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.
[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 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.
Choosing the Coulomb gauge, for which ∇⋅A = 0, makes A into a transverse field.
The Fourier expansion of the vector potential enclosed in a finite cubic box of volume V = L3 is then where
The best known example of quantization is the replacement of the time-dependent linear momentum of a particle by the rule Note that the Planck constant is introduced here and that the time-dependence of the classical expression is not taken over in the quantum mechanical operator (this is true in the so-called Schrödinger picture).
is the electric constant, which appears here because of the use of electromagnetic SI units.
The introduction of the Planck constant is essential in the transition from a classical to a quantum theory.
The factor is introduced to give the Hamiltonian (energy operator) a simple form, see below.
The classical Hamiltonian has the form The right-hand-side is easily obtained by first using (can be derived from Euler equation and trigonometric orthogonality) where k is wavenumber for wave confined within the box of V = L × L × L as described above and second, using ω = kc.
Substitution of the field operators into the classical Hamiltonian gives the Hamilton operator of the EM field, The second equality follows by use of the third of the boson commutation relations from above with k′ = k and μ′ = μ.
Note again that ħω = hν = ħc|k| and remember that ω depends on k, even though it is not explicit in the notation.
The notation ω(k) could have been introduced, but is not common as it clutters the equations.
Since harmonic oscillator energies are equidistant, the n-fold excited state
; can be looked upon as a single state containing n particles (sometimes called vibrons) all of energy hν.
By mathematical induction the following "differentiation rule", that will be needed later, is easily proved, Suppose now we have a number of non-interacting (independent) one-dimensional harmonic oscillators, each with its own fundamental frequency ωi .
The quantized EM field has a vacuum (no photons) state
To consider the action of the number operator of mode (k, μ) on a n-photon ket of the same mode, we drop the indices k and μ and consider Use the "differentiation rule" introduced earlier and it follows that A photon number state (or a Fock state) is an eigenstate of the number operator.
The zero of energy can be shifted, which leads to an expression in terms of the number operator, The effect of H on a single-photon state is Thus the single-photon state is an eigenstate of H and ħω = hν is the corresponding energy.
Introducing the Fourier expansion of the electromagnetic field into the classical form yields Quantization gives The term 1/2 could be dropped, because when one sums over the allowed k, k cancels with −k.
The effect of PEM on a single-photon state is Apparently, the single-photon state is an eigenstate of the momentum operator, and ħk is the eigenvalue (the momentum of a single photon).
Since the photon propagates with the speed of light, special relativity is called for.
The spin operators satisfy the usual angular momentum commutation relations Indeed, use the dyadic product property because ez is of unit length.
In this manner, By inspection it follows that and therefore μ labels the photon spin, Because the vector potential A is a transverse field, the photon has no forward (μ = 0) spin component.
The classical approximation to EM radiation is good when the number of photons is much larger than unity in the volume
For example, the photons emitted by a radio station broadcast at the frequency ν = 100 MHz, have an energy content of νh = (1 × 108) × (6.6 × 10−34) = 6.6 × 10−26 J, where h is the Planck constant.
The energy content of this volume element at 5 km from the station is 2.1 × 10−10 × 0.109 = 2.3 × 10−11 J, which amounts to 3.4 × 1014 photons per
The waves emitted by this station are well-described by the classical limit and quantum mechanics is not needed.
This article incorporates material from the Citizendium article "Quantization of the electromagnetic field", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.