Non-Classical Selection of the Physical Solutions
Pp. 110-127 (18)
The energy parameter in the stationary Schrödinger equation is primarily
continuous. Schrödinger has selected the discrete quantum states by means of
boundary conditions. This, however, is mathematics for classical systems like
strings and pipes. And, more important, it hides the intrinsic discrete structure of
that equation. The latter is represented by the recursion relations between solutions
to different values of the energy parameter. These follows from Whittaker’s integral
expressions of the solutions and hold true for all solutions, not only for
Schrödinger’s eigensolutions. The physically relevant solutions are distinguished by
certain mathematical properties as well as physical criteria, in particular, by their
compliance with the absence of perpetua mobilia. This non-classical approach is
exemplified by means of the model system harmonic oscillator. For this, the chapter
starts with a sketch of the classical harmonic oscillator, including quite general
topics such as the separation of internal and external system parameters and
Huygens’ principle, which will reappear in the quantum realm.
Boundary conditions, Eigensolutions, Energy parameter, External
system parameter, Harmonic oscillator, Huygens’ principle, Internal system
parameter, Perpetuum mobile, Stationary Schrödinger equation, Selection
problem, Wave function, Weber’s equation, Whittaker’s integral expressions.
Taraz State Pedagogical University, Kazakhstan.