Wave Equations and the Conservation of Energy

A. Lees

Summary There are two conditions which a wave equation should satisfy, one corresponding to the conservation of probability, the other expressing a relation which, in the case in which the Hamiltonian does not involve the time explicitly, corresponds to the conservation of energy. It is known that the first condition is satisfied by Schrödinger's non-relativistic equation and by Dirac's equation, but not by the second-order relativistic equation. We prove that the second condition also is satisfied in the same two cases. These facts seem to provide a stronger reason for rejecting the second-order relativistic equation than those usually given.

Peter J. Riggs

Joe L. Ferguson

Francesco R. Ruggeri

In a number of notes (Parts I,II,III) we considered the problem of photons A moving in the positive x direction striking a medium at x=0 and reflecting B and refracting C. In (1) conservation of energy: I(A)/c1 = I(B)/c1 + I(C)/c2 ((1)), where I(A) is intensity which has the form of energy * velocity i.e. a flux, is given. Thus I(A)/c1 = n1 E , I(B)=n2 E and I(C)=n3 E, where E is constant energy of a photon and ni are the number of photons. Conservation of energy/or photon is a conservation law in terms of “particles” and is furthermore a time averaged result. A single photon either reflects or refracts. In both cases conservation of particle number and energy exists and so one may wonder why one needs a time averaged “particle” type conservation equation in the first place. We argue that it demonstrates the role of flux in this problem - a role which differs from that of hydrodynamics in classical physics. ((1)) shows immediately that there is neither conservation of flux i.e. I(A) values or momentum even in a steady state scenario (i.e. time averaged scenario) represented by ((1)). This is different from an incompressible liquid moving from a wide pipe to a narrow one. In such a case flux is conserved yet microscopic particles change velocity, momentum and energy. As noted in Part I, one may write: I(A)/c1 - I(B)/c1 = I(C)/c2 , multiply by E (energy) and use E=pc. This yields a “kind” of momentum conservation equation, but not in terms of particle numbers (i.e. not in a particle picture). Rather I(A)p is momentum multiplied by flux which is unusual. ((1)) is a conservation of particle number and energy, but does not include conservation of flux or momentum flux which one might expect in a steady state flow equation and which appears in classical fluid dynamics. As noted in Part I to obtain these additional conservation/balance equations one is forced to use square root probabilities i.e. I(A)=AA with A representing a square root flux or dynamic probability. Using the idea of differences of squares, one may write: A + B = C ((2a)) and Ap1 -p1B = p2C ((2b)) and solve for B and C in terms of A. In other words a hydrodynamical approach applies to the square root probabilities A,B,C and not to I(A)=AA etc. Ultimately squaring, one obtains I(A),I(B) and I(C) showing that the square root probability or wave/dynamical probability represents balance and conservation. In ((2a)) and ((2b)) i.e. with square root probabilities one has dropped the view of particles and changed to a square root flux/ new dynamical probability picture together with its balances. This is in fact the wave picture because A,B,C are linked with exp(ip1x),exp(-ip1x) and exp(ip2x) (which for photons are associated with the wave picture of the electric and magnetic fields). Thus even though there is particle wave duality, in terms of conservation waves (square root probabilities) yield particle conservation equations. For example, it is the wave picture which allows one to solve for B and C in terms of A. The wave picture may be associated with a steady state (time removed) equilibrium scenario in which dynamical (square root) probabilities add in space (not by time grouping) and it contains both conservation of probability and momentum probability while momentum (of the particles) may not be conserved in the particle picture as is the case for the reflection/refraction problem. Particle number/energy are also guaranteed.

J. Formanek, R.J. Lombard, J. Mareš

Edward J. Gillis

The assumption that wave function collapse is induced by the interactions that generate decoherence leads to a stochastic collapse equation that does not require the introduction of any new physical constants and that is consistent with conservation laws. The collapse operator is based on the interaction energy, with a variable timing parameter related to the rate at which individual interactions generate the branching process. The approximate localization of physical systems follows from the distance-dependent nature of the interactions. The equation is consistent with strict conservation of momentum and orbital angular momentum, and it is also consistent with energy conservation within the accuracy allowed by the limited forms of energy that can be described within nonrelativistic theory. A relativistic extension of the proposal is outlined.