William Arthur Atkins. Scientific Thought: In Context. Editor: K Lee Lerner & Brenda Wilmoth Lerner. Volume 2, Gale, 2009.

## Introduction

The Heisenberg uncertainty principle, also known as the principle of indeterminacy or the Heisenberg position-momentum uncertainty relation, is a quantum mechanics theory that limits just how precisely physicists can measure atomic particles like electrons. The uncertainty principle forces them instead to make probable estimates for pairs of intricately linked physical characteristics at any given time. These pairs of characteristics, such as position and momentum or time and energy, are known as conjugates.

It is possible to determine, for instance, the position of a particle precisely if the particle’s momentum is not specified at the same time. However, if one measures position accurately then one must sacrifice an accurate knowledge of momentum. Thus, it is impossible to know the exact position and momentum of a particle at any given point in time.

## Historical Background and Scientific Foundations

Werner Karl Heisenberg (1901-1976) attended Ludwig Maximilian University in Munich, Germany. In 1920 he went on to the University of Munich to study mathematics and physics from Arnold Sommerfeld (1868-1951), Wilhelm Wien (1864-1928), and Ernst Pringsheim (1859-1917) and, later, at the University of Göttingen from Max Born (1882-1970), James Franck (1882-1964), and David Hilbert (1862-1943).

At this time the leading atomic theory was that of Danish physicist Niels Bohr (1885-1962), Sommerfeld, and their colleagues. The Bohr-Sommerfeld quantum theory describes an atom as a tiny positive nucleus surrounded by negative electrons that orbit like planets around the sun. The theory was accurate when describing a hydrogen atom with one orbiting electron; it was unable to describe more complicated atoms, however.

In the early 1920s it became apparent that the theory did not satisfy all areas of physics, especially those of spectroscopy (the study of light emitted and absorbed by atoms), optics (the nature of light, and whether it consists of particles or waves), and atomic and molecular physics (the properties of atoms and molecules).

After Heisenberg was awarded his doctorate in 1923 from the University of Munich, he became a teaching assistant for Max Born. During this time he began to develop a theory of quantum mechanics, basing his work on matrix equations developed in the nineteenth century by the British mathematician Arthur Cayley (1821-1895).

Heisenberg was awarded the Nobel Prize for physics in 1932 for his theory of quantum mechanics, published in 1925; his principle of uncertainty, announced in 1927; and for the discovery of different (allotropic) forms of hydrogen. Today, the Heisenberg uncertainty principle is a central principle of modern physics.

### The Science

In the everyday world measurements are easy to make. If you want to measure the dimensions of a table, a tape measure will be sufficient. You will not need to consider quantum forces because the table is so big when compared to these small and immaterial forces. When measuring small particles like atoms, however, quantum forces cannot be ignored.

Just how small is an atom? Its diameter is about 10^{-11} meters, or 1 hundred-billionth of a meter. The atom’s nucleus has a diameter of about 10^{-15} of a meter, or 1 million-billionth of a meter. The width of a human hair, in comparison, is about 1 million carbon atoms, while an average human cell contains about 100 trillion atoms. Atoms are the smallest parts of an element that can be divided but still retain their unique identifying properties. They consist of a dense, positively charged nucleus surrounded by one or more negatively charged electrons. Measuring them accurately is not easy. To measure an electron, for instance, you can’t use a tape measure. For this you need a high-powered atomic force microscope that uses light or some other form of electromagnetic radiation.

Electromagnetic radiation is waves of energy produced by the acceleration or oscillation (movement back and forth) of an electric charge. With both electric and magnetic components, electromagnetic radiation ranges from extremely high frequencies (short wavelengths) to extremely low frequencies (long wavelengths). From high to low frequency, the forms of electromagnetic radiation are gamma rays, x rays, ultraviolet radiation, visible light, infrared radiation, microwaves, and radio waves.

Electromagnetic radiation is made up of photons, which have enough energy and momentum that when they strike an electron they change its direction. Atomic force microscopes measure the exchange of energy that occurs when the two collide, and this, Heisenberg discovered, changes the physical characteristics of the electron being observed. Thus one may be able to get an accurate measurement of momentum, but acquire only a poor measurement of position.

The uncertainty principle explains the problem. Heisenberg stated that the process of measuring the position *x* of a particle disturbs the particle’s momentum *p,* so that *Dx* × *Dp* is greater than or equal to *h,* where *Dx* is the uncertainty (difference) of the position of a particle along a spatial dimension, *Dp* is the uncertainty of the momentum, and *h* is Planck’s constant.

