Quantum mechanics

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Quantum mechanics, also referred to as quantum physics, is a physical theory that describes the behavior of matter at short length scales.

Table of contents

1 Introduction 2 Description of the theory 3 Mathematical formulation 4 Interactions with other theories of physics 5 Applications 6 Philosophical debate 7 History 8 Some quotations 9 External Links

Introduction

The quantum theory provides a quantitative explanation for two types of phenomena that classical mechanics and classical electrodynamics cannot account for:

• Some observable physical quantities, such as the total energy of a blackbody, take on discrete rather than continuous values. This phenomenon is called quantization, and the smallest possible intervals between the discrete values are called quanta (singular: quantum, from the Latin word for "quantity", hence the name "quantum mechanics.") The size of the quanta typically varies from system to system.
• Under certain experimental conditions, microscopic objects like atoms or electrons exhibit wave-like behavior, such as interference. Under other conditions, the same species of objects exhibit particle-like behavior ("particle" meaning an object that can be localized to a particular region of space), such as scattering. This phenomenon is known as wave-particle duality.

The foundations of quantum mechanics were established during the first half of the 20th century by the work of Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Paul Dirac, and others. Some fundamental aspects of the theory are still being actively studied. Quantum mechanics has also been adopted as the underlying theory of many fields of physics and chemistry, including condensed matter physics, quantum chemistry, and particle physics.

Description of the theory

Quantum mechanics describes the instantaneous state of a system with a wave function that encodes the probability distribution of all measurable properties, or observables. Possible observables for a system include energy, position, momentum, and angular momentum. Quantum mechanics does not assign definite values to the observables, instead making predictions about their probability distributions. The wavelike properties of matter are explained by the interference of wave functions.

Wave functions can change as time progresses. For example, a particle moving in empty space may be described by a wave function that is a wave packet centered around some mean position. As time progresses, the center of the wave packet changes, so that the particle becomes more likely to be located at a different position. The time evolution of wave functions is described by the Schrödinger equation.

Some wave functions describe probability distributions that are constant in time. Many systems that would be treated dynamically in classical mechanics are described by such static wave functions. For example, an electron in an unexcited atom is pictured classically as a particle circling the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric probability cloud surrounding the nucleus.

When a measurement is performed on an observable of the system, the wavefunction turns into one of a set of wavefunctions that are called eigenstates of the observable. This process is known as wavefunction collapse. The relative probabilities of collapsing into each of the possible eigenstates is described by the instantaneous wavefunction just before the collapse. Consider the above example of a particle moving in empty space. If we measure the particle's position, we will obtain a random value '\'x. In general, it is impossible for us to predict with certainty the value of x which we will obtain, although it is probable that we will obtain one that is near the center of the wave packet, where the amplitude of the wave function is large. After the measurement has been performed, the wavefunction of the particle collapses into one that is sharply concentrated around the observed position x''.

During the process of wavefunction collapse, the wavefunction does not obey the Schrödinger equation. The Schrödinger equation is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, the eigenstate to which the wavefunction collapses is probabilistic, not deterministic. The probabilistic nature of quantum mechanics thus stems from the act of measurement.

One of the consequences of wavefunction collapse is that certain pairs of observables, such as position and momentum, can never be simultaneously ascertained to arbitrary precision. This effect is known as Heisenberg's uncertainty principle.

Mathematical formulation

In the mathematically rigorous formulation developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called state vectors) residing in a complex separable Hilbert space (called the state space.) The exact nature of the Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, plays a central role.

Each observable is represented by a densely-defined Hermitian linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. During a measurement, the probability that a system collapses to each eigenstate is given by the absolute square of the inner product between the eigenstate vector and the state vector just before the measurement. We can therefore find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.

The details of the mathematical formulation are contained in the article Mathematical formulation of quantum mechanics.

Interactions with other theories of physics

The fundamental rules of quantum mechanics are very broad. They state that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators. These must be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical (i.e. non-quantum) physics when a system becomes large, which is known as the classical or correspondence limit. One may therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit.

When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field rather than a fixed set of particles. The first complete quantum field theory, quantum electrodynamics, provides a fully relativistic description of the electromagnetic interaction.

The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical 1/r Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. The quantum field theory of the strong nuclear force is quantum chromodynamics, which describes the interactions of the subnuclear particles, the quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory.

It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research.

Semi-classical approximations are techniques that make it possible to formulate a quantum problem with some physical quantities replaced by their classical analogues, in an effort to reduce the complexity of the model. Even within non-relativistic quantum mechanics, a fully microscopic treatment generally requires large-scale numerical computations. Analytic quantum solutions that describe the system behavior in terms of known mathematical functions are available only for a small class of systems, of which the harmonic oscillator and the hydrogen atom are the most important representatives.

Even the helium atom, containing just one more electron than hydrogen, defies all attempts at a fully analytic treatment in quantum mechanics. In such a situation, approximate semi-classical results can provide valuable insights. The necessary methods rely on a detailed understanding of the corresponding classical mechanics, allowing in particular for the existence of chaos. The study of these approximations belongs to the field of quantum chaos.

Applications

Much of modern technology operates under quantum mechanical principles. Examples include the laser, the electron microscope, and magnetic resonance imaging. Most of the calculations performed in computational chemistry rely on quantum mechanics.

Many of the phenomena studied in condensed matter physics are fully quantum mechanical, and cannot be satisfactorily modeled using classical physics. This includes the electronic properties of solids, such as superconductivity and semiconductivity. The study of semiconductors has led to the invention of the diode and the transistor, which are indispensable for modern electronics.

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks with much greater efficiency than classical computers. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum states over arbitrary distances.

Philosophical debate

Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate and many interpretations. See interpretation of quantum mechanics for more detail.

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 nature of quantum mechanics predictions cannot be explained in terms of some other deterministic theory, and do not simply reflect our limited knowledge. Quantum mechanics provides probabilistic results because the physical universe is itself probabilistic rather than deterministic.

Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement. 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. John Stewart Bell's theoretical solution to the EPR paradox, and its later experimental verification, disproved a large class of such hidden variable theories and persuaded the majority of physicists that quantum mechanics is not an approximation to a nominally classical hidden-variable theory.

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 non-deterministic behavior governed by probabilities because we can observe only the universe we inhabit.

The Bohm interpretation postulates the existence of a non-local, universal wavefunction (Schrödinger equation) which allows distant particles to interact instantaneously. It is not popular among physicists largely because it is considered very inelegant.

History

In 1900, Max 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.

The phrase "quantum physics" was first used in Johnston's Planck's Universe in Light of Modern Physics.

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, beginning 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.

Recently, there has been much interest in quantum information.

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 Bohr

God does not play dice with the cosmos.

Albert Einstein

Einstein, don't tell God what to do.

Niels Bohr in response to 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

If that turns out to be true, I'll quit physics.

Max von Laue, Nobel Laureate 1914, of de Broglie's thesis on electrons having wave properties.

Anyone wanting to discuss a quantum mechanical problem had better understand and learn to apply quantum mechanics to that problem.

Willis Lamb, Nobel Laureate 1955

External Links

• A history of quantum mechanics
• George W Mackey, "The mathematical foundations of quantum mechanics", New York, W. A. Benjamin, 1963.
• David Mermin on the future directions of physics