quantum mechanics (redirected from Quantum mechanic)

quantum mechanics: see quantum theory.

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quantum mechanics

Branch of mathematical physics that deals with atomic and subatomic systems. It is concerned with phenomena that are so small-scale that they cannot be described in classical terms, and it is formulated entirely in terms of statistical probabilities. Considered one of the great ideas of the 20th century, quantum mechanics was developed mainly by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, and Max Born and led to a drastic reappraisal of the concept of objective reality. It explained the structure of atoms, atomic nuclei (see nucleus), and molecules; the behaviour of subatomic particles; the nature of chemical bonds (see bonding); the properties of crystalline solids (see crystal); nuclear energy; and the forces that stabilize collapsed stars. It also led directly to the development of the laser, the electron microscope, and the transistor.

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quantum mechanics

The branch of physics developed in the first part of the 20th century that was highly successful in explaining the behavior of atoms, molecules and nuclei. Developed between 1900 and 1930 and combined with the general and special theory of relativity, it revolutionized the field of physics. The new concepts, which were the particle properties of radiation, the wave properties of matter, quantization of physical properties and the idea that one can no longer know exactly where a single particle such as an electron is at any one instance were necessary to explain all of the new experimental evidence that was available at the time. For example, quantum mechanics explains the behavior of semiconductors which are used to make the myriad devices we use every day. See wave-particle duality.

Following are the important contributors to the foundation of quantum mechanics and the principles they uncovered.

   Year  Researcher    Quantum Mechanics Concept
   1901  Planck        Blackbody radiation
   1905  Einstein      Photoelectric effect
   1913  Bohr          Quantum theory of spectra
   1922  Compton       Scattering of photons
                        off electrons
   1924  Pauli         Exclusion principle
   1925  de Broglie    Matter waves
   1926  Schroedinger  Wave equation
   1927  Heisenberg    Uncertainty principle
   1927  Davison and
          Germer       Wave properties of electrons
   1927  Born          Interpretation of the
                        wavefunction

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quantum mechanics [′kwän·təm mi′kan·iks]

(physics)

The modern theory of matter, of electromagnetic radiation, and of the interaction between matter and radiation; it differs from classical physics, which it generalizes and supersedes, mainly in the realm of atomic and subatomic phenomena. Also known as quantum theory.

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Quantum mechanics

The modern theory of matter, of electromagnetic radiation, and of the interaction between matter and radiation; also, the mechanics of phenomena to which this theory may be applied. Quantum mechanics, also termed wave mechanics, generalizes and supersedes the older classical mechanics and Maxwell's electromagnetic theory. Atomic and subatomic phenomena provide the most striking evidence for the correctness of quantum mechanics and best illustrate the differences between quantum mechanics and the older classical physical theories. Quantum mechanics is needed to explain many properties of bulk matter, for instance, the temperature dependence of the specific heats of solids.

The formalism of quantum mechanics is not the same in all domains of applicability. In approximate order of increasing conceptual difficulty, mathematical complexity, and likelihood of future fundamental revision, these domains are the following: (i) Nonrelativistic quantum mechanics, applicable to systems in which particles are neither created nor destroyed, and in which the particles are moving slowly compared to the velocity of light. Here a particle is defined as a material entity having mass, whose internal structure either does not change or is irrelevant to the description of the system. (ii) Relativistic quantum mechanics, applicable in practice to a single relativistic particle (one whose speed equals or nearly equals c); here the particle may have zero rest mass, in which event, its speed must equal c. (iii) Quantum field theory, applicable to systems in which particle creation and destruction can occur; the particles may have zero or nonzero rest mass. This article is concerned mainly with nonrelativistic quantum mechanics, which apparently applies to all atomic and molecular phenomena, with the exception of the finer details of atomic spectra. Nonrelativistic quantum mechanics also is well established in the realm of low-energy nuclear physics. See Atomic structure and spectra, Nuclear physics, Quantum field theory, Relativistic quantum theory

Planck's constant

The quantity 6.626 × 10^-34 joule-second, first introduced into physical theory by Max Planck in 1901, is a basic ingredient of the formalism of quantum mechanics. Planck's constant commonly is denoted by the letter h; the notation ħ = h/2π also is standard.