Thus determining the exact position and momentum of a moving particle at any given point in time is limited to the precision of each measurement—with the product of these uncertainties equaling Planck’s constant or greater. The more precise the measurement of position, the less precise the measurement of momentum, and vice versa. In large-scale measurements (such as tables), these errors are negligible, but they cannot be ignored in minute-scale measurements (such as atoms).

Heisenberg’s theory is based on measurable radiation emitted by the atom, but other atomic or subatomic particles cannot be measured precisely. For example, Heisenberg stated that it is impossible to assign a position to an electron in three-dimensional space and in one-dimensional time. Heisenberg did not consider space and time to be two separate entities but one combined entity called space-time. Similarly, speed and position are not two separate things when dealing with atoms. They are inversely related, so that one can only be determined at the expense of the other one.

In accordance with his uncertainty principle, Heisenberg stated that numbers—ordinarily used to represent physical quantities—cannot be used to represent quantities involving miniscule materials such as atomic and subatomic particles. Instead, he used mathematical structures called matrices. In 1925, matrix mechanics—developed by Heisenberg, Max Born, and Pascual Jordan (1902-1980)—became the first complete formulation of quantum mechanics to describe the behaviors of subatomic particles.

### Influences on Science and Society

The uncertainty principle has had a great impact on fundamental physics, chemistry, and other sciences. It has also been highly controversial. In fact, when confronted with Heisenberg’s theory, Einstein famously scoffed, “God does not play dice with the universe,” meaning that scientists should, instead, find logical explanations for behaviors found within the subatomic world.

The uncertainty principle involves four variables: the electron’s position, momentum, energy, and time.

When Heisenberg was developing his theory, scientists held that the precision of any measurement was limited only by the accuracy of the measuring instrument.

Heisenberg, however, showed that at the atomic level and below an instrument’s precision is irrelevant because quantum mechanics limits the precision possible when two canonically conjugate (specifically paired, inversely linked) variables are measured at the same time. For instance, momentum and position are canonic conjugates, as are energy and time. By proposing that certain aspects of subatomic reality are unknown, accepting and defining the bounds of uncertainty, Heisenberg freed scientists to use statistics to determine subatomic processes for the first time.

Although some physicists initially interpreted the uncertainty principle as a violation of fundamental physical laws, others realized its importance. Ultimately, it became the fundamental premise of quantum mechanics, which explained particles at atomic and subatomic levels, something that had been impossible with classical mechanics and electromagnetism.

## Modern Cultural Connections

The Heisenberg uncertainty principle contradicts much earlier scientific dogma. Before its introduction, the central ideas of physics and chemistry were that (1) all matter consists of discrete particles with properties of gravitational mass and electrical charge, (2) light is a continuous electromagnetic wave that travels at a constant speed, (3) continuous electromagnetic fields are created by discrete charged particles, and (4) local interactions of electromagnetic charges are limited by the velocity of electromagnetic waves.

However, the quantum theory alters these ideas by showing that (1) all matter, besides being composed of discrete particles, also has wave properties, (2) light has discrete particle properties, (3) discrete statistical fields exist, and (4) instantaneous nonlocal matter interactions exist, and are called instant action at a distance.

The uncertainty principle also implies that space-time is smoothly curved in the macroscopic world. However, when viewed on increasingly smaller scales, quantum fluctuations occur as observations approach the Planck scale (10^{-33} cm). These effects are observed when theoretical physicists attempt to join the equations of general relativity (which involves the force of gravity) with those of quantum mechanics (which concerns the strong force, the weak force, and the electromagnetic force). So far such attempts have not produced a unified field theory that would unite all of the fundamental forces between elementary particles.

The uncertainty principle states that well-defined atomic orbits are not possible, the uncertainty in the electron’s position is a basic atomic property, and observing any physical characteristic produces a disturbance—an interaction between the observer and the observed—that cannot be eliminated by more accurate measuring devices. The uncertainty principle also says the observer must be regarded as an inherent part of the measurement.

Quantum mechanics has improved on classical mechanics as the theoretical basis of all of physics. For most macroscopic systems, classical mechanics approximates most physical measurements, but in the atomic realm, only quantum mechanics—based on the Heisenberg uncertainty principle—can be used.