The passage of an electron through a cavity resonator is considered for the purpose of studying the appearance of quantum mechanical effects in an essentially classical experiment as the frequency of the electromagnetic field becomes high. Both the electron and the field must be represented, initially, by wave packets---in space and momentum coordinates for the former, and in electric field and vector potential coordinates for the latter---in order to correspond to classically meaningful systems. The velocity of the electron at the output of the cavity is investigated. It is found that the expectation value of velocity has a quantum-mechanical correction term. A more significant effect, however, is the nonvanishing of the expectation value of the square of the deviation from the expectation value (mean square deviation from the mean) of the velocity. This deviation is a random phenomenon which will produce noise in an electron beam. An expression for the mean square deviation, in which the results of electron and field quantization are separately apparent, is derived, and its significance is discussed. An order-of-magnitude calculation is made for the ratio of minimum possible mean square deviation of velocity to velocity increment due to the field, and the frequency for which this ratio becomes unity is calculated for a particular set of conditions to be of the order of 3\ifmmode\times\else\texttimes\fi{}${10}^{12}$/sec.
The passage of an electron through a cavity resonator is considered for the purpose of studying the appearance of quantum mechanical effects in an essentially classical experiment as the frequency of the electromagnetic field becomes high. Both the electron and the field must be represented, initially, by wave packets---in space and momentum coordinates for the former, and in electric field and vector potential coordinates for the latter---in order to correspond to classically meaningful systems. The velocity of the electron at the output of the cavity is investigated. It is found that the expectation value of velocity has a quantum-mechanical correction term. A more significant effect, however, is the nonvanishing of the expectation value of the square of the deviation from the expectation value (mean square deviation from the mean) of the velocity. This deviation is a random phenomenon which will produce noise in an electron beam. An expression for the mean square deviation, in which the results of electron and field quantization are separately apparent, is derived, and its significance is discussed. An order-of-magnitude calculation is made for the ratio of minimum possible mean square deviation of velocity to velocity increment due to the field, and the frequency for which this ratio becomes unity is calculated for a particular set of conditions to be of the order of 3\ifmmode\times\else\texttimes\fi{}1012/sec.
The study of the appearance of quantum effects in an essentially classical experiment---involving the passage of an electron through a resonant cavity---as the frequency of the electromagnetic field increases, was begun in a previous paper, in which the electron velocity was examined. In the present paper, an analysis is made of the electric field in the same experiment. Both the expectation value of the field, and the expectation value of the square of the deviation from the expectation value (the dispersion) of the field are calculated to the lowest order in the electron charge. The expectation value of the field is seen to differ from the classical value by a small amount, the difference being due to the wave properties of the electron. The dispersion in the field is shown to consist of three parts: one due to the quantum-mechanical properties of the field alone, the second due to the wave properties of the electron, and the third due to the interaction between the quantum-mechanical properties of the field and the classical properties of the electron. The orders of magnitude of the three terms are shown to decrease in the order in which they are enumerated. A calculation is then made of the effect on the field of a number of electrons passing through the cavity, the interaction between the electrons being neglected. The number of electrons required in a "bunch" to produce a field comparable to the dispersion in the field is calculated, and, for a typical set of experimental conditions, it is seen that the highest frequency at which this requirement can be met is of the order of 3\ifmmode\times\else\texttimes\fi{}1012/sec.
We develop a quantum field-theoretic method to describe the effect of frequency mixing using Josephson junctions or using an electron beam traversing a succession of microwave cavities. One of our purposes is to discuss the basic mechanism of current modulation and the subsequent radiation in the complete quantum mechanical context. We use the Heisenberg picture, so that the field operators depend on time and state vectors are fixed. It is essential to take quantum mechanical expectation values only for the quantities that are actually observed. Thus, in the case of frequency mixing, we do not evaluate the current but rather the radiated power, which is observable. An evaluation of the current would destroy correlations between Fourier components of the current produced in the first cavity or in the first junction. Leaving the current as an operator permits the correlations produced in the first cavity (junction) to affect the radiation in the second cavity (junction). Another purpose of the approach is to find the possible dependence of the output radiation on the quantum state of the incident radiation preexisting in cavities or in Josephson junctions. In the case of frequency mixing using two microwave cavities where the fields are treated as independent, we show that it is possible to distinguish the case where both cavities are in definite phase-certain coherent states from the phase-uncertain number states by the value of the wave function renormalization constant. However, if the relative phase of the two cavities cannot be defined, this distinction disappears. This indicates that the radiation occurs at the beat frequency even when the cavity fields are taken to be in energy eigenstates, so that no phase whatever can be associated with the fields in those cavities and thus a classical velocity modulation picture does not apply.
We investigate the dynamics of a charged particle interacting with a multimode quantized electromagnetic field and obtain an analytic solution for the full electron--field system. This framework enables the calculation of position expectation values and uncertainties for arbitrary wave packets and field states, allowing us to identify quantum corrections to the corresponding classical motion. While the corrections to the position expectation value are weak and largely insensitive to the quantum state of the field, the wave packet broadening exhibits a pronounced dependence on the field state. In particular, the quantum uncertainty of the radiation is directly imprinted onto the spatial uncertainty of the particle. We illustrate these effects for Gaussian wave packets interacting with coherent, Fock, and squeezed states, including bright squeezed vacuum. The interaction with a finite-duration laser pulse is also analyzed as a multimode example. Our results provide a transparent analytic route toward understanding how quantum fluctuations of light influence electron dynamics in strong-field settings.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
The quantum theory of radiation, formulated in wave-mechanical language but in terms of running waves, is shown to lead under suitable conditions to the classical theory of emission by a particle as a first approximation. Non-relativistic mechanics is used, and the limitations of the method are brought out in terms of classical theory. The characteristic quantum phenomenon of the Compton effect requires that the Compton recoil shall exceed the spread in momentum of the particle, and so that the wave-packet shall greatly exceed in diameter the wave-length of the incident radiation; this condition is realizable on any scale of magnitude. For quasi-classical motion just the reverse is required; furthermore, if the motion is to last for a number of waveperiods, the wave-length must greatly exceed mch (m=massoftheparticle). The classical packet-motion arises from combinations between quantum-states of the particle-field system; this fact suggests that the photon is not a fundamental constituent of the field but only one possible aspect of its action. The infinite "zero-energy" associated apparently even with the lowest energy-state of the field implies no electromagnetic field intensities at any point. Whereas for problems on energy or momentum it is customary to represent an incident wave-train by a single typical quantum state, to obtain motional phenomena of classical type a more complete expression such as that given in the paper must be used. The physically empty field likewise corresponds to a mixture of quantum states but without correlation among their amplitudes.
Effects of quantum deviation of a two-level atom at coherent scattering on an inhomogeneous optical potential created by crossed electromagnetic fields are considered. The region of interaction is formed by a lowfrequency quantized standing wave from a micromaser and a coherent traveling optical wave generated by an optical fiber located inside a cavity. The atom interacts with both fields under the conditions of two-photon two-wave resonance. It is shown that two effects of quantum deviation of translational motion of the atom can be observed. Interaction with the standing wave is caused under these conditions by a harmonic potential the character of scattering of the atom on which depends significantly on the initial conditions of preparation of the atom and quantized mode. The other effect—deviation of the atom by the classical traveling wave—is also completely quantum mechanical under these conditions and is produced by the noncommutative contribution of the kinetic energy operator of the atom and the interaction energy.