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 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.
Quantum mechanics seems to use complex numbers in a fundamentally different way when compared to classical physics. Selected episodes from early stages of the theory are presented to identify different approaches to the use of complex numbers. If for Heisenberg it seemed quite natural to use complex exponentials, Schrödinger clearly struggled with the complex nature of his wave function and tried to assign physical meaning only to its real component. Dirac was probably the first to stress that complex quantities in quantum mechanics cannot be split up into pure real and imaginary parts. Possible educational implications are outlined.
Till 1925 the idea of what the quantum theory would be was unknown. During 1925 three different but equivalent versions of quantum theory were proposed – Schrödinger-proposed wave mechanics; Heisenberg-developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie’s matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner (hence called wave function and denoted it as ψ). This function is generally complex and assumed to contain information about a system. Schrödinger set up a linear and time-dependent wave-like equation, called Schrödinger wave equation, to describe the wave aspect of a particle taking account of de Broglie’s relation for wavelength. Physically, |ψ(X, t)|2, where ψ(X, t) is the solution of the Schrödinger equation, is interpreted as position probability density. That is, |ψ(X, t)|2 is the probability density of observing a particle at position X at time t. ψ does not give exact outcomes of observations but helps us to know all possible events and their probabilities . Further, the probability interpretation allows us to find the average or expected result of a set of measurements on a quantum system.
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.
Schrödinger's original interpretation of the Schrödinger equation had many attractive features lost in later interpretations of the quantum theory. But that interpretation runs into a number of formidable well-known and not so well-known objections. I argue, following the methodological precepts of Paul Feyerabend, that we need not regard any of these objections as fatal, provided we are prepared to opt for a number of bold and rather radical mathematical and theoretical conjectures. These would amount jointly to the conjecture that a fully time-symmetric consistently classically interpreted non-second-quantized analogue of existing quantum field theory would ( pace Jaynes, Tomonaga, Bell, and others) ultimately prove predictively equivalent to orthodox second-quantized theory. Introduction Schrödinger initially proposed his equation as a classical theory of matter waves directly analogous to Maxwell's theory of electromagnetic waves. |ψ| 2 represented a classical charge density functioning in the ordinary classical way as a source of electromagnetic fields, and acted on by these fields via the potential term in the matter–wave equation. This is a theory of coupled classical fields with no probabilities entering into its interpretation, and from a modern point of view it can be thought of in terms of the coupled Dirac and Maxwell fields, without second-quantization and interpreted in a purely classical manner.
This paper proposes a physical interpretation of the imaginary unit i in the Schrödinger equation, arguing that its complex structure is not a mathematical artifact but a direct signature of the zero field’s internal phase dynamics. In standard quantum mechanics, i enables oscillatory, unitary time evolution—yet its physical meaning remains unaddressed. Within the framework of R-theory (Resonance Theory), the universe is understood as a coherent resonant field with two inseparable components: a real, space-time-manifested aspect and an imaginary, non-local phase domain. This paper identifies i as the mathematical projection of this hidden imaginal axis. The wavefunction’s complex nature—traditionally viewed as formal—is reinterpreted as a physical necessity. The real amplitude corresponds to manifested energy, while the imaginary component expresses the latent coherence through which all energy oscillates between potential and presence. In this view, the Schrödinger equation becomes a dynamical law describing rotations within the zero field, where i functions as the operator of coherence rather than an abstract algebraic device. The implications are significant: Quantum oscillation reflects the continuous exchange between real and imaginal components of the zero field. Complexity is elevated from a mathematical convenience to an ontological principle. Consciousness emerges as the experiential side of the same imaginal phase that maintains physical coherence. Reality itself is intrinsically complex—built from orthogonal dimensions of manifestation and potential. This paper reframes the imaginary dimension as the fundamental axis through which the universe sustains resonance. What physics calls “imaginary” is revealed as the inner structure of reality, the phase without space that allows energy, matter, and awareness to exist as coherent oscillations within the zero field.