Quantum Mechanics May Not Need Imaginary Numbers After All

The mathematical language of quantum mechanics has relied on complex numbers for nearly a century, even though its founder, Erwin Schrödinger, was never fully comfortable with that choice. Complex numbers are mathematical quantities that include an imaginary component and are widely used to simplify calculations in physics. A new theoretical study revisits one of Schrödinger’s earliest ideas and argues that quantum mechanics can be described using only real numbers, meaning ordinary numbers without imaginary parts, while still reproducing all known energy predictions. The work revisits a largely overlooked fourth-order wave equation, an equation involving higher levels of mathematical change, proposed by Schrödinger himself in 1926 and examines its physical meaning and practical consequences in clearer terms.

The study was carried out by Professor Nicos Makris from Southern Methodist University and Professor Gary Dargush from the University at Buffalo. They show that Schrödinger’s original fourth-order, real-valued matter-wave equation, a mathematical description of how particles behave like waves, leads to exactly the same energy values as the familiar second-order, complex-valued Schrödinger equation that is taught in textbooks today. However, the real-valued equation also predicts an additional set of energy values that mirror the known ones. The research is published in the peer-reviewed journal Physics Open.

According to Professor Makris, the motivation for revisiting this idea comes directly from Schrödinger’s own writings. Schrödinger was uneasy about the heavy reliance on complex numbers, writing that “the use of a complex wave function” carried “a certain crudeness”. A wave function is a mathematical tool that describes the likelihood of finding a particle in a particular place. He suggested that the physical state of a system might instead be represented by a real function and how it changes over time. In the present study, the researchers return to this early formulation and test what it predicts using modern mathematical tools and computing methods, while staying faithful to Schrödinger’s original reasoning.

Professor Makris and Professor Dargush show that the fourth-order equation correctly reproduces the energy levels of well-known quantum systems, including the harmonic oscillator, a simple model often compared to a mass on a spring, particles confined in simple energy wells, which are regions that trap particles much like a bowl traps a marble, and the hydrogen atom, the simplest atom made of one proton and one electron. These systems are often used as basic examples in physics because their behavior is well understood. For every positive energy level predicted by the standard Schrödinger equation, the real-valued version produces a matching negative counterpart. As Professor Dargush explained, “Schrödinger’s 4th-order, real-valued matter-wave equation… produces the precise eigenvalues of Schrödinger’s 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues.” Eigenvalues here refer to the allowed energy values that a quantum system can have.

To reach this conclusion, Professor Makris and Professor Dargush developed a variational framework, a mathematical approach that finds solutions by minimizing or balancing quantities, that reshapes the wave equations into a form that can be solved using only real numbers. In simple terms, this approach turns the problem into one that resembles calculations already used in engineering and classical physics. These equations were then solved numerically, meaning with the help of computers, using a method that breaks the problem into small, manageable pieces, allowing computers to calculate accurate results even when the energy landscape changes abruptly. The numerical results closely matched known solutions, confirming that the real-valued description works across several different quantum systems.

One striking feature of the fourth-order equation is that it explicitly depends on how the energy landscape changes from place to place, referring to how forces acting on a particle vary across space, rather than only on its overall shape. The commonly used Schrödinger equation avoids this complexity, which makes it easier to apply. Professor Makris and Professor Dargush explain that this simplicity comes at a cost. As Professor Makris noted, “Schrödinger’s classical 2nd-order, complex-valued matter-wave equation… is a simpler description of the matter-wave, since it does not involve the spatial derivatives of the potential, at the expense of missing the negative, mirror energy levels.” Spatial derivatives describe how a quantity changes with position, a concept similar to how steepness describes changes in a hill’s slope.

The physical meaning of these negative energy levels is still unclear. Professor Makris and Professor Dargush compare this situation to classical vibration problems in engineering, where mathematical equations often predict extra solutions that are not physically meaningful and are therefore ignored. Whether the negative quantum energy levels predicted here correspond to real physical effects or should be treated in a similar way remains an open question that the study does not attempt to answer.

Instead, the work focuses on showing that the mathematics itself is sound and that the calculations can be carried out reliably. By demonstrating that a fully real-number-based description can reproduce all known quantum energy values, the study challenges the widespread belief that complex numbers are essential to quantum mechanics. It also brings renewed attention to a question that Schrödinger himself raised but ultimately set aside nearly a hundred years ago.

In conclusion, the Professor Makris and Professor Dargush team argue that non-relativistic quantum mechanics, the version of quantum theory that applies to everyday speeds rather than near-light-speed motion, allows for a real-valued description that is mathematically complete and numerically accurate. Although the interpretation of the additional mirror energy levels remains unresolved, the study provides a solid basis for further investigation. As the researchers emphasize, “there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative, mirror energy levels,” a finding that invites fresh discussion about the fundamental mathematical foundations of quantum theory.

Journal Reference

Makris N., Dargush G.F., “A real-valued description of quantum mechanics with Schrödinger’s 4th-order matter-wave equation.” Physics Open, 2025. DOI: https://doi.org/10.1016/j.physo.2025.100262

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