What does Quantum Mechanics Mean?

Patrice Ayme gave a long comment to my previous post that effectively asked me to explain in some detail the significance of some of my comments on my conference talk involving quantum mechanics. But before that, I should explain why there is even a problem, and I apologise if the following potted history seems a little turgid. Unfortuately, the background situation is important. 

First, we are familiar with classical mechanics, where, given all necessary conditions, exact values of the position and momentum of something can be calculated for any future time, and thanks to Newtom and Leibniz, we do this through differential equations involving familiar concepts such as force, time, position, etc. Thus suppose we shot an arrow into the air and ignored friction and we wanted to know where it was, when. Velocity is the differential of position with respect to time, so we take the velocity and integrate it. However, to get an answer, because there are two degrees of freedom (assuming we know which direction it was shot) we get two constants to the two integrations. In classical mechanics these are easily assigned: the horizontal constant depends on where it was fired from, and the other constant comes from the angle of elevation. 

Classical mechanics reached a mathematical peak through Lagrange and Hamilton. Lagrange introduced a term that is usually the difference between the potential and kinetic energy, and thus converted the problem from forces to one of energy. Hamilton and Jacobi converted the problem to one involving action, which is the time integral of the Lagrangian. The significance of this is that in one sense action summarises all that is involved in our particle going from A to B. All of these variations are equivalent, and merely reflect alternative ways of going about the problem, however the Hamilton Jacobi equation is of special significance because it can be mathematically transformed into a mathematical wave expression. When Hamilton did this, there were undoubtedly a lot of yawns. Only an abstract mathematician would want to represent a cannonball as a wave.

So what is a wave? While energy can be transmitted by particles moving (like a cannon ball) waves transmit energy without moving matter, apart from a small local oscillation. Thus if you place a cork on the sea far from land, the cork basically goes around in a circle, but on average stays in the same place. If there is an ocean current, that will be superimposed on the circular motion without affecting it. The wave has two terms required to describe it: an amplitude (how big is the oscillation?) and a phase (where on the circle is it?).

Then at the end of the 19th century, suddenly classical mechanics gave wrong answers for what was occurring at the atomic level. As a hot body cools, it should give radiation from all possible oscillators and it does not. To explain this, Planck assumed radiation was given off in discrete packets, and introduced the quantum of action h. Einstein, recognizing the Principle of Microscopic Reversibility should apply, argued that light should be absorbed in discrete packages as well, which solved the problem of the photoelectric effect. A big problem arose with atoms, which have positively charged nuclei and electrons moving around it. To move, electrons must accelerate, and hence should radiate energy and spiral into the nucleus. They don’t. Bohr “solved” this problem with the ad hoc assumption that angular momentum was quantised, nevertheless his circular orbits (like planetary orbits) are wrong. For example, if they occurred, hydrogen would be a powerful magnet and it isn’t. Oops. Undeterred, Sommerfeld recognised that angular momentum is dimensionally equivalent to action, and he explained the theory in terms of action integrals. So near, but so far.

The next step involved the French physicist de Broglie. With a little algebra and a bit more inspiration, he represented the motion in terms of momentum and a wavelength, linked by the quantum of action. At this point, it was noted that if you fired very few electrons through two slits at an appropriate distance apart and let them travel to a screen, each electron was registered as a point, but if you kept going, the points started to form a diffraction pattern, the characteristic of waves. The way to solve this was if you take Hamilton’s wave approach, do a couple of pages of algebra and quantise the period by making the phase complex and proportional to the action divided by (to be dimensionally correct bcause the phase must be a number), you arrive at the Schrödinger equation, which is a partial differential equation, and thus is fiendishly difficult to solve. About the same time, Heisenberg introduced what we call the Uncertainty Principle, which usually states that you cannot know the product of the position and the momentum to better than h/2π. Mathematicians then formulated the Schrödinger equation into what we call the state vector formalism, in part to ensure that there are no cunning tricks to get around the Uncertainty Principle.