Uncertainty principle

In classical physics the observables characterizing a given system are assumed to be simultaneously measurable (in principle) with arbitrarily small error. For instance, it is thought possible to observe the initial position and velocity of a particle and therewith, using Newton's laws, to predict exactly its future path in any assigned force field. According to the uncertainty principle, accurate measurement of an observable quantity necessarily produces uncertainties in one's knowledge of the values of other observables. In particular, for a single particle relation (1a)

(1a) 

(1b) 

holds, where Δx represents the uncertainty (error) in the location of the x coordinate of the particle at any instant, and Δp_x is the simultaneous uncertainty in the x component of the particle momentum. Relation (1a) asserts that under the best circumstances, the product ΔxΔp_x of the uncertainties cannot be less than about 10^-34 joule second.

The uncertainty relation (1b) is derived and interpreted somewhat differently than relation (1a); it asserts that for any system, an energy measurement with error ΔE must be performed in a time not less than Δt ∼ ħ/ΔE. If a system endures for only Δt seconds, any measurement of its energy must be uncertain by at least ΔE ∼ ħΔt. See Uncertainty principle

Wave-particle duality

It is natural to identify such fundamental constituents of matter as protons and electrons with the mass points or particles of classical mechanics. According to quantum mechanics, however, these particles, in fact all material systems, necessarily have wavelike properties. Conversely, the propagation of light, which, by Maxwell's electromagnetic theory, is understood to be a wave phenomenon, is associated in quantum mechanics with massless energetic and momentum-transporting particles called photons. The quantum-mechanical synthesis of wave and particle concepts is embodied in the de Broglie relations, given by Eqs. (2a) and (2b).

(2{\it a}) 

(2{\it b}) 

These give the wavelength λ and wave frequency f associated with a free particle (a particle moving freely under no forces) whose momentum is p and energy is E; the same relations give the photon momentum p and energy E associated with an electromagnetic wave in free space (that is, in a vacuum) whose wavelength is λ and frequency is f. See Photon

The wave properties of matter have been demonstrated conclusively for beams of electrons, neutrons, atoms (hydrogen, H, and helium, He), and molecules (H₂). When incident upon crystals, these beams are reflected into certain directions, forming diffraction patterns. Diffraction patterns are difficult to explain on a particle picture; they are readily understood on a wave picture, in which wavelets scattered from regularly spaced atoms in the crystal lattice interfere constructively along certain directions only. See Electron diffraction, Neutron diffraction

The particle properties of light waves are observed in the photoelectric effect and the compton effect. See Compton effect, Photoemission

Complementarity

Wave-particle duality and the uncertainty principle are thought to be examples of the more profound principle of complementarity, first enunciated by Niels Bohr (1928). According to the principle of complementarity, nature has “complementary” aspects; an experiment which illuminates one of these aspects necessarily simultaneously obscures the complementary aspect. To put it differently, each experiment or sequence of experiments yields only a limited amount of information about the system under investigation; as this information is gained, other equally interesting information (which could have been obtained from another sequence of experiments) is lost. Of course, the experimenter does not forget the results of previous experiments, but at any instant, only a limited amount of information is usable for predicting the future course of the system.

Quantization

In classical physics the possible numerical values of each observable, meaning the possible results of exact measurement of the observable, generally form a continuous set. For example, the x coordinate of the position of a particle may have any value between -∞ and +∞. In quantum mechanics the possible numerical values of an observable need not form a continuous set, however. For some observables, the possible results of exact measurement form a discrete set; for other observables, the possible numerical values are partly discrete, partly continuous; for example, the total energy of an electron in the field of a proton may have any positive value between 0 and +∞, but may have only a discrete set of negative values, namely, -13.6, -13.6/4, -13.6/9, -13.6/16 eV,…. Such observables are said to be quantized; often there are simple quantization rules determining the quantum numbers which specify the allowable discrete values. Spectroscopy, especially the study of atomic spectra, probably provides the most detailed quantitative confirmation of quantization.

Probability considerations

The uncertainty and complementarity principles, which limit the experimenter's ability to describe a physical system, must limit equally the experimenter's ability to predict the results of measurement on that system. Suppose, for instance, that a very careful measurement determines that the x coordinate of a particle is precisely x = x₀. This is permissible in nonrelativistic quantum mechanics. Then, formally, the particle is known to be in the eigenstate corresponding to the eigenvalue x = x₀ of the x operator. Under these circumstances, an immediate repetition of the position measurement again will indicate that the particle lies at x = x₀. Knowing that the particle lies at x = x₀ makes the momentum p_x of the particle completely uncertain, however, according to relation (1a). A measurement of p_x immediately after the particle is located at x = x₀ could yield any value of p_x from -∞ to +∞.

