Mathematical Framework

1

In the previous section, some attempt was made to outline a few basic concepts that started to initiate the transition from classical to quantum physics. As such, these discussions were still anchored in a physical interpretation of the various effects, which the pioneers of quantum theory then sought to describe within a mathematical framework, which was more reflective of quantum mechanics than Newtonian mechanics. Today, many of the papers discussing the development of this framework will simply start from a position already predicated on so much mathematical abstraction that, for many, it can represent an impenetrable barrier to further understanding. This problem is often compounded by the impression of there being little attempt to anchor any of the abstractions to any meaningful physical description of the quantum universe.

While many of us will start our journey into science armed with a basic understanding of the mathematics underpinning Newtonian mechanics, we may quickly arrive at a situation where the complexity of the system cannot be resolved by applying Newton’s laws to individual particles. For example, applying Newton’s laws of motion to a volume of gas containing billions of ‘particles’ is not really a practical approach and so Newtonian mechanics gives way to the statistical mechanics of thermodynamics. Likewise, while Einstein had his great insight into special relativity in 1905, it required another 10 years to develop general relativity, primarily because Einstein had to learn and develop his ideas within the mathematical framework of differential geometry encompassing both matrices and tensors. History also suggests that Heisenberg, who is often associated with the initial development of matrix mechanics, did not actually know what a matrix was, when he first started to develop his ideas.

Within the historical context of the last 100 years, the description of many physical systems has become increasingly abstracted, as science progressed beyond the realm of ‘human experience’ and, in many ways, any intuitive understanding of the underlying physics. As such, physics and mathematics have become joined in an ever closer partnership, which can often appeared to undermine the traditional role of experimental verification. However, human experience might still tell us that it is possible to construct a seemingly logical hypothesis, which is mathematically consistent, but which is eventually shown not to be fully consistent with subsequent observations. We might characterise this human experience in terms of the following quote:

"For every problem,
there exists a simple and elegant solution,
which is absolutely wrong.
J. Wagoner "

 So, at this point, we shall continue to follow the basic chronology of developments, which were to influence the fledgling science of quantum mechanics at the start of 1925. In this context, Newton’s great work called ‘Mathematical Principles of Natural Philosophy’, i.e. the ‘Principia’, was still considered to underpin much of the established view of science, as it had since its publication in 1687. Of course, within the subsequent 250 year period, there had been many developments, especially in the field of mathematics that had begun to modify the presentation of Newtonian physics. However, before introducing some of the mathematical ideas, which would help reformulate Newtonian mechanics, it might be worth simply summarising a few basic Newtonian concepts that would be later challenged by quantum mechanics:

[1]      1

In [1], we see a summary of some key equations normally associated with a physical particle of mass [m], which are anchored in position [x] and the change in position with time [t]. In this context, the idea of kinetic energy [T] is an attribute that we might associate  with the mass [m] in isolation. However, we known that in order to define the total energy of a particle, we need to include the idea of potential energy [U], but this requires a description of a system of two or more particles, e.g. gravitational or charge potential. As such, the actual definition of potential energy depends on the nature of the system, but can be generalised as being a function of position, i.e. U(x).

[2]      2

One other footnote that may be worth mentioning, as a scalar quantity, potential energy is usually considered to be negative with respect to kinetic energy. As such, the total energy may be zero, if the system as a whole contains positive kinetic energy and an equal amount of negative potential energy. In the following introduction of Lagrange and Hamilton mechanics, the basic results in [1] and [2] essentially remain unaffected, although the methods by which the results are obtained may appear to be expressed within a greater amount of mathematical abstraction. Whether you like these methods often depends on familiarity plus your preference for mathematical, as opposed to physical descriptions.