To build on these amazing results I think that other exact solutions of the Schroedinger equation should be tested in the same way. I have the second edition of Atkins, “Molecular Quantum Mechanics” in which he gives solutions for free translation, particle in a box, square well, harmonic oscillator, particle on a ring and sphere, and the H atom. In Eq. (5.5.2) of the second edition he gives the basic result that if [H, A] = 0, then (d / dt) <A> = 0, the constant of motion result of quantum mechanics. In this equation:

H = p squared / (2m) + V

where V is the potential energy. However no one seems to have ever tried to work out [x squared, p squared] and it has just been found to be zero for all the wavefunctions of the particle in a box and harmonic oscillator. On page 93 he gives the results:

[p, x squared] psi = – 2 i h bar x psi

[p squared, x] psi = – 2 i h bar p psi

We know also from note 175(6) that:

[x squared, p squared] psi = 2 h bar squared psi + 4 i h bar x p psi

On page 93 he gives the standard Heisenberg uncertainty principle. When [A,B] = 0, then it is possible to prepare a system in a state where delta A = delta B = 0, and both A and B can be specified precisely. So we have found that both x squared and p squared can be specified precisely for all wavefunctions of the particle in a box and harmonic oscillator – well known exact solutions of the Schroedinger quation. However for the same wavefunctions x and p cannot be specified precisely. The standard Copenhagen interpretation becomes absurd. It claims that if x is specified precisely, then p is unknowable. However, if x squared is specified precisely, then p squared is fully knowable (i.e. specified precisely). Since p squared is obtained from an unknowable p, how can it be fully knowable if p is completely unknowable? In the causal interpretation (of which ECE is a part), then these results are logical outcomes of the Schroedinger operator p – no more and no less.

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