Is science sometimes in danger of getting tunnel vision? Recently published ebook author, Ian Miller, looks at other possible theories arising from data that we think we understand. Can looking problems in a different light give scientists a different perspective?

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Archive for October, 2018
In my last post, I mentioned that in my alternative interpretation of quantum mechanics, I rejected the Born interpretation as being fundamental, and instead I argued that because the wave must keep up with the particle it had to transmit energy. That means the square of the amplitude of the wave should equal that energy, as it does in general wave physics. There is a further difference of definition. I use a term, the wave displacement, to reflect at a coordinate the value of ψ.ψ*, whereupon the amplitude becomes the maximum value of that displacement. This does not mean that Born is wrong because if we think of energy density at a point, that should be roughly proportional to probability, but there are differences. The wave has nodes, and there is a lot of arm-waving in answer to the question, in a stationary state such as an orbital, how does an electron cross a nodal surface, because as soon as it is on it, the probability of it being there is non-zero, but you cannot go from plus to minus, i.e. crest to trough, without going through zero. The problem goes away if the square of the amplitude reflects an energy rather than probability.

There is one further difference. The phase of the wave is given by exp(2πiS/h). That is generally agreed in all textbooks, although it is usually written differently, using %u0127 instead. Now, what is action? Basically, it is the time integral of the Lagrange function, which in turn is an energy (usually the difference between the kinetic and potential energy).  What is important here is that it is something that increases with time, which is why the phase proceeds as an oscillation. The stationary wave, as in the particle in a box, is actually really two running waves proceeding in equal and opposite directions. The reason that is relevant is that from Euler, who developed the mathematics of complex numbers,  exp(iπ) = -1. Thus when S = nh  the value of the phase is 1; when S = nh/2, n odd, the value of the phase is -1.  All other values of S lie somewhere between. In the textbooks, you will see that ψ is always complex.  That is not exactly true and at the antinodes, it becomes real, with A, A the amplitude.
So, what does that mean? Let us assume, for the moment, that the wave only means something physically when real. You may well say, that is just another assumption, but I argue it is not an additional one to what we already have because if so, then it is easy to derive the Uncertainty Principle, which, as an aside, now becomes a physical principle and has nothing to do with the observer. Physicists would generally accept that. As for what it means for general quantum mechanics, apart from requiring the Uncertainty Principle, which was already required, not much because you cannot know the phase of a quantal matter running wave.

However, for a stationary wave, such as the particle in a box, or the wave for a stationary state, as in a free atom or molecule, the antinode defines the amplitude of the wave, and the square of the amplitude of the wave is proportional to the energy of the system. If so, the problem for the chemical bond reduces itself to finding the location of the antinode, and applying the electric field coupling at that point. That is relatively easy because when the nature of the motion is independent of time, the action can be represented as ∫pdq, p the momentum, q the generalized coordinate, and from our discussion on Euler, the action must be quantized over a period, which gives the de Broglie relation pλ = h. So, it is comforting to know that, coming from a somewhat different route that is not conventional, we arrive at a totally conventional relationship. The important point here is the quantization of action gives the covalent radius and that radius gives the bond energy through calculating the potential energy at that point. This means that a good approximation to the bond length and energy of the hydrogen molecule is essentially no more complicated than mental arithmetic, or with a hand held calculator, solely from thinking of the electric field. Errors are about 0.3%.  Better agreement can be found by including some of the more minor contributions.

For molecules in general, there is a little more to it than that because the nature of the waves between the nuclei change for atoms other than hydrogen but the need for computation is still within the hand-held calculator's ability. There are still deviations from observed values, but the results should be close enough to be useful, and as an example, the dissociation of the Sb2 molecule is calculated to be 298.1 kJ/mol (obs 299.2).  In fairness, that is one of the better results, and bismuth, boron and sodium are relatively poorly behaved, but that is in part because their atomic ionization potentials do not behave well either. If interested, details can be obtained in an ebook:
Posted by Ian Miller on Oct 7, 2018 10:02 PM BST