Role of Space Time in Jainas Syadavada and Quantum Theory

Author(s): Filita Bharucha
Publisher: Shri Satguru Publications

Page 27

Role of Quantum Theory in Deviant Logic
given to the logical connectives, this distributive law remains valid.
Experimental Propositions of Quantum Mechanics: It is well known that in classical mechanics the state of a mechanical system at any time is completely known if one specifies in generalized coordinates 'q' and in generalized momenta 'p' at that time, where is the number of degrees of freedom of the mechanical system. There can be obtained by solving Hamilton's equations of motion with the initial conditions specified by the values of 'q' and 'p' at time t = o. Quantum mechanical state of microscopic systems is represented by a certain wave-function y (or a state vector 1x>) which can be explained as a linear combination of eigenfunctions ui (or eigenvectors |i>1) of an operator representing an observable. The wave-function y (or the state vector x>) is obtained by solving a differential equation known as Schrodinger equation which is of first order in time, the initial condition being the value of y at time t = 0. In what follows we accept quantum mechanics in its conventional form as the correct representation of the microworld and accordingly build up our formalism.
17
Let an observable in quantum mechanics be denoted by A with the corresponding Hermitian operator denoted by the same letter A for convenience. We know from quantum mechanics that A must possess sufficient number of eigenstates, often infinite, that any state vector whatever can be expanded in terms of the corresponding eigen-vectors. The spectrum of the corresponding eigenvalues may be discreet (finite or denumerably infinite, degenerat or non-degenerate) of continuous. For our analysis we shall assume that the eigenvalue spectra of all the operators we shall introduce, be discrete, finite and non-degenerate. Let the eigenvectors of A be noted by |a> where i = 1, 2........., n ( < ∞). If A represented an observable, then any arbitrary state vector 1x> can be expanded as