Quantum Physics


Why it is Hard to Understand – And, Therefore, Explain – Quantum Math

Authors: Jean Louis Van Belle

If mathematics is the queen of science, then physics might well be the king. It successes are obvious. However, as a science, physics may have failed in one regard, and that is to explain what its basic concepts – such as state vectors, wavefunctions, and transformation matrices – actually represent. When studying quantum mechanics, it is, effectively, hard to keep up the initial enthusiasm, and those who branch out to other fields – which is most of us – quickly end up going through the motions only: we regurgitate models and equations and know how to solve the standard problems, so as to pass the exam, but then forget about them as soon as possible. This paper explores a very intuitive sentiment about the issue: the wavefunction is a rather ‘flat’ mathematical object – it is two-dimensional, basically – so it can’t do the trick, perhaps. In contrast, Maxwell’s equations have real vectors in them, which is why a deeper or more intuitive understanding of electromagnetism comes relatively easily. Indeed, when everything is said and done, we are just human beings living in three-dimensional space, and that is why vector equations (or systems of vector equations), as a mathematical tool, make sense to us. This paper further explores this sentiment. It also offers a way out by, predictably, presenting yet another possible physical interpretation of the wavefunction. More importantly (for the reviewer of this paper, at least), this paper offers a sensible response to the mainstream view that three-dimensional physical interpretations of the wavefunction cannot make any sense because of the weird 720° symmetry of the wavefunction when describing spin-1/2 particles (fermions or – for all practical purposes – electrons). The author does so by analyzing (1) Dirac’s belt trick more in detail – and what it implies in terms of the interaction between the observer and the object – as well as (2) Feynman’s derivation of the transformation matrices for spin-1/2 two-state systems.

Comments: 14 Pages.

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Submission history

[v1] 2018-06-13 11:18:39

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