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arXiv:0912.3996

The mystery of square root of minus one in quantum mechanics, and its demystification

C.P. Kwong

Published Dec 20, 2009• 1 citation
arXivHTMLPDF

Abstract

To most physicists, quantum mechanics must embrace the imaginary number i = square root of minus one is at least a common belief if not a mystery. We use the famous example pq -qp = h/(2 pi i) to demonstrate the possible elimination of i when constructing this noncommutative relationship. We then discuss the role of i in the formulation of Schroedinger's wave equation. Common to the original development of these two quantum theories was the use of complex exponential to represent the fundamental variables (i.e., p, q, and the wave function). Understanding this complex function from the right perspective, as we suggest in this essay, removes the mysteries surrounding the complex nature of quantum mechanics.

DOI
10.48550/arxiv.0912.3996

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To most physicists, quantum mechanics must embrace the imaginary number i = square root of minus one is at least a common belief if not a mystery. We use the famous example pq -qp = h/(2 pi i) to demonstrate the possible elimination of i when constructing this noncommutative relationship. We then discuss the role of i in the formulation of Schroedinger's wave equation. Common to the original development of these two quantum theories was the use of complex exponential to represent the fundamental variables (i.e., p, q, and the wave function). Understanding this complex function from the right perspective, as we suggest in this essay, removes the mysteries surrounding the complex nature of quantum mechanics.
This chapter demonstrates the deduction of quantum physics and, in particular, presents its new applications to eigenvalue problems. Specifically, it derives three of the five axiom presumptions of quantum mechanics (QM), for example, deducing the Schrödinger equation in two general ways. It transforms these three axiom presumptions into three theorems of QM, not only resolving the difficulty in understanding them but also yielding new theories and discoveries. For instance, this paper addresses the puzzle of the origin of wave-particle duality, derives operators, eigenvalues, and eigenstates, deduces commutation relations for coordinate and momentum as well as for time and energy, and reveals that QM is essentially a generalized (mechanics) theory of the complex square root of the (real density function of) classical statistical mechanics. The realization that QM is a generalized theory of the complex square root of classical statistical mechanics represents both a groundbreaking discovery and a revolutionary advancement in physics. This insight profoundly influences philosophical perspectives on the development of modern physics, resolves numerous puzzles in QM and quantum information, and provides QM with a scientifically solid foundation. This foundation eliminates the need for basic axiom presumptions and dispels the so-called “strange incomprehensible properties” of QM, as both classical statistical mechanics and its complex square root that have been scientifically validated. Additionally, this chapter uncovers the reason for not taking the time derivative of spatial coordinates in the Schrödinger equation. Consequently, it offers a solution to the puzzle of the origin of first quantization and primarily deduces quantum physics without the strange and incomprehensible properties currently associated with it.
In this paper, some of Schrödinger's initial struggles with the complex nature of his wave function are outlined. It is shown that he first attached physical meaning only to its real component and even tried to avoid the explicit appearance of the imaginary unit i in his fundamental (time-dependent) equation. This attitude is quite understandable, since he was committed to the classical framework of wave theory, not only ontologically, but also formally. Around one year after the publication of his famous papers on wave mechanics, Schrödinger seems to have accepted that the wave function must be complex and that the physical interpretation is to be related to its absolute square. This is justified by him both due to equivalence considerations with matrix mechanics and in analogy with Maxwell's equations. It is argued that an awareness of the struggles Schrödinger went through to accept a complex wave function can be a relief to students who themselves wonder why ψ needs to be complex when they encounter the topic for the first time.
In this contribution I outline some of Schrödinger’s initial resistance to the complex nature of his wave function. It is shown how he first attached physical meaning only to its real component and even tried to avoid the explicit appearance of the imaginary unit in his fundamental (timedependent) equation. This attitude is quite understandable, since he was committed to the classical framework of wave theory, not only ontologically, but also formally. Around two years after the publication of his famous papers on wave mechanics, Schrödinger seems to have accepted that the wave function must be complex and that the physical interpretation is to be related to its absolute square. This is justified by him both due to equivalence considerations with matrix mechanics and in analogy with Maxwell’s equations. This episode can be of special pedagogical relevance as students often struggle with the complex nature of the wave function on their first encounter with it.
In 1927, Nobel prize winning physicist, E. Schrodinger, in correspondence with Ehrenfest, wrote the following about the new theory: “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Psi is surely fundamentally a real function.” This seemingly simple issue remains unexplained almost ninety years later. In this dissertation I elucidate the physical and theoretical origins of the complex requirement. I identify a freedom/constraint situation encountered by vectors when, employed in accordance with adopted quantum representational methodology, and representing angular momentum states in particular. Complex vectors, quite simply, provide more available adjustable variables than do real vectors. The additional variables relax the constraint situation allowing the theory’s representational program to carry through. This complex number issue, which lies at the deepest foundations of the theory, has implications for important issues located higher in the theory. For example, any unification of the classical and quantum accounts of the settled order of nature, will rest squarely on our ability to account for the introduction of the imaginary unit.