H.M. França

Classical aspects of the Pauli-Schrödinger equation

Abstract We derive a Pauli-Schrodinger type equation from the classical Liouville equation, for a neutral particle with arbitrary spin and magnetic dipole. Our derivation does not apply to a general classical phase-space distribution. Nevertheless, in certain particular cases we show that there is a correspondence between the classical equations and the Pauli-Schrodinger equation. Consequently, the results of the Stern-Gerlach, and also the Rabi type molecular beam experiments, can be interpreted classically, that is, in such a way that the particles have well-defined and continuous trajectory, and also continuous orientation of the spin vector. Theoretical and experimental implications of this conclusion are briefly commented.

Non-Heisenberg states of the harmonic oscillator

The effects of the vacuum electromagnetic fluctuations and the radiation reaction fields on the time development of a simple microscopic system are identified using a new mathematical method. This is done by studying a charged mechanical oscillator (frequency Ω0)within the realm of stochastic electrodynamics, where the vacuum plays the role of an energy reservoir. According to our approach, which may be regarded as a simple mathematical exercise, we show how the oscillator Liouville equation is transformed into a Schrödinger-like stochastic equation with a free parameter h′ with dimensions of action. The role of the physical Plancks constant h is introduced only through the zero-point vacuum electromagnetic fields. The perturbative and the exact solutions of the stochastic Schrödinger-like equation are presented for h′>0. The exact solutions for which h′<h are called sub-Heisenberg states. These nonperturbative solutions appear in the form of Gaussian, non-Heisenberg states for which the initial classical uncertainty relation takes the form 〈(δx2)〉〈(δp)2〉=(h′/2)2,which includes the limit of zero indeterminacy (h → 0). We show how the radiation reaction and the vacuum fields govern the evolution of these non-Heisenberg states in phase space, guaranteeing their decay to the stationary state with average energy hΩ0/2 and 〈(δx)2〉〈(δp)2〉=h2/4 at zero temperature. Environmental and thermal effects-are briefly discussed and the connection with similar works within the realm of quantum electrodynamics is also presented. We suggest some other applications of the classical non-Heisenberg states introduced in this paper and we also indicate experiments which might give concrete evidence of these states.

Radiative noise in circuits with inductance

Abstract Using the general idea of the fluctuation–dissipation theorem we study a new contribution to the voltage fluctuations which is associated with the presence of radiation resistance . We consider the particular case of a solenoid immersed in a cavity with equilibrium radiation at temperature T . We prove that these new fluctuations are generated by the random magnetic field present in the cavity. These magnetic voltage fluctuations are shown to be experimentally distinguishable from the voltage fluctuations associated with the well known Nyquist noise. Accordingly we suggest feasible experiments to measure this magnetic noise. All the calculations are made within the context of Stochastic Electrodynamics, a theory in which the vacuum zero-point field is taken as a real electromagnetic field. We also study the average energy of an RLC circuit in thermodynamic equilibrium with the radiation.

Critical assessment of the Schrödinger picture of quantum mechanics

We provide an example in which the Heisenberg and the Schrodinger pictures of quantum mechanics give different results, thus confirming the statement of P.A.M. Dirac that the two pictures may lead to inequivalent results. We consider a onedimensional nonrelativistic charged harmonic oscillator (frequency ω0 and mass m), and take into account the action of the radiation reaction and the vacuum electromagnetic forces on the charged oscillator. We show that the Heisenberg picture gives the correct value, ¯ hω0/2, for the ground state energy of the harmonic oscillator in both cases of classical and quantized vacuum fields. In the case of the Schrodinger picture, considering classical vacuum fields, and using a simple calculation for the classical radiation reaction force that is valid in the limit of large mass ( mc 2 � ¯ hω0), we obtain the value ¯ hω0 for the ground state energy of the harmonic oscillator. We show that the vacuum electromagnetic forces play a very important role in the understanding of this discrepancy.  2002 Elsevier Science B.V. All rights reserved.

Anomalous paramagnetic behavior: the role of zero-point electromagnetic fluctuations

Abstract The interaction of a microscopic magnetic dipole and the inductor of a RLC circuit without batteries, is described using the approach of stochastic electrodynamics. The purpose of this study is to clarify the effects of the current fluctuations on the paramagnetic behaviour of a sample of magnetic material which is close to a thin solenoid. A suppression is predicted in the average magnetization, even in the case in which the circuit temperature is arbitrarily close to the absolute zero.

Resonant paramagnetic enhancement of the thermal and zero-point Nyquist noise

Abstract The interaction between a very thin macroscopic solenoid, and a single magnetic particle precessing in a external magnetic field B 0 , is described by taking into account the thermal and the zero-point fluctuations of stochastic electrodynamics. The inductor belongs to a RLC circuit without batteries and the random motion of the magnetic dipole generates in the solenoid a fluctuating current I dip ( t ), and a fluctuating voltage e dip ( t ), with spectral distribution quite different from the Nyquist noise. We show that the mean square value 〈 I dip 2 〉 presents an enormous variation when the frequency of precession approaches the frequency of the circuit, but it is still much smaller than the Nyquist current in the circuit. However, we also show that 〈 I dip 2 〉 can reach measurable values if the inductor is interacting with a macroscopic sample of magnetic particles (atoms or nuclei) which are close enough to its coils.

Einstein-Ehrenfest's radiation theory and Compton-Debye's kinematics

Einstein and Ehrenfests radiation theory is modified in order to take into account the effects of the random zero-point fields, characteristic of classical stochastic electrodynamics, in a system of classical molecules interacting with thermal radiation. This is done by replacing the Einstein concept of “random spontaneous emission” by the concept of stimulated emission by the random zero-point fields. As a result, Compton and Debyes kinematic relations are obtained within the realm of a completely classical theory, that is, without having to consider the wave-particle duality for the molecules or the radiation.

Towards A Classical Re-Interpretation of the Schrodinger Equation According to Stochastic Electrodynamics

We study the statistical evolution of a charged particle moving in phase space under the action of vacuum fluctuations of the zero-point electromagnetic field. Our starting point is the Liouville equation, from which we derive a classical stochastic Schrodinger like equation for the probability amplitude in configuration space. It should be stressed that we are not deriving the Schrodinger wave equation. An equation formally identical to the Schrodinger equation used in Quantum Mechanics is obtained as a particular case of the classical stochastic Schrodinger like equation. An inconsistency appearing in the standard Schrodinger equation, when vacuum electromagnetic fluctuations and radiation reaction are taken into account, is clearly identified and explained. The classical stochastic Schrodinger like equation, however, is consistently interpreted within the realm of Stochastic Electrodynamics.

Maxwell electromagnetic theory, Planck's radiation law, and Bose—Einstein statistics

We give an example in which it is possible to understand quantum statistics using classical concepts. This is done by studying the interaction of chargedmatter oscillators with the thermal and zeropoint electromagnetic fields characteristic of quantum electrodynamics and classical stochastic electrodynamics. Plancks formula for the spectral distribution and the elements of energy hw are interpreted without resorting to discontinuities. We also show the aspects in which our model calculation complement other derivations of blackbody radiation spectrum without quantum assumptions.