Heisenberg Uncertainty Principle

FIELDS OF STUDY: Particle Physics; Quantum Physics; Classical Mechanics

ABSTRACT: This article describes the Heisenberg uncertainty principle and examines the history of quantum mechanics leading up to it. It states that the energy and the location of an electron in an atom cannot both be determined with great accuracy. It enabled different theories to be combined into a single model for the behavior of electrons in atoms.

Principal Terms

• accuracy: the extent to which measurements of a property differ from its actual value.
• eigenstate: the state for which the value of a measurable, observable operator (change agent) has one exact mathematical solution, with no uncertainty.momentum: an intrinsic property of matter, the product of mass and velocity.
• position: in quantum mechanics, an electron’s location in space relative to the nucleus of an atom.
• precision: the extent to which different measurements of the same property differ from one another.
• probability density function: the math function that describes the probability of an electron being found in a defined region of space about an atomic nucleus.
• quantum mechanics: a branch of physics based on the theory that energy is not continuous but rather is composed of discrete particle-like packets called "quanta"; "quantum" and "photon" are synonyms.
• quantum state: the energy level and various specific attributes of an electron.

Atomic Structure

The uncertainty principle, proposed by Werner Heisenberg (1901–76) in 1927, came at a time when physicists the world over were working to reconcile the new concepts of atomic structure and quantum theory with earlier ideas and observations. The principle was poorly accepted at first. However, it soon proved to be one of the most important concepts in the transition from classical mechanical theory to the modern quantum theory of atomic structure.

In 1897, J. J. Thomson (1856–1940) showed that the cathode rays observed in a cathode-ray tube were in fact streams of negatively charged particles with little mass. Those particles were later called "electrons." Not long after, Ernest Rutherford (1871–1937), Hans Geiger (1882–1945), and Ernest Marsden (1889–1970) identified protons as particles with more mass than electrons and the opposite (positive) charge. In 1911, they conducted the famous "alpha scattering" experiment. In this experiment, they directed a stream of alpha particles at a target of very thin gold foil. They noted that most of the particles passed directly through the foil as though it were not there. However, some particles seemed to have struck something very dense inside the foil and bounced off it. These observations were the first real evidence that atoms are composed of a very small, dense nucleus with a much larger, very diffuse cloud of electrons around it.

To describe the atom more completely, a third, neutral particle, termed the "neutron," had to exist. Because neutrons have no electrical charge, it was not possible to observe them directly by electrical means. It was not until 1932, when James Chadwick (1891–1974) demonstrated indirectly that they exist, that the basic principles of atomic structure were resolved. The resulting model was based on classical mechanics. It showed an atom with a very small, dense nucleus surrounded by electrons in specific orbits, much like planets orbiting the sun. In this model, the electrons would radiate energy constantly and would eventually fall into the nucleus as the orbit decayed.

By the early 1900s, many experiments had shown how atoms absorb and emit light. Studies also showed that only very specific wavelengths and energies along the electromagnetic spectrum were absorbed and emitted by atoms. Electrons can move into higher or lower energy states. In 1900, Max Planck (1858–1947) proposed that the energy involved in those transitions could only be transmitted or absorbed in discrete units, or "quanta." In 1913, Niels Bohr (1885–1962) amended the planetary model. He assumed that atoms could exist in static states where their electrons do not constantly radiate energy. He also assumed that electrons follow unique waveforms about the nucleus, not circular orbits. From this, Bohr developed a theory that successfully explained the observed spectrum of the hydrogen atom.

Another key observation from spectral data was that streams of electrons exhibit "wave-particle duality." Electrons can behave as particles under some conditions and as electromagnetic waves under others. Many physicists worked to reconcile these behaviors in terms of the motion of electrons about the nucleus. One important breakthrough came from Albert Einstein (1879–1955), who realized that the energy of an electron is equal to the product of its mass and the speed of light squared. These were key steps in the founding of quantum mechanics. Quantum mechanics seeks to describe the behavior of electrons in atoms using math.

One problem with early quantum theory was that wave-particle duality does not lend itself to precision and accuracy. Math solutions based on the electromagnetic properties of photons did not agree with solutions based on the particle behavior of electrons. This was mainly because scientists did not realize that electrons and photons could be the same thing. In 1924, Louis de Broglie (1892–1987) suggested that the wavelength of a photon is inversely proportional to the product of its momentum and the mass of an electron. Planck’s constant, h, is the constant of proportionality.

Another problem with early quantum theory was that the math could only be resolved for simple atomic structures with one electron and one proton, and only at one energy level. Later calculations based on an initial calculation became more and more inaccurate with respect to observations. In 1927, Heisenberg explained this failure with what is now known as the uncertainty principle.

The Uncertainty Principle

In essence, the Heisenberg uncertainty principle states that it is not possible to determine both the exact energy level and the exact position of an electron in an atom at once. A single electron is so small that any measurement of its energy affects its location. Likewise, any measurement of its location affects its energy. In systems that can be described fully by classical mechanics, such effects are not relevant. For example, measuring the speed of a baseball using radar affects neither the speed nor the position of the baseball when it is measured. However, if the "baseball" were a particle of similar size to the wavelength of the electromagnetic waves from the radar gun, it would experience significant effects on its position and energy. The motion of an electron in an atom is affected by any interaction with electromagnetic radiation of a wavelength appropriate to carry out measurements. This is known as the Compton effect. It was named for Arthur Compton (1892–1962), who observed that x-rays reflected from a surface have lower energy, by consistent discrete amounts, than the incoming x-rays. Electromagnetic waves consist of photons, and photons can be the same thing as electrons. The interaction between light and an electron can therefore be thought of as two physical entities colliding, and the laws of conservation of energy and momentum apply.

Many physicists disliked the uncertainty principle because it means that precise and accurate solutions, such as those obtainable in classical mechanics, are not possible. Further progress in quantum mechanics had to rely on statistics and probabilities rather than exact solutions.

The Schrödinger Equation

Erwin Schrödinger (1887–1961) worked on the same problem at about the same time as Heisenberg. Instead of treating the electron as a particle and making matrices of when, where, and how fast it was moving, he looked at the electron as a wave. He also related the matrix-based approach to his wave-mechanics-based approach. The math, while quite complex, elegantly described the behavior of electrons in atoms.

Thus, according to the uncertainty principle, an electron in a specific quantum state (with a certain energy and specific, unique values for certain attributes) exists somewhere in a range of places rather than at one specific place. Each of those regions is described by an "eigenfunction," a math function for which there can be one or more real solutions. Each of those real solutions, or "eigenvalues," corresponds to a specific eigenstate. The probability distribution function for a quantum state describes the probability of finding an electron in a specific region of space about the nucleus. These regions, and their 3-D shapes, define what are often termed "atomic orbitals." In turn, the orbitals and their electrons determine all normal chemical and molecular behavior.

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