2008-02-27# This month history: Heisenberg's uncertainty principle |

n February 1927, the young Werner Heisenberg developed a key piece of quantum theory, the uncertainty principle, with profound implications.

Werner Heisenberg was born in December 1901 in Germany, into an upper-middle-class academic family. He liked mathematics and technical gadgets as a boy, and his teachers considered him gifted. In 1920 he began studies at the University of Munich, and published four physics papers within two years under the guidance of mentor Arnold Sommerfeld. Heisenberg became professional friends with Wolfgang Pauli, who was just one year older than Heisenberg and also a student at Munich.

He earned his doctorate in 1923, with a thesis on a problem in hydrodynamics, though he nearly failed due to his poor performance on the required experimental questions on the oral examination. After receiving his doctorate, he worked as an assistant to Max Born at Göttingen, then spent a year working with Niels Bohr at his institute in Copenhagen.

The prevailing quantum theory in the early 1920s modeled the atom as having electrons in fixed quantized orbits around a nucleus. Electrons could move to higher or lower energy by absorbing or emitting a photon of the right wavelength. The model worked well for hydrogen, but ran into problems with larger atoms and with molecules. Physicists realized a new theory was necessary.

Heisenberg objected to the current model because he claimed that since one couldn’t actually observe the orbit of electrons around a nucleus, such orbits couldn’t really be said to exist. One could only observe the spectrum of light emitted or absorbed by atoms. Starting in 1925, Heisenberg set to work trying to come up with a quantum mechanics that relied only on properties that could, at least in theory, be observed.

With help and inspiration from several colleagues, Heisenberg developed a new approach to quantum mechanics. Basically, he took quantities such as position and velocity, and found a new way to represent and manipulate them. Max Born identified the strange math in Heisenberg’s method as matrices. The new formulation accounted for many observed properties of atoms.

Shortly after Heisenberg came up with his matrix-based quantum mechanics, Erwin Schrödinger developed his wave formulation. The absolute square of Schrödinger’s wave function was soon interpreted as the probability of finding a particle in a certain state. Schrödinger’s wave formulation, which he soon proved was mathematically equivalent to Heisenberg’s matrix methods, became the more popular approach, partly because physicists were more comfortable with it than with the unfamiliar matrix mathematics. The unpopularity of his own method annoyed Heisenberg, especially because a lot was at stake at the time as he and other young scientists were beginning to look for their first jobs as professors as an older generation of scientists was retiring.

Though others may have found the wave approach easier to use, Heisenberg’s matrix mechanics led him naturally to the uncertainty principle for which he is well known. In matrix mathematics, it is not always the case that a x b = b x a, and for pairs of variables that don’t commute, such as position and momentum, or energy and time, an uncertainty relation arises.

Heisenberg conducted a thought experiment as well. He considered trying to measure the position of an electron with a gamma ray microscope. The high-energy photon used to illuminate the electron would give it a kick, changing its momentum in an uncertain way. A higher resolution microscope would require higher energy light, giving an even bigger kick to the electron. The more precisely one tried to measure the position, the more uncertain the momentum would become, and vice versa, Heisenberg reasoned. This uncertainty is a fundamental feature of quantum mechanics, not a limitation of any particular experimental apparatus.

Heisenberg outlined his new principle in 14-page a letter to Wolfgang Pauli, sent February 23, 1927. In March he submitted his paper on the uncertainty principle for publication.

Niels Bohr pointed out some errors in Heisenberg’s thought experiment, but agreed the uncertainty principle itself was correct, and the paper was published.

The new principle had deep implications. Before, it had been thought that if you knew the exact position and momentum of a particle at any given time, and all the forces acting on it, you could, at least in theory, predict its position and momentum at any time in the future. Heisenberg had found that not to be true, because you could never actually know a particle’s exact position and momentum at the same time.

The uncertainty principle soon became part of the basis for the widely accepted Copenhagen interpretation of quantum mechanics, and at the Solvay conference in Brussels that fall, Heisenberg and Max Born declared the quantum revolution complete.

In the fall of 1927, Heisenberg took a position as a professor at the University of Leipzig, making him the youngest full professor in Germany. In 1932 he won the Nobel Prize for his work on quantum mechanics. He continued his scientific research in Germany. During World War II, though he was not a member of the Nazi party, he was a patriotic German citizen, and he became a leader in the German fission program, which failed in its effort to build at atomic bomb. Heisenberg’s actions and motivations have been the subject of controversy ever since. He died in 1976.

Reference/further reading: David Cassidy, Uncertainty: the Life and Science of Werner Heisenberg (New York: W.H. Freeman, 1992).

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## This month history: Heisenberg's uncertainty principle