The purpose of this note is to investigate dynamical features of the 2x2 matrix approach contained in Einstein´s 1905 energy momentum equation E2 = p2 + m2 (1) In particular, we wish to investigate the existence of physical oscillations consistent with the model. The model was based on conversion of (1) into a 2x2 matrix eigenvalue system with various matrices as operators. The matrices and eigenvectors obey quantum mechanical math with exp(-iHt) as a time evolution operator. It is thus possible to calculate O(t)=exp(iHt) O exp(-iHt) where O is an operator. This leads to oscillatory motion along and perpendicular to the x axis in the one dimensional approach used. This oscillatory motion is considered in sme detail and comparisons with transformations of light from a stationary to moving frame are made.
The purpose of this note is to investigate dynamical features of the 2x2 matrix approach contained in Einstein´s 1905 energy momentum equation E2 = p2 + m2 (1) In particular, we wish to investigate the existence of physical oscillations consistent with the model. The model was based on conversion of (1) into a 2x2 matrix eigenvalue system with various matrices as operators. The matrices and eigenvectors obey quantum mechanical math with exp(-iHt) as a time evolution operator. It is thus possible to calculate O(t)=exp(iHt) O exp(-iHt) where O is an operator. This leads to oscillatory motion along and perpendicular to the x axis in the one dimensional approach used. This oscillatory motion is considered in sme detail and comparisons with transformations of light from a stationary to moving frame are made.
In a series of notes we examined a 2x2 matrix model which is naturally incorporated into Einstein´s 1905 energy momentum equation p2 + m2 = E2 (1). We showed that it was possible to obtain energy, velocity and momentum matrices, calculate expectation, dispersion values and establish exp(-iHt) as the time evolution operator. From this, we calculated zitterbewegung and the time dependence of other matrices. The purpose of this note is to see if this formalism, established for a relativistic model can also be applied to the nonrelativistic case. Several cases are considered. Next, we try to apply the model to the case of a classical spring, which is related to the production of phonons.In this case, we apply the mathematics of the model to the classical spring, not the model itself. It is found that one can obtain a velocity matrix and find an energy matrix with time evolution of exp(-iHt). Furthermore, it appears that time dependent operator calculations predict classical periodic motion with a period of sqrt(k/m), where k is the spring constant and m the mass.
In a series of notes, it was seen that Einstein’s 1905 energy momentum equation E2 = p2 + m2 (1) can be written as an eigenvalue equation linear in E, p and m. This equation applies to particles and leads to zitterbewegung results which appear in the literature. [ In a note, it was seen how the 2x2 matrix eigenvalue equation was general and could be applied to a photon moving in the y direction in a frame moving along the x axis with speed v. It was then argued that zitterbewegung should be applied to photons as well. In this note, we try to investigate this further. In addition, we attempt to clarify x and y direction motion of photons and their relation to the 2x2 model. We also examine zitterbewegung in EM energy density equations such as: energy density= .5(E2 +B2) and try to apply it to the cases of a photon moving in the y direction in a box moving in the x direction and to a photon moving in the x direction in a similar box. In such cases, the E field carries effects of the translational motion which are part of the zitterbewegung.
