Some thoughts on unification of quantum and classical mechanics

Anselm of Aosta was a Benedictine monk in the 11th Century who rose through the ranks of the church to become Archbishop of Canterbury and eventually a Saint.  Anselm thought that belief in God was more than just an article of faith but was also rational. He sought to prove the existence of God by means of logic.  The essence of his argument was that if we postulate that God exists then we can, through a series of logical steps, prove that god  exists.  Such a circular argument is of course invalid, although it is often used to reinforce an argument.  It is simply not the case that by assuming something to be true you can prove that therefore it must be true.

This has important implications for quantum physics because quantum theory is itself based on an assumption.  It was originally proposed by John W Nicholson.  He observed that the units of Planck’s constant we the same as those for angular momentum.  So he reasoned that the orbital angular momentum of electron in the hydrogen atom was related to Planck’s constant.  He went one step further and argued that the orbital angular momentum could only take on values which were an integer multiple of Planck’s constant.  He thus “quantised” angular momentum.  Niels Bohr then used this  assumption to develop his mathematical model for the hydrogen atom.

This idea that angular momentum is quantised is a cornerstone of quantum theory.  Not only did Bohr use it to create a mechanical model for the atom, but so did Louis de Broglie and Erwin Schrödinger.   De Broglie argued that the electron could be viewed as a wave, and showed that his waves formed a harmonic series when viewed in the context of the atom.  However his waves were the result of assuming that the wavelength is related to Planck’s constant, which in turn he assumed to be quantised in exactly the same way as Bohr.  Schrödinger developed a set of equations to describe de Broglie’s waves and this too incorporates Nicholson’s assumption.  Indeed the simplest way to derive Schrödinger’s wave equation is to substituted quantised angular momentum into the canonical form of an undamped second order differential equation.

The implication for quantum theory then is that it is not possible to prove that quantum theory is correct using anything that derives from quantum theory itself, since to do so would be to follow the path of St Anselm.  In order to prove that quantum theory is correct it is necessary to prove that angular momentum is somehow quantised, and that cannot be done by first assuming that angular momentum is quantised.  This means that there is no way to manipulate de Broglie’s waves or Schrödinger’s equations or indeed anything that derives from them in a way that shows that angular momentum is quantised.

The only such valid proof must therefore spring from classical mechanics, which itself presents a problem. In effect this is exactly what Niels Bohr tried to do.  He argued that our understanding of classical mechanics was wrong in that it should take account of the idea that angular momentum is not continuous, but is quantised in the way that Nicholson had suggested.  The problem is that despite numerous attempts nobody has ever been able to prove that this assumption is correct.  Doing so would effectively unite quantum and classical theory.

De Broglie himself was to spend the better part of 40 years in the attempt and failed.  He was trying to show what he called the ’causal link’ between his wave mechanics and classical mechanics.  Einstein tried to do so using his field theory and also failed.  The problem has largely been overlooked since then with physicists simply accepting the quantisation of angular momentum as a sort of received wisdom, interestingly this is somewhat akin to an act of faith.

While it is not possible to prove that a postulate is true based on the truth of the assumption, the obverse is not the case.  It is possible to disprove postulate by first assuming that it is true and then showing that this leads to a contradiction, a paradox or an absurdity.  Such proofs are referred to as Reductio ad Absurdum and are commonplace throughout mathematics, for example Euclid’s proof that the square root of two is irrational.  Indeed the so called ‘scientific method’ is itself based on the underlying idea of reductio ad absurdum.  The scientific method requires that we first put forward an assumption or postulate and based on this develop a model.  The model is then tested against experimental or empirical data and if it fails the postulate underpinning the model is deemed to be incorrect.  In this case the absurdity is the failure of the experiment used to test the model, but otherwise the logic is essentially the same.  It should be noted that the scientific method can never prove a postulate to be true, it can only ever be used to show that a postulate is false.

The assumption that angular momentum is quantised is just such a case of reductio ad absurdum..  Using this assumption we can derive Bohr’s model for the hydrogen atom, however the model requires that changes in the energy level of the atom occur when the electron moves from one orbit to another without ever occupying anywhere in between the two orbits. This was quickly dubbed the “quantum leap” and is clearly physically impossible and should be sufficient to render the model invalid and also the assumption that lies behind it.  That is angular momentum cannot be quantised, at least not in the way that Nicholson and Bohr describe.

