磁場を量子化して、量子化された電磁場を得る方法である。 2つ目の方法は、古典電磁気学と解析力学によって「古典的な電磁場は、無限個の古典的な調和振動子の集まりと等価である」ことを示し、その調和振動子を量子力学の知識によって量子化する。すると無限個の量子的な調和振動子を得られるが、それを量子化された電磁場と考える。以下ではこちらの方法について述べる。 In fisica, la quantizzazione del campo elettromagnetico è la descrizione del campo elettromagnetico, responsabile delle onde elettromagnetiche, nel formalismo della meccanica quantistica. La trattazione è, sotto alcuni aspetti, simile a quella dell'oscillatore armonico, anche se più complicata data la maggiore complessità delle equazioni che descrivono il campo. The quantization of the electromagnetic field, means that an electromagnetic field consists of discrete energy parcels, 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 Planck's 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. He applied a technique which is now generally called second quantization, 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 Planck's 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 belonging to mode is introduced below, and it is shown that it has the following properties: 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.
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