In a series of notes it was shown that a 2x2 matrix energy eigenvalue equation is naturally contained in Einstein´s 1905 energy momentum equation E2 = p2 + m2 . This equation, which appears to be related to the Dirac equation, but contains no spatial features, is linear in energy and describes bouncing of a particle back and forth with an overall average motion of v to the right. It appears to be quantum mechanical in nature as the energy matrix acts as an evolution operator, expectation values of operators are taken and one has an energy eigenvector representing the square roots of probability. Knuth [18 ] has used a model which yields directly probabilities for a particle bouncing back and forth with an average v to the right. It was shown that these probabilities, the squares of the terms of the energy eigenvector, can be obtained by maximizing Shannon´s entropy subject to the constraint the f1-f2= v, where f1 is the probability to move forward and f2, that to move backward. In this note, we try to find a link between the 2x2 matrix equation and the Schrodinger (or Dirac) equation in a region with a potential V(r). The Schrodinger equation, unlike the matrix equation, is spatial. Next, we try to see if there are any features of zitterbewegung that might apply in this case. In the 2x2 matrix model, Schrodinger´s zitterbewegung can be derived, so it seems reasonable to ask whether there might be zitterbewegung in the more complicated case of a bound state in a potential. Finally, we raise the issue of a possible link between the Schrodinger equation and statistical mechanics. In the 2x2 model, the forward and backward postulated motion of a particle was connected to maximum entropy. It would be interesting if a similar idea could be applied to bound states. We do not present any calculations related to such entropy maximization, but mention work in the literature in which Tsallis entropy is maximized subject to constraints to obtain the ground state wavefunction without solving the single particle temperature=0 Schrodinger equation.
In a number of notes , a linear 2x2 matrix Dirac type equation was examined as a solution of Einstein’s 1905 energy momentum equation E2=p2+m2 (1). This model was able to reproduce Schrodinger zitterbewegung.To do this, however, one had to identify the 2x2 energy matrix H as being part of the evolution operator exp(-iHt). It was seen that the identity matrix together with H could transform any two vector into the eigenvector of H, which is taken as the stable solution. This idea of an evolution operator thus seems to be important. In addition, the model led to a 2-eigenvector which was interpreted as the square root of two components of probability, one in which a particle moves forward the other in which it moves backward Overall, the particle moves forward with a velocity of v along the x axis. Knuth has also proposed such a model. It was found that the probabilities which came from this quantum mechanical type matrix solution matched those obtained by maximizing statistical entropy. In a note , the 2x2 matrix model was applied to a system of classical particles which could have energy e1 or e2. It was argued that the 2x2 model predicted zitterbewegung for this statistical system. The object of this note is to explore whether the 2x2 model can be used to predict zitterbewegung in a statistical system with a potential.
Previously a 2x2 matrix equation, linear in energy, was obtained from Einstein’s energy momentum equation E2=p2+m2 which reproduced Schrodinger zitterbewegung [1]. A similar equation was applied to the case of a gas with two allowable energy levels and to a gas in a potential field, the goal beng to obtain zitterbewegung results in statistical mechanics. The starting point for applying the 2x2 matrix equation to statistical mechanics was the fact that the 2x2 matrix approach yields an eigenvector which is the square root of two probabilities which sum to one. In this note, we wish to see if we can extend ideas further. First, we wish to to see if there are stronger reasons for applying the 2x2 matrix approach to statistical mechanics, and second, we wish to find further analogies between the approach and statistical mechanics. We consider relationships between physical constraint equations and the 2x2 matrix approach., We also consider functions analogous to the rest mass and energy in the statistical case, where the analogous energy also represents the frequency in zitterbewegung.
A 2x2 matrix model of Einstein´s energy momentum equation appears to have links to statistical mechanics. First, it seems to suggest using dp (change in momentum) as a constant when counting energy levels in statistical mechanical calculations of the partition function. It also has further links to the de Broglie wavelength which is used to justify the use of equally spaced dp for energy state counting. It also appears that a 2x2 matrix model is consistent with Knuth´s statistical model. The two component positive energy eigenvector represents square roots of forward and backward motion probabilities. These probabilities maximize entropy. Thus, it appears that the 2x2 model, which uses quantum mechanical formalism, is also equivalent to a statistical mechanical model. In other words, at least in this model, there appears to be a link between statistical mechanics and quantum mechanics . It should be noted, however, that in this note very little has been shown about this link other than the equivalency of the energy eigenvector with maximum entropy. It is also suggested that zitterbewegung could lead to statistical fluctuations and that they may account for issues in statistical mechanics when interparticle separations reach the size of the de Broglie wavelength.