Harmonic oscillators and photons

Abstract

Prerequisites: Chapters 2–5, 9, 10, 12, and 13. For additional background on vector calculus, electromagnetism and modes see Appendices C and D. In this section, we return to the harmonic oscillator and consider it in a mathematically more elegant way. This approach leads to the introduction of “raising” and “lowering” operators that take us from one harmonic oscillator state to another. The introduction of these operators allows us to rewrite the harmonic oscillator mathematics quite economically. We then show that the electromagnetic field for a given mode can also be described in a manner exactly analogous to a harmonic oscillator. In this case, we describe the states of this generalized harmonic oscillator in terms of the number of photons per mode, with that number corresponding exactly to the quantum number for the corresponding harmonic oscillator state. The raising and lowering operators are now physically interpreted as “creation” and “annihilation” operators for photons and are key operators for describing electromagnetic fields. By this process, we can describe the electromagnetic field quantum mechanically, rather than our previous semiclassical use of quantum mechanical electron behavior but with classical electric and magnetic fields. We say we have “quantized” the electromagnetic field. This quantization is then the basis for all of quantum optics. This approach also prepares us for discussions in subsequent chapters of fermion operators and of the full description of stimulated and spontaneous emission.