The Schrödinger equation expresses the energy in terms of a wave function ψ. That immediately raised the question, what does ψ mean? The square of a wave amplitude usually indicats the energy transmitted by the wave. Because ψ is complex, Born interpreted ψ.ψ* as indicating the probability that you would find the particle at the nominated point. The state vector formalism then proposed that ψ.ψ* indicates the probability that a state will have probabilities of certain properties at that point. There was an immediate problem that no experiment could detect the wave. Either there is a wave or there is not. De Broglie and Bohm assumed there was and developed what we call the pilot wave theory, but almost all physicists assume, because you cannot detect it, there is no actual wave.

What do we know happens? First, the particle is always detected as a point, and it is the sum of the points that gives the diffraction pattern characteristic of waves. You never see half a particle. This becomes significant because you can get this diffraction pattern using molecules made from 60 carbon atoms. In the two-slit experiment, what are called weak measurements have shown that the particle always goes through only one slit, and not only that, they do so with exactly the pattern predicted by David Bohm. That triumph appears to be ignored. Another odd feature is that while momentum and energy are part of uncertainty relationships, unlike random variation in something like Brownian motion, the uncertainty never grows

Now for the problems. The state vector formalism considers ψ to represent states. Further, because waves add linearly, the state may be a linear superposition of possibilities. If this merely meant that the probabilities merely represented what you do not know, then there would be no problem, but instead there is a near mystical assertion that all probabilities are present until the subject is observed, at which point the state collapses to what you see. Schrödinger could not tolerate this, not the least because the derivation of his equation is incompatible with this interpretation, and he presented his famous cat paradox, in which a cat is neither dead nor alive but in some sort of quantum superposition until observed. The result was the opposite of what he expected: this ridiculous outcome was asserted to be true, and we have the peculiar logic applied in that you cannot prove it is not true (because the state collapses if you try to observe the cat). Equally, you cannot prove it is true, but that does not deter the mystics. However, there is worse. Recall I noted when we integrate we have to assign necessary constants. When all positions are uncertain, and when we are merely dealing with probabilities in superposition, how do you do this? As John Pople stated in his Nobel lecture, for the chemical bonds of hydrocarbons, he assigned values to the constants by validating them with over two hundred reference compounds. But suppose there is something fundamentally wrong? You can always get the right answer if you have enough assignable constants.The same logic applies to the two-slit experiment. Because the particle could go through either slit and the wave must go through both to get the diffraction pattern, when you assume there is no wave it is argued that the particle goes through both slits as a superposition of the possibilities. This is asserted even though it has clearly been demonstrated that it does not. There is another problem. The assertion that the wave function collapses on observation, and all other probabilities are lost actually lies outside the theory. How does that actually happen? That is called the measurement problem, and as far as I am aware, nobody has an answer, although the obvious answer, the probabilities merely reflected possibilities and the system was always just one but we did not know it is always rejected. Confused? You should be. Next week I shall get around to some from my conference talk that caused stunned concern with the audience.

Science in Action – or Not

For my first post in 2019, I wish everyone a happy and prosperous New Year. 2018 was a slightly different year than most for me, in that I finally completed and published my chemical bond theory as an ebook; that is something I had been putting off for a long time, largely because I had no clear idea what to do with the theory. There is something of a story behind this, so why not tell at least part of it in my first blog post for the year? The background to this illustrates why I think science has gone slightly off the rails over the last fifty years.

The usual way to get a scientific thought over is to write a scientific paper and publish it in a scientific journal. These tend to be fairly concise, and primarily present a set of data or make one point. One interesting point about science is that if it is not in accord with what people expect, the odds on are it will be ignored, or the journals will not even accept it. You have to add to what people believe to be accepted. As the great physicist Enrico Fermi once said, “Never underestimate the joy people derive from hearing something they already know.” Or at least think they know. The corollary is that you should never underestimate the urge people have to ignore anything that seems to contradict what they think they know.

