For nearly a decade, Sheng Ran, an assistant professor of physics, has struggled with the legacy of Erwin Schrödinger, the Austrian physicist who died in 1961. One hundred years ago, Schrödinger essentially created the field of quantum mechanics by publishing an equation describing how the quantum states of physical systems evolve over time. 

After years of research, Ran published an updated framework to quantum mechanics in Physical Review A, a journal of the American Physical Society. “I believe this is an important development in the field,” Ran said.

Just as Newton’s second law predicts the future position of a ball or planet if one knows the forces acting on it, Schrödinger’s equation predicts wave functions, such as the probability that an electron will occupy a particular position over time.

While the equation has been a cornerstone of physics education, Ran wondered if it could tell an even fuller story. During his postdoctoral work at the University of California, San Diego, he became convinced that the conventional view of quantum mechanics was missing a dimension.

“The Schrödinger equation predicts the evolution of a wave function over time, but I thought it should also be possible to predict how the wave function evolves with energy,” said Ran, a member of the Center for Quantum Leaps. “The equation is still valid, but it’s incomplete.”

In standard quantum mechanics, quantum states are described by momentum and energy instead of space and time. “The equation describing momentum-energy is a mathematical re-expression of the same quantum state,” Ran said. 

Rather than replacing quantum mechanics, the new framework extends its structure by introducing an additional layer of dynamics. It describes the quantum state in two complementary parts: one evolving in time, and another evolving in energy.

The work can be viewed as restoring a fundamental symmetry between space-time and momentum-energy — two descriptions that are mathematically equivalent in standard quantum mechanics but treated asymmetrically in the way that dynamics are defined.

In the new framework, energy becomes a dynamic factor in the evolution of wave functions. “In standard quantum mechanics, energy is not treated as an evolution parameter,” Ran said. “Half of the picture is frozen. My equation allows us to describe how a wave function evolves with respect to energy itself.”

As a researcher who builds quantum materials in his lab, Ran often works with the momentum-energy picture of quantum systems. Years ago, he began to imagine life in that world. “Someone living in that system wouldn’t think in terms of time — like having a meeting at 2 p.m. or going to the gym at 3. Instead, they would think in terms of energy: What happens at 5 electron volts, what happens at 100 electron volts?” That thought led to a realization. “In that world, the natural description would be evolution with respect to energy,” Ran said. “And I realized that standard quantum mechanics doesn’t provide that.”

Ran believes his framework could be especially useful for understanding physics in extreme situations, including near black holes or at the beginning of the Big Bang, where changes in time are trivial compared with changes in energy.

The framework also suggests new structural effects that do not appear in standard quantum mechanics. For example, it naturally produces a uniform background energy contribution under certain conditions, resembling features associated with dark energy. It also predicts exponential boundary relations that resemble mathematical structures found near black hole horizons.

“My framework provides a new way to think about extreme but very important scenarios,” he said.

Time will tell if the approach reshapes textbooks. Ran has faced skepticism from some physicists when describing his new approach, but he hopes the new paper will spark new conversations. “I’m writing several more papers on this topic that will be submitted soon,” he said. “This is just the very beginning.”