Several authors have used the Heisenberg picture to show that the atomic transitions, the stability of the ground state and the position-momentum commutation relation [x,p]=ih, can only be explained by introducing radiation reaction and vacuum electromagnetic fluctuation forces. Here we consider the simple case of a nonrelativistic charged harmonic oscillator, in one dimension, to investigate how to take into account the radiation reaction and vacuum fluctuation forces within the Schrodinger picture. We consider the effects of both classical zero-point and thermal electromagnetic vacuum fields. We show that the zero-point electromagnetic fluctuations are dynamically related to the momentum operator p=-ih d/dx used in the Schrodinger picture. Consequently, the introduction of the zero-point electromagnetic fields in the vector potential A_x(t) used in the Schrodinger equation, generates ``double counting'', as was shown recently by A.J. Faria et al. (Physics Letters A 305 (2002) 322). We explain, in details, how to avoid the ``double counting'' by introducing only the radiation reaction and the thermal electromagnetic fields into the Schrodinger equation.
Several authors have used the Heisenberg picture to show that the atomic transitions, the stability of the ground state and the position-momentum commutation relation [x,p]=ih, can only be explained by introducing radiation reaction and vacuum electromagnetic fluctuation forces. Here we consider the simple case of a nonrelativistic charged harmonic oscillator, in one dimension, to investigate how to take into account the radiation reaction and vacuum fluctuation forces within the Schrodinger picture. We consider the effects of both classical zero-point and thermal electromagnetic vacuum fields. We show that the zero-point electromagnetic fluctuations are dynamically related to the momentum operator p=-ih d/dx used in the Schrodinger picture. Consequently, the introduction of the zero-point electromagnetic fields in the vector potential A_x(t) used in the Schrodinger equation, generates double counting'', as was shown recently by A.J. Faria et al. (Physics Letters A 305 (2002) 322). We explain, in details, how to avoid the double counting'' by introducing only the radiation reaction and the thermal electromagnetic fields into the Schrodinger equation.
Dalibard, Dupont-Roc and Cohen-Tannoudji (J. Physique 43 (1982) 1617; 45 (1984) 637) used the Heisenberg picture to show that the atomic transitions, and the stability of the ground state, can only be explained by introducing radiation reaction and vacuum fluctuation forces. Here we consider the simple case of nonrelativistic charged harmonic oscillator, in one dimension, to investigate how to take into account the radiation reaction and vacuum fluctuation forces within the Schrodinger picture. We consider classical vacuum fields and large mass oscillator.
The role of vacuum fluctuations and radiation reaction in the process of spontaneous emission is examined with particular reference to the Heisenberg picture of quantum electrodynamics. In this picture the radiation reaction is found to be responsible for spontaneous emission if the equations are ordered normally. However, in the case of nonnormally ordered equations there appears to be an explicit contribution to the observable effects from vacuum fluctuations as well. We point out that this contribution from the vacuum field is an artifact of the approximation methods employed for eliminating the free-field variables from the nonnormally ordered equations These approximations are the same as those used in the master-equation approach, in which spontaneous emission appears to be due solely to the vacuum fluctuations. We show that the vacuum field does not contribute explicitly in any ordering if it is eliminated nonperturbatively and the approximation methods are invoked only after its elimination from the equations of motion.
A Schrödinger type equation for a mathematical probability amplitude ψ(x,t), is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M, moving in a potential V(x). The particle phase space probability density is denoted W(x,p,t) and the entire system is immersed in the "vacuum" zero-point electromagnetic radiation . We show that the generalized Liouville equation is reduced to a non-quantized Liouville equation in the equilibrium limit where the small radiative corrections cancel each other approximately. Our derivation will be based on a simple Fourier transform of the non-quantized phase space probability distribution W(x,p,t). For convenience, we introduce in this Fourier transform an auxiliary constant α, with dimension of action, and an auxiliary coordinate denoted by y. We shall prove that α is equal to the Planck's constant present in the momentum operator of the Schrödinger equation. Moreover, we shall show that this momentum operator is deeply related with the ubiquitous zero-point electromagnetic radiation. It is also important to say that we do not assume that the mathematical amplitude ψ(x,t) is a de Broglie matter-wave, in other words, the wave-particle duality hypothesis is not used within our work. The implications of our study for the standard interpretation of the photoelectric effect is discussed by considering the main characteristics of the phenomenon. We also mention, briefly, the effects of the zero-point radiation in the tunnelling phenomenon and in the Compton's effect.
This chapter reviews the classical description of radiation. It describes how classical electromagnetic radiation interacts with a quantum mechanical particle. The general rule is that, even though the wave functions and the electromagnetic potentials change with the gauge, all physically observable quantities are independent of the gauge. The chapter also describes the process of the absorption of light by a system in terms of the effect of an external classical electromagnetic field on the system. The total emission rate is the sum of the induced and the spontaneous rates. Spontaneous emission is just the quantum mechanical version of the classical phenomenon of radiation from an accelerating charge. A very important consequence of the fact that the electromagnetic field is quantized is that there are vacuum fluctuations of the field, analogous to the zero point motion of a harmonic oscillator. Einstein gave a simple statistical argument to determine the rate at which spontaneous emission takes place.
In this paper, we dwell on three issues: (1) revisit the relation between vacuum fluctuations and radiation reaction in atom-field interactions, an old issue that began in the 1970s and settled in the 1990s with its resolution recorded in monographs; (2) the fluctuation–dissipation relation (FDR) of the system, pointing out the differences between the conventional form in linear response theory (LRT) assuming ultra-weak coupling between the system and the bath, and the FDR in an equilibrated final state, relaxed from the nonequilibrium evolution of an open quantum system; (3) quantum radiation from an atom interacting with a quantum field: We begin with vacuum fluctuations in the field acting on the internal degrees of freedom (idf) of an atom, adding to its dynamics a stochastic component which engenders quantum radiation whose backreaction causes quantum dissipation in the idf of the atom. We show explicitly how different terms representing these processes appear in the equations of motion. Then, using the example of a stationary atom, we show how the absence of radiation in this simple cases is a result of complex cancellations, at a far away observation point, of the interference between emitted radiation from the atom and the local fluctuations in the free field. In so doing we point out in Issue 1 that the entity which enters into the duality relation with vacuum fluctuations is not radiation reaction, which can exist as a classical entity, but quantum dissipation. Finally, regarding issue 2, we point out for systems with many atoms, the co-existence of a set of correlation-propagation relations (CPRs) describing how the correlations between the atoms are related to the propagation of their (retarded non-Markovian) mutual influence manifesting in the quantum field. The CPR is absolutely crucial in keeping the balance of energy flows between the constituents of the system, and between the system and its environment. Without the consideration of this additional relation in tether with the FDR, dynamical self-consistency cannot be sustained. A combination of these two sets of relations forms a generalized matrix FDR relation that captures the physical essence of the interaction between an atom and a quantum field at arbitrary coupling strength.
Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.