The Schrödinger Equations and Their Consequences

Baxter H. Armstrong, Edwin Albert Power

It is shown that the condition of finiteness at the origin, usually imposed on solutions to the nonrelativistic Schrödinger equation for the hydrogen atom (and some other systems) is not necessary. The quadratically integrable l = 0 “anomalous” solution which is usually excluded in this fashion is not a solution to the Schrödinger equation at the origin, and consequently does not need to be excluded by a boundary or other type condition. The relativistic generalization through the Dirac equation of this anomalous solution is not quadratically integrable. The solution generalized according to the relativistic Schrödinger equation is quadratically integrable and obeys the equation at the origin. Consequently, a boundary or other type condition may still be needed in a general relativistic theory.

Ralph deLaubenfels

Diego R. Alcoba

Four basic theorems involving the correlation matrices are demonstrated here. One of these theorems establishes, as a necessary and sufficient condition, that the solutions of the fourth-order modified contracted Schr\"odinger equation correspond uniquely to those of the Schr\"odinger equation. The complete equivalence between these two equations is demonstrated. Another equation equivalent to the Schr\"odinger equation but involving only the correlation matrices is also obtained as a consequence of the second theorem.

Vladimir M. Shtelen

We show that the free Schrödinger equation admits Lorentz space-time transformations when corresponding transformations of the ψ-function are nonlocal.Some consequences of this symmetry are discussed.

Felix Alexandrovich Berezin, Mikhail Shubin