Free Particle Quantum Mechanics, Conservation and the Kronecker Delta

 In classical physics, one states in words the existence of conservation of momentum and energy. An example would be an elastic collision of two particles. In particular, this conservation is imposed, it does not mathematically appear unless one describes the process by say a Kronecker delta, i.e.  delta( momentum vector before, momentum vector after) or delta(energy before, energy after). Even in such a case, the Kronecker delta is isolated and so one still has to manually ensure that energies and momenta chosen in a calculation ensure conservation.

   A Kronecker delta is mathematically equivalent to the dot product of two orthonormal states in linear algebra. This implies an operator which may obtain the particular value in question, momentum vector or energy, when operating on the state. A Hermitian matrix operator is guaranteed to have real eigenvalues and orthogonal eigenvectors, which is required. Thus, mathematically, one may postulate energy and momentum states and corresponding operators because this automatically creates the Kronecker delta which describes conservation in  classical processes. This approach holds both nonrelativistically and relativistically.

    The question becomes: How does one specifically describe the operator and its state? For both a relativistic and nonrelativistic free particle classical action, which describes motion,  one may write: A= -Et+px where E,p are the relativistic free particle values in one case and E=p dot p/2mo and p=mov in the other. One may note that E and p are values which may be conserved in interactions. The action shows that the operators -d/dt partial and d/dx partial yield E and p respectively, but one does not have an eigenstate. This may be obtained using  id/dt partial  exp(-iEt+ipx) = E exp(-iEt+ipx) and -id/dx partial exp(-iEt+ipx) = p exp(-iEt+ipx). Because one uses partial derivatives, t and x are treated as independent relative to the time and space translation operators d/dt partial and d/dx partial.

   As a result, we argue that by mathematically linking a Kronecker delta associated with classical conservation with linear algebra, i.e. momentum and energy states and finding a corresponding operator by examining the classical action leads to the quantum free particle wavefunction exp(-iEt+ipx). One may apply this to an elastic two particle collision to see that the Kronecker delta and its associated state/operator form describe conservation of both energy and momentum.One may ask whether exp(ip dot r) has any physical relevance. Two-slit interference experiments using particles with rest mass shows that it does.