Quantum mechanics is a physical theory formulated in the first half of the twentieth century. It successfully describes the behavior of matter at small distance scales.
It explains and quantifies three effects that classical physics cannot account for:
- The values of some measurable variables of a system, most notably the total energy of a bounded system, can attain only certain discrete values determined by the system. (The smallest possible jumps in the values of those observables are called "quanta" (Latin quantum, quantity), hence the name quantum mechanics.)
- Matter exhibits properties of waves (see wave-particle duality).
- Certain pairs of observables, for example the position and momentum of a particle, can never be simultaneously ascertained to arbitrary precision (see Heisenberg's uncertainty principle).
For large systems, the correspondence principle imposes the requirement that quantum mechanics must agree with the predictions of classical physics.
Description of the theory
In quantum mechanics, all of these are resolved by describing the instantaneous state of a system with a wave function that encodes the probability distributions of all observables.
Quantum mechanics makes predictions only about these probability distributions and not about the precise values of observables. The wavelike nature of matter is readily explained as interference effects between probability waves.
Many systems that were formerly seen as changing over time (for instance, an electron circling a proton) are now described as static (a proton surrounded by a "probability cloud" describing the likelihood of locating the electron at a specific place).
If the probability distributions do change over time, then the Schrödinger equation is used to describe the corresponding evolution of the wave function.
Mathematical formulation
In the mathematically rigorous formulation developed by Paul Dirac and John von Neumann, a system is described by a complex separable Hilbert space (typically a space of square integrable "wave"-functions), a state of the system is a unit vector in that space, and every observable is represented by a self-adjoint densely defined linear operator on that space. Given a state and such an operator, the probabilities for the various outcomes of the corresponding observation can be calculated. The time evolution of a system is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the energy observable, plays a prominent role. The probability distribution of an observable in a given state can be computed from the spectral decomposition of the corresponding operator. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. After the measurement is conducted, the system's state will be an eigenstate corresponding to the measured eigenvalue. Heisenberg's uncertainty principle becomes a theorem about non-commuting operators.
The details of the mathematical formulation are contained in the article mathematical formulation of quantum mechanics.
Extensions
The orginal formulation of quantum mechanics was not compatible with special relativity. However, the principles of quantum mechanics can and have been extended into quantum field theories, which are consistent with special relativity. Quantum mechanics as such omits the electromagnetic force, the strong nuclear force, and gravity. The quantum field theory describing electromagnetism is quantum electrodynamics; it is, at least in principle, capable of explaining chemical interactions as well as the interaction of matter and electromagnetic radiation. The quantum field theory describing the strong nuclear force is quantum chromodynamics, which describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force can be unified, in their quantised forms, into a single quantum field theory: electroweak theory. The unification of quantum mechanics with gravity and hence with general relativity has eluded researchers so far (see Theory of everything).
Applications
Quantum mechanical explanations for the behavior of transistors and diodes underly all of modern technology. Quantum mechanical principles are required by the laser, the electron microscope, and magnetic resonance imaging used in medicine. Most calculations performed in computational chemistry are quantum mechanical.
Researchers are seeking robust methods of directly manipulating quantum states. If successful, this will pave the way for quantum computers, which can perform certain computational tasks much more efficiently than classical computers; and quantum cryptography, which will allow guaranteed secure transmission of information.
Philosophical debate
Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate.
The Copenhagen interpretation, due largely to Niels Bohr, was the standard interpretation of quantum mechanics when it was first formulated. According to it, the probabilistic results provided by quantum mechanics are irreducible, and do not simply reflect our limited knowledge.
Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism. He held that quantum mechanics must be incomplete, and produced a series of objections to the theory. The most famous of these was the EPR paradox.
The many worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a "multiverse" composed of mostly independent parallel universes. While the multiverse is deterministic, we perceive nondeterministic behavior governed by probabilities because we can observe only the universe we inhabit.
History
In 1900, Planck introduced the idea that energy is quantized, in order to derive a formula for the observed frequency dependence of the energy emitted by a black body. In 1905, Einstein explained the photoelectric effect by postulating that light energy comes in quanta called photons. In 1913, Bohr explained the spectral lines of the hydrogen atom, again by using quantization. In 1924, Louis de Broglie put forward his theory of matter waves.
These theories, though successful, were strictly phenomenological: there was no rigorous justification for quantization. They are collectively known as the old quantum theory.
Modern quantum mechanics was born in 1925, when Heisenberg developed matrix mechanics and Schrödinger invented wave mechanics and the Schrödinger equation. Schrödinger subsequently showed that the two approaches were equivalent.
Heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation took shape at about the same time. In 1927, Paul Dirac unified quantum mechanics with special relativity. He also pioneered the use of operator theory, including the influential bra-ket notation. In 1932, John von Neumann formulated the rigorous mathematical basis for quantum mechanics as operator theory.
In the 1940s, quantum electrodynamics was developed by Feynman, Dyson, Schwinger, and Tomonaga. It served as a role model for subsequent quantum field theories.
The many worlds interpretation was formulated by Everett in 1956.
Quantum chromodynamics had a long history, begining in the early 1960s. The theory as we know it today was formulated by Polizter, Gross and Wilzcek in 1975. Building on pioneering work by Schwinger, Higgs, Goldstone and others, Glashow, Weinberg and Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force.
Some quotations
- I do not like it, and I am sorry I ever had anything to do with it.
- Erwin Schrödinger, speaking of quantum mechanics
- Those who are not shocked when they first come across quantum mechanics cannot possibly have understood it.
- Niels Henrik David Bohr
- God does not play dice with the universe.
- Albert Einstein
- I think it is safe to say that no one understands quantum mechanics.
- Richard Feynman
- It's always fun to learn something new about quantum mechanics
- Benjamin Schumacher
Further information:
- A history of quantum mechanics
- George W Mackey, "The mathematical foundations of quantum mechanics", New York, W. A. Benjamin, 1963.