To get around this slight inconvenience, physicists will often say that the Bohr model is obsolete and that our view of the world has moved on, that reality is not what it seems, that particles do not exist until they are observed etc.  However it is a false premise to proceed along these lines when the underlying assumption has already been shown to be invalid.  What all of these machinations amount to is simply another way of trying to describe the quantum leap but without using the words “quantum” or “leap”.  The electron for example is described as a wave function which “collapses” when it is observed to reveal the position or the velocity of the electron itself, which is now viewed as being in its particle form rather that its wave form.  The process of collapsing is just another way of describing the instantaneous transformation of the electron into a particle which exists at some point in space by denying that it existed as a particle prior to this transformation.  In reality all such descriptions are simply euphemisms for the quantum leap.

We do however know that something is quantised and that this is connected with the discrete energy levels of the atom.  This comes about because there is a relationship between the various energy levels which is linked to a harmonic series.  In the case of de Broglie these harmonics are thought to be related to the orbit of the electron as standing waves, existing as multiples of the base orbital frequency.

Wherever we see a harmonic series in nature there must be an accompanying sampling process or quantisation taking place.  This comes about when we consider the Fourier representation of a harmonic series.   The Fourier representation of a harmonic series is unity at the base or fundamental frequency and at every integer multiple of the base frequency and has a value of zero everywhere else.  For real entities this extends along both positive and negative jω axes to infinity.  Such a function is commonly referred to as Dirac Comb.   The Fourier transform of a Dirac Comb is itself another Dirac Comb, only this time in either the time domain or the space domain.  Such a Dirac comb is a sampling function or, if you prefer, a quantisation function.

If we apply classical Newtonian mechanics to the electron orbiting the hydrogen atom we obtain a stable atom, but the energy of the electron is fixed, there are no different energy levels which cause the adsorptions or emissions that we observe.  What all of this amounts to is that we have missed something.  It means that there is something about classical mechanics which we have not quite got right.  That there is something about classical mechanics as we currently understand it that does not behave as we would expect in the extreme conditions that exist within the atom.  Bohr argued that this deficiency was that we had failed to take account of the idea that angular momentum is quantised.   When Bohr postulated that angular momentum is quantised, he was in fact postulating that this is the deficiency in our understanding of classical mechanics.  But the quantisation of angular momentum results in an absurdity and so cannot be a valid assumption.

All of which begs the question that if angular momentum is not quantised but we know that something must be quantised, what exactly is the variable of quantisation inherent in the structure of the atom?

A possible clue lies in the timeline of events leading up to Bohr’s model.  For the better part of 300 years it was assumed that Newton’s version of classical mechanics was complete.  Then in 1905 Einstein showed that it was not, he showed that time and distance and mass varied according to the relative velocity of the observer and the observed.   Bohr’s model was first published in 1912 and even at the time was acknowledged to have problems.  Bohr chose to ignore relativity, which at the time was not well understood and even rejected by some physicists.  Given the timing of the discovery of relativity and the attempts to describe the dynamics of the atom, it seems highly likely that relativity lies at the heart of any misconceptions we might have about classical mechanics.

In summary: quantum theory can only be validated from the perspective of classical theory and not from within quantum theory itself.  This means that there has to be something wrong in our current understanding of classical mechanics, since as it is currently understood the mechanics of the atom simply do not work.  The presence of a harmonic series in the mechanics of the atom means that something is sampled or quantised.  Niels Bohr sought to suggest it is angular momentum that is quantised into units of Planck’s constant.  However this assumption leads to an absurdity, the quantum leap, and so cannot itself be valid.  Despite this the assumption has been carried forward without question into subsequent models of the atom.  It is highly likely that whatever is quantised it is connected in some way with the effects of relativity.

We will ultimately gauge the success of any new postulates and models for the atom based on the scientific method, which requires that whatever we postulate as the deficiency in classical mechanics is tested by experiment. In the meantime we can suggest a few pointers to a successful model.  First, of course, it must have an infinite number of stable states which return the correct values for the energy levels of the atom by matching those predicted by the Rydberg formula.  The dynamics should be such that the orbital radius of the electron is the same in all the various energy levels, since anything other than this would require the existence of the quantum leap or its latter day equivalent.  It should also address the issues of which Bohr was unaware or chose to ignore; these include an explanation for the existence and the value of the Fine Structure Constant, an explanation for the existence of and value of Zero Point energy, an explanation as to why the orbiting electron does not emit synchrotron radiation and it must be seen to fully take into account the effects of relativity.

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