2X2 Matrix Model And The Schrodinger Equation
Francesco R. Ruggeri
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.
•Aug 10, 2018•0 citations