The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.
Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically. Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983). Quantum theory of particles and fields There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable (
x ,
v^) , with the commutation relation [
x ,
v^] =
iℏ/
m , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion
K(
x,v,t) of the classical system, The comparison with the quantization in the space (
x,p) is done with the usual Schrödinger’s equation for the Hamiltonian
H(
x,p,t), and with the commutation relation [
x ,
p^] =
iℏ. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (
x ,
v^) quantization has fewer oscillations than the (
x ,
p^) quantization, the average energy of the system is higher in (
x ,
p^) quantization than on the (
x ,
v^) quantization, and the Boltzmann- Shannon entropy on the (
x ,
p^) quantization is higher than on the (
x ,
v^) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.
In our preceding treatments of the interaction of radiation with matter, we assumed that the electric field is classical. In most cases, this assumption is validated by experiment. There are several instances, however, for which a classical field fails to give experimentally observed results, whereas a quantized field succeeds. This is true, for example, of spontaneous emission, a phenom enon which was described phenomenologically in our earlier work (see Chaps. 2 and 7). Another case involves the 2 and 2p \ states in the hydrogen atom, which are predicted to have the same energy not only by the simple theory of Chap. 1, but also by the more sophisticated Dirac theory. In both cases, a classical electric field was assumed. Experimentally, the two levels differ by approximately the Lamb shift, 1057 MHz. A fully quantized treatment of the field and atom system gives impressive agreement with the experimentally observed shift. Derivation of the fluctuations in intensity of a laser near threshold requires the quantum theory o f radiation. In addition there are other problems for which the concept of a “particle,” the photon, is either necessary or convenient. Despite a popular impression to the contrary, however the explanation of the photoelectric effect is not one of these (see Prob. 2-10).
. Each transition dipole in a single atom has a radiation reaction field which acts on itself and on the other transition dipoles of the atom. The self interaction on a given transition gives rise to spontaneous emission and radiative decay on that transition. Here we emphasize that the mutual radiative interaction of different transition dipoles can sometimes have important consequences. The mutual radiative interaction can substantially alter radiative decay rates and inhibit the radiative decay of certain expectation values. The hydrogen atom and the harmonic oscillator are cited as examples and a possible experimental test of mutual radiation reaction involving a single electron in a Penning trap is discussed. Introduction Modern quantum mechanics began with Heisenberg's picture of a bound charge as a “virtual orchestra” of transition dipoles oscillating at different frequencies. It is known from modern quantum theory that each of these dipoles generates a radiation reaction field which, acting on the dipole itself, gives rise to spontaneous emission and radiative relaxation (Milonni, 1976). In this paper we consider the mutual interaction of different transition dipoles within a single “atom”. We begin by considering some simple cases of mutual radiation reaction in classical electrodynamics and proceed to the quantum theory of mutual radiation reaction (MRR), citing the hydrogen atom and the charged harmonic oscillator as examples. We point out that the equations describing MRR are not at all new, but can be traced to the earliest treatments of the interaction of atoms with the quantized electromagnetic field.