More generally, suppose the system is known to be in the eigenstate corresponding to the eigenvalue α of the observable A. Then for any observable B, which is to some extent complementary to A, that is, for which an uncertainty relation of the form of relations (1) limits the accuracy with which A and B can simultaneously be measured, it is not possible to predict which of the many possible values B = β will be observed. However, it is possible to predict the relative probabilities P_α(β) of immediately thereafter finding the observable B equal to β, that is, of finding the system in the eigenstate corresponding to the eigenvalue B = β.

To the eigenvalues correspond eigenfunctions, in terms of which P_α ≲ β can be computed. In particular, when α is a discrete eigenvalue of A, and the operators depend only on x and p_x, the probability P_α(β) is postulated as in

(3) 

Eq. (3), where u(x,α) is the eigenfunction corresponding to A = α; v(x,β) is the eigenfunction corresponding to B = β; and the * denotes the complex conjugate. The integral in Eq. (3) is called the projection of u(x,α) on u(x,β). The quantity |u(x,α)|² dx is the probability that the system, known to be in the eigenstate A = α, will be found in the interval x to x + dx. See Eigenvalue (quantum mechanics)

Wave function

When the system is known to be in the eigenstate corresponding to A = α, the eigenfunction u(x,α) is the wave function; that is, it is the function whose projection on an eigenfunction v(x,β) of any observable B gives the probability of measuring B = β. The wave function ψ(x) may be known exactly; in other words, the state of the system may be known as exactly as possible (within the limitations of uncertainty and complementarity), even though ψ(x) is not the eigenfunction of a known operator. This circumstance arises because the wave function obeys Schrödinger's wave equation. Knowing the value of ψ(x) at time t = 0, the wave equation completely determines ψ(x) at all future times. In general, however, if ψ(x,0) = u(x,α), that is, if ψ(x,t) is an eigenfunction of A at t = 0, then ψ(x,t) will not be an eigenfunction of A at later times t > 0.

A system described by a wave function is said to be in a pure state. Not all systems are described by wave functions, however. For example, a beam of hydrogen atoms streaming out of a small hole in a hydrogen discharge tube can be regarded as a statistical ensemble or mixture of pure states oriented with equal probability in all directions.

Schrödinger equation

Equation (4) describes a plane wave

(4) 

of frequency f, wavelength λ, and amplitude A(λ), propagating in the positive x direction. The previous discussion concerning wave-particle duality suggests that this is the form of the wave function for a beam of free particles moving in the x direction with momentum p = p_x, with Eq. (2) specifying the connections between f, λ, and E, p. Differentiating Eq. (4), it is seen that Eqs. (5) hold. Since for a free particle E = p²/2m, it follows also that Eq. (6) is valid.

(5{\it a}) 

(5{\it b}) 

(6) 

See W ave motion

Equation (6) holds for a plane wave of arbitrary λ, and therefore for any superposition of waves of arbitrary λ, that is, arbitrary p_x. Consequently, Eq. (6) should be the wave equation obeyed by the wave function of any particle moving under no forces, whatever the projections of the wave function on the eigenfunctions of p_x. Equations (5) and (6) further suggest that for a particle whose potential energy V(x) changes, in other words, for a particle in a conservative force field, ψ(x,t) obeys Eq. (7).

(7) 

Equation 7 is the time-dependent Schrödinger equation for a one-dimensional (along x), spinless particle. Noting Eq. (5b), and observing that Eq. (7) has a solution for the form of Eq. (8), it is inferred that ψ(x) of Eq. (8) obeys the time-dependent Schrödinger equation, Eq. (9).

(8) 

(9) 

See F orce

Equation (9) is solved subject to reasonable boundary conditions, for example, that ψ must be continuous and must not become infinite as x approaches ±∞. These boundary conditions restrict the values of E for which there exist acceptable solutions ψ(x) to Eq. (9), the allowed values of E depending on V(x). In this manner, the allowed energies of atomic hydrogen listed in the earlier discussion of quantization are obtained.

The forms of Eqs. (5a), (7), and (9) suggest that the classical observable p_x, must be replaced by the operator (ħ/i) (∂/∂x). With this replacement, Eq. (10) holds.

(10) 

In other words, whereas the classical canonically conjugate variables x and p_x are numbers, obeying the commutative law in Eq. (11a), the quantum-mechanical quantities x and p_x are noncommuting operators, obeying Eq. (11b).

(11{\it a}) 

(11{\it b}) 

Correspondence principle

Since classical mechanics and Maxwell's electromagnetic theory accurately describe macroscopic phenomena, quantum mechanics must have a classical limit in which it is equivalent to the older classical theories. Although there is no rigorous proof of this principle for arbitrarily complicated quantum-mechanical systems, its validity is well established by numerous illustrations.