My problem was I believed the general approach to chemical bond theory was wrong in the sense it was not useful. The basic equations could not be solved, and progress could only be made through computer modelling, together with as John Pople stated in his Nobel lecture, validation, which involved “the optimization of four parameters from 299 experimentally derived energies”. These validated parameters only worked for a very narrow range of molecules; if they were too different the validation process had to be repeated with a different set of reference molecules. My view of this followed another quote from Enrico Fermi: I remember my friend Johnny von Neumann used to say, “with four parameters I can fit an elephant and with five I can make him wiggle his trunk.” (I read that with the more modern density functional theory, there could be up to fifty adjustable parameters. If after using that many you cannot get agreement with observation, you should most certainly give up.)

Of course, when I started my career, the problem was just plain insoluble. If you remember the old computer print-out, there were sheets of paper about the size of US letter paper, and these would be folded in a heap. I had a friend doing such computations, and I saw him once with such a pile of computer paper many inches thick. This was the code, and he was frantic. He kept making alterations, but nothing worked – he always got one of two answers: zero and infinity. As I remarked, at least the truth was somewhere in between.

The first problem I attacked was the energy of electrons in the free atoms. In standard theory, the Schrödinger equation, when you assume that an electron in a neutral atom sees a charge of one, the binding energy is far too weak. This is “corrected”througha “screening constant”, and each situation had its own “constant”. That means that each value was obtained by multiplying what you expect by something to give the right answer. Physically, this is explained by the electron penetrating the inner electron shells and experiencing greater electric field.

What I came up with is too complicated to go into here, but basically the concept was that since the Schrödinger equation (the basis of quantum mechanics) is a wave equation, assume there was a wave. That is at odds with standard quantum mechanics, but there were two men, Louis de Broglie and David Bohm, who had argued there was a wave that they called the pilot wave. (In a recent poll of physicists regarding which interpretation was most likely to be correct, the pilot wave got zero votes.) I adopted the concept (well before that poll) but I had two additional features, so I called mine the guidance wave.

For me, the atomic orbital was a linear sum of component waves, one of which was the usual hydrogen-like wave, plus a wave with zero nodes, and two additional waves to account for the Uncertainty Principle. It worked to a first order using only quantum numbers. I published it, and the scientific community ignored it.

When I used it for chemical bond calculations, the results are accurate generally to within a few kJ/mol, which is a fraction of 1% frequently. Boron, sodium and bismuth give worse results.  A second order term is necessary for atomic orbital energies, but it cancels in the chemical bond calculations. Its magnitude increases as the distance from a full shell increases, and it oscillates in sign depending on whether the principal quantum number is odd or even, which results when going down a group of elements, that the lines joining them zig zag.

Does it matter? Well, in my opinion, yes. The reason is that first it gives the main characteristics of the wave function in terms only of quantum numbers, free f arbitrary parameters. More importantly, the main term differs depending on whether the electron is paired or not, and since chemical bonding requiresthe pairing of unpaired electrons, the function changes on forming bonds. That means there is a quantum effect that is overlooked in the standard calculations. But you say, surely they would notice that? Recall what I said about assignable parameters? With four of them, von Neumann could use the data to calculate an elephant! Think of what you could do with fifty!

As a postscript, I recently saw a claim on a web discussion that some of the unusual properties of gold, such as its colour, arise through a relativistic effect. I entered the discussion and said that if my paper was correct, gold is reasonably well-behaved, and its energy levels were quite adequately calculated without needing relativity, as might be expected from the energies involved. This drew almost derision – the paper was dated, an authority has spoken since then. A simple extrapolation from copper to silver to gold shows gold is anomalous – I should go read a tutorial. I offered the fact that all energy levels require enhanced screening constants, therefore Maxwell’s equations are not followed. These are the basic laws of electromagnetism. Derision again – someone must have worked that out. If so, what is the answer? As for the colour, copper is also coloured. As for the extrapolation, you should not simply keep drawing a zig to work out where the zag ends. The interesting point here was that this person was embedded in “standard theory”. Of course standard theory might be right, but whether it is depends on whether it explains nature properly, and not on who the authority spouting it is.

Finally, a quote to end this post, again from Enrico Fermi. When asked what characteristics Nobel prize winners had in common: “I cannot think of a single one, not even intelligence.”