Werner Heisenberg (1901

“The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.”

– Heisenberg, uncertainty paper, 1927

This is a brief explanation of the “uncertainty relation“ that connects a subatomic particle’s position and its momentum (which is mass multiplied by velocity). This concept has deep effects on key ideas like causality and the ability to predict how an atomic particle will behave in the future.

Due to both the scientific and philosophical weight of what may sound like a simple uncertainty relation, scientists refer to it as an uncertainty principle. It is also sometimes more clearly described as the “principle of indeterminacy.“ This page explores how Heisenberg’s uncertainty relation and principle first came about.

The beginnings of the uncertainty principle are shaped as much by personal dynamics as by physics itself. Heisenberg’s path to this idea grew out of a disagreement that started in early 1926. On one side were Heisenberg and his closest collaborators, who supported the “matrix“ version of quantum mechanics. On the other side stood Erwin Schrödinger and his group, who promoted the newer “wave mechanics.“

“I knew of [Heisenberg's] theory, of course, but I felt discouraged, not to say repelled, by the methods of transcendental algebra, which appeared difficult to me, and by the lack of visualizability.”

– Schrödinger in 1926

Many physicists were hesitant to embrace “matrix mechanics“ at first because it was highly abstract and involved unfamiliar math. When Schrödinger introduced his version of wave mechanics in early 1926, it was quickly accepted. This approach used more traditional ideas and math, and it seemed to eliminate the confusing quantum jumps and breaks. French physicist Louis de Broglie had earlier proposed that matter, like light, could show wave-like behavior. Building on this idea, which had support from Einstein, Schrödinger explained the quantum energy levels of electrons in atoms as the result of vibration frequencies in electron “matter waves“ circling the nucleus. Just like a piano string produces a steady note, an electron wave would have a specific, fixed energy. Compared to Heisenberg’s matrix mechanics, which required complex calculations, Schrödinger’s model made atomic behavior easier to calculate and visualize.

“I had no faith in a theory that ran completely counter to our Copenhagen conception.”

– Heisenberg, recollection

In May 1926, Schrödinger published a demonstration showing that wave mechanics and matrix mechanics produced the same results – they were, mathematically, two forms of the same theory. However, he also claimed that wave mechanics was the better approach. This sparked a strong backlash, especially from Heisenberg, who stood by the idea of quantum jumps and rejected a theory based on smooth, continuous waves.

But this debate wasn’t only about theory. Careers were on the line. The young physicists behind matrix mechanics were now seeking teaching roles, just as older theorists were leaving their posts at German universities. At the same time Heisenberg was under pressure from his family to secure one of those academic positions, his major contribution – matrix mechanics – seemed to be losing ground to Schrödinger’s rising wave mechanics.

“The more I think about the physical portion of Schrödinger's theory, the more repulsive I find it...What Schrödinger writes about the visualizability of his theory 'is probably not quite right,' in other words it's crap.”

– Heisenberg, writing to Pauli, 1926

Heisenberg had just started working as Niels Bohr’s assistant in Copenhagen when Schrödinger arrived in October 1926 to debate the competing quantum theories. The discussions between them and Bohr were intense but ended without a clear winner. It became clear that neither theory fully explained atomic behavior on its own. Both camps continued searching for a better physical understanding of quantum mechanics that matched their ideas.

After Schrödinger proved that wave and matrix mechanics gave equivalent results, and Max Born offered a statistical explanation of the wave function, further progress came from Pascual Jordan in Göttingen and Paul Dirac in Cambridge. They developed unified mathematical expressions known as “transformation theory,” which laid the groundwork for modern quantum mechanics. The next challenge was to interpret what these equations meant for real-world situations – whether atomic events behaved like waves, particles, or both.

As Bohr later pointed out, although quantum mechanics governs atomic behavior, scientists still interact with larger, visible systems where Newton’s classical physics holds. What was needed was a way to link the predictions of quantum theory to measurements taken in everyday lab settings.

While reviewing Dirac’s and Jordan’s papers and corresponding with Wolfgang Pauli, Heisenberg spotted a fundamental issue related to measuring certain variables. His analysis revealed that whenever someone tried to measure both the position and momentum of a particle at the same time, uncertainty always appeared. The same was true for energy and time. According to Heisenberg, these limits were not due to flaws in the tools or human error – they were built into the nature of quantum mechanics itself.

Heisenberg shared his findings in a detailed 14-page letter to Pauli in February 1927. That letter later became a published paper, introducing what would become known as the uncertainty principle.