HAMILTONIAN MATRIX REPRESENTATIONS FOR THE DETERMINATION OF APPROXIMATE WAVE FUNCTIONS FOR MOLECULAR RESONANCES

Wave functions obtained using a standard complex Hamiltonian matrix diagonalization procedure are square integrable and therefore constitute only approximations to the corresponding resonance solutions of the Schrödinger equation.   The nature of this approximation is investigated by means of explicit calculations using the above method which employ accurate diabatic potentials of the B 1Σ+ - D’ 1Σ+ vibronic resonance states of the CO molecule.  It is shown that expanding the basis of complex harmonic oscillator functions gradually improves the description of the exact resonance wave functions out to ever larger internuclear distances before they take on their unwanted bound-state characteristics.  The justification of the above matrix method has been based on a theorem that states that the eigenvalues of a complex-scaled Hamiltonian H (ReiΘ) are associated with the energy position and linewidth of resonance states (R is an internuclear coordinate and Θ is a real number).  It is well known, however, that the results of the approximate method can be obtained directly using the unscaled Hamiltonian H (R) in real coordinates provided a particular rule is followed for the evaluation of the corresponding matrix elements.  It is shown that the latter rule can itself be justified by carrying out the complex diagonalization of the Hamiltonian in real space via a product of two transformation matrices, one of which is unitary and the other is complex orthogonal, in which case only the symmetric scalar product is actually used in the evaluation of all matrix elements.  There is no limit on the accuracy of the above matrix method with an un-rotated Hamiltonian, so that exact solutions of the corresponding Schrödinger equation can in principle be obtained with it.  This procedure therefore makes it unnecessary to employ a complex-scaled Hamiltonian to describe resonances and shows that any advantages which have heretofore been claimed for its use are actually non-existent

THE CLOCK PUZZLE AND THE INCOMPATIBILITY OF PROPORTIONAL TIME DILATION AND REMOTE NON-SIMULTANEITY

The slowing down of the rates of clocks by virtue of their motion (time dilation) and the conclusion that events which are simultaneous for one such clock may occur at different times for another (remote non-simultaneity) are two of the most well-known consequences of Einstein's Special Theory of Relativity (STR).  A simple algebraic puzzle is presented which shows that these two predictions are fundamentally incompatible with each other.  Both effects are derived from the Lorentz transformation (LT), which is the cornerstone of STR, thereby proving that the theory is not physically viable and is therefore in need of comprehensive revision.  Another version of the Lorentz transformation (Global Positioning System-LT) is presented which satisfies both of Einstein's postulates of relativity, but which does away with the space-time mixing characteristic of the original LT.  The GPS-LT is in agreement with all experimental data as yet observed regarding the variation of clock rates with motion, as expressed in a Universal Time-dilation Law (UTDL). It is also consistent with the Relativistic Velocity Transformation (RVT) and is therefore able to explain many of the effects previously looked upon as unique successes of the LT. Â

MAXWELL'S EQUATIONS, THE RELATIVITY PRINCIPLE AND THE OBJECTIVITY OF MEASUREMENT

The interpretation of the velocity parameter in Maxwell's equations and the Lorentz Force law is examined.  It is shown that the predicted trajectories of electrons in an electromagnetic field depend on the state of motion of the observer.  This ambiguity can be removed by assuming that the origin of the field is the unique reference system for determining the electron velocity in these equations.  The same convention is recommended for the definition of the momentum of particles in general.  Accordingly, it is assumed that observers will only disagree on the results of measurements because they employ different systems of physical units in their respective rest frames. This situation needs to be recognized in applying Galileo's Relativity Principle (RP), particularly with respect to the transformation properties of physical laws in different rest frames

Expressing the Units of Electricity and Magnetism Directly in the MKS System

Since the unit of electric charge can be chosen independently of the value of the permittivity of free space ε0, it is shown that all electromagnetic quantities can also be assigned units directly in the MKS system.  For example, the unit of electric charge can be 1 J as long as ε0 has units of 1 N.  A table is given that makes a comprehensive comparison of the standard units in the Giorgi system with those in two such direct MKS schemes.  A simple procedure is also described for changing the numerical values of the units in a systematic manner by dividing the various electromagnetic quantities into five distinct classes.  This allows one to equate the value of ε0 to 1/4π, for example, similarly as for the Gaussian system of units, while still retaining the same formulas as in the Giorgi system.

