Alpha Institute for Advanced Studies

<{x, p}> = 0 = <[ x squared, p squared]>

for all wavefunctions of quantum mechanics. I will continue to test this by hand and Horst Eckardt by computer in the next few days. The symbol { , } is the anticommutator and the symbol [ , ] the commutator. The symbol < > is the expectation value. There is therefore a fundamental relation between commutators and anti-commutators in quantum mechanics. I advise staying well clear of the Heisenberg uncertainty principle and Copenhagen interpretation and to use the original Schroedinger equation and Schroedinger’s own interpretation. In the above equation p is the Schroedinger operator

p = – i h bar partial / partial x

which gives the familiar

[x, p] psi = i h bar psi

where psi is the wavefunction. The anticommutators of angular momentum or rotation generator are zero as is well known:

{J sub X, J sub Y} psi = 0
et cyclicum

so <{x, p}> psi is a related fundamental property. In general {x, p} psi is not zero but its expectation value is zero. Another major problem with Copenhagen is that the expectation value of [x, p] is imaginary valued, i.e. it is i h bar, so has no real part and no physical meaning. In the so called uncertainty principle it is always asserted that the expectation value of relevance is h bar, and that has given rise to the uncertainty relations and endless confusion.

2x partial psi / partial x = – psi

where
p = – i h bar partial / partial x

In this note:

x squared + y squared = r squared = constant

a175thpapernotes8.pdf

m sub J = 0, plus or minus n, n = 1, 2, 3, ………..

So I can request Dr Horst Eckardt, co author of UFT 175, to carry out this research. Whatever the result Copenhagen interpretation is obscure and meaningless. This is because numerical research for UFT 175 has already shown that the anti commutator vanishes for all wavefunctions of the particle in a box and harmonic oscillator, meaning that the commutator of x squared and pp vanishes, so pp can be specified precisely (Copenhagen parlance) if x is specified precisely. Since pp is derived from p, the assertion that p and x cannot be specified simultaneously become untenable, Q.E.D. The particle on a ring is the third well known exact solution of the Schroedinger equation. After evaluation, multiply each anti-commutator by – i h bar. In the first instance assume

x = y

I used the Cartesian representation rather than cylindrical polar, because the latter introduces yet more confusion into an already obscure Heisenberg uncertainty principle for the particle on a ring (for a discussion of thi p oint see Atkins, “Molecular Quantum Mechanics”, page 62, footnote, second edition, OUP, 1983).

a175thpapernotes8.pdf

[J squared, J sub Z] psi = 0
[J sub X, J sub Y] = i h bar J sub Z
et cyclicum

so J squared and J sub Z are knowable and J sub Y and J sub Z are absolutely unknowable. In ECE theory these angular momentum results are simply relations between rotation generators, and there is nothing that is unknowable. All is Baconian and causal. Rotation generators are the same as angular momentum generators within h bar and no one would claim that two rotation generators are unknowable. Anti commutators of rotation generators are zero. There are books full of unscientific rubbish generated by trying to base a philosophy on some results but not others. The hallmark of the twentieth century is the fall back into mediaeval darkness, in more ways than one. Technology has no beneficial effect on the weaknesses of the human mind. Technology is beneficial only in a materialistic sense, so it is a mistake to try to use physicists to generate technology.

a175thpapernotes7.pdf

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.