CRITIQUE OF THE TREATMENT OF EINSTEIN'S SPECIAL THEORY OF RELATIVITY IN ISAACSON'S BIOGRAPHY

Consideration is given to the properties expected for inertial systems, that is, objects which are not subject to unbalanced external forces. Isaacson points out correctly that Einstein based his version of relativity theory on the motion of such inertial systems. It is overlooked, however, in both his discussion and also the original work of Einstein himself that, consistent with Newton's First Law of Motion (Law of Inertia), the rates of inertial clocks must remain constant so long as no unbalanced force is applied to them. As a consequence, it can be safely concluded that the ratio of any two such rates must also be constant. This in turn leads to a prediction about the relationship between elapsed times measured by two inertial clocks (Newtonian Simultaneity), namely they must always occur in strict proportion to one another (Δt=QΔt'). The latter result is shown to stand in direct contradiction to the predictions of the Lorentz transformation derived by Einstein in his famous 1905 paper, namely the occurrence of space-time mixing and remote non-simultaneity (RNS). It is furthermore pointed out that the frequency of sound waves is independent of the state of motion of the source because, as Einstein argued in a 1907 paper in which he derived the gravitational red shift, the number of wave crests emitted by the source per unit time is not changed thereby. This fact has a definite bearing on Einstein's postulate of relativity according to which he assumed that the speed of light in free space is independent of the state of motion of both the observer and the source. It indicates instead that the postulate must be reformulated to state that the speed of light relative to its source is always the same in free space, a version that Einstein also carefully considered, as clearly mentioned in Isaacson's narrative. Accordingly, it becomes clear that Einstein was incorrect in his claim that the classical (Galilean velocity transformation) is not applicable to light. His relativistic velocity transformation only has validity for specific cases in which the speed of light is measured under different circumstances by a single observer, such as in the famous Fresnel/Fizeau light-drag experiment in which the motion of light waves in refractive media is investigated. The failure of the Lorentz transformation necessitates a rethinking of some of its well-known predictions such as time dilation and Lorentz-FitzGerald length contraction. Experimental evidence is presented which indicates instead that the slowing down of clocks upon acceleration is accompanied by isotropic length expansion. The applicability of these theoretical developments for  the Global Positioning System of navigation is discussed. Opportunities for carrying out new experiments on this basis are also outlined in the present critique. 

FREQUENCY VARIATIONS, THE SPEED OF SOUND AND THE GRAVITATIONAL RED SHIFT

The relationships between frequency, wavelength and speed of waves is considered for both sound and light waves. Einstein's prediction of the gravitational red shift was based on an important observation about the dependence of measured frequencies of light waves on gravitational potential.  He pointed out that the only reason observers located on the earth's surface and the lower potential of the sun could differ on their value of a given light frequency is because their respective timing devices do not run at the same rate, even though they are otherwise completely equivalent ("gleich beschaffen").  This amounts to assuming that the unit if time varies with gravitational potential.  The analogous conclusion holds for all other physical properties.  In effect, this means that there is a conversion factor between the units of each quantity for any pair of rest frames that the respective stationary observers located there can use to quantitatively relate the measurements of one another.   How this factor, referred to as S, can be determined in general is discussed.  Furthermore, it is argued on the basis of experiments with atomic clocks that an analogous conversion factor (Q) can also be determined for each pair of rest frames.   On basis of the Law of Causality it can be assumed that the rate of an inertial clock cannot change spontaneously.  This conclusion (Newtonian Simultaneity)  rules out any occurrence of remote non-simultaneity, contrary to what is predicted on the basis of the Lorentz transformation