Quantization (physics)

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In physics, quantisation (in American English quantization) is the systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. It is a procedure for constructing quantum mechanics from classical mechanics. A generalization involving infinite degrees of freedom is field quantization, as in the "quantization of the electromagnetic field", referring to photons as field "quanta" (for instance as light quanta). This procedure is basic to theories of atomic physics, chemistry, particle physics, nuclear physics, condensed matter physics, and quantum optics.

Contents

Historical overview

In 1901, when Max Planck was developing the distribution function of statistical mechanics to solve the ultraviolet catastrophe problem, he realized that the properties of blackbody radiation can be explained by the assumption that the amount of energy must be in countable fundamental units, i.e. amount of energy is not continuous but discrete. That is, a minimum unit of energy exists and the following relationship holds for the frequency . Here, is called Planck's constant, which represents the amount of the quantum mechanical effect. It means a fundamental change of mathematical model of physical quantities.

In 1905, Albert Einstein published a paper, "On a heuristic viewpoint concerning the emission and transformation of light", which explained the photoelectric effect on quantized electromagnetic waves. [1] The energy quantum referred to in this paper was later called "photon".  In July 1913, Niels Bohr used quantization to describe the spectrum of a hydrogen atom in his paper "On the constitution of atoms and molecules".

The preceding theories have been successful, but they are very phenomenological theories.  However, the French mathematician Henri Poincaré first gave a systematic and rigorous definition of what quantization is in his 1912 paper "Sur la théorie des quanta". [2] [3]

The term "quantum physics" was first used in Johnston's Planck's Universe in Light of Modern Physics.  (1931).

Canonical quantization

Canonical quantization develops quantum mechanics from classical mechanics. One introduces a commutation relation among canonical coordinates. Technically, one converts coordinates to operators, through combinations of creation and annihilation operators. The operators act on quantum states of the theory. The lowest energy state is called the vacuum state.

Quantization schemes

Even within the setting of canonical quantization, there is difficulty associated to quantizing arbitrary observables on the classical phase space. This is the ordering ambiguity: classically, the position and momentum variables x and p commute, but their quantum mechanical operator counterparts do not. Various quantization schemes have been proposed to resolve this ambiguity, [4] of which the most popular is the Weyl quantization scheme. Nevertheless, the Groenewold–van Hove theorem dictates that no perfect quantization scheme exists. Specifically, if the quantizations of x and p are taken to be the usual position and momentum operators, then no quantization scheme can perfectly reproduce the Poisson bracket relations among the classical observables. [5] See Groenewold's theorem for one version of this result.

Covariant canonical quantization

There is a way to perform a canonical quantization without having to resort to the non covariant approach of foliating spacetime and choosing a Hamiltonian. This method is based upon a classical action, but is different from the functional integral approach.

The method does not apply to all possible actions (for instance, actions with a noncausal structure or actions with gauge "flows"). It starts with the classical algebra of all (smooth) functionals over the configuration space. This algebra is quotiented over by the ideal generated by the Euler–Lagrange equations. Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the Peierls bracket. This Poisson algebra is then ℏ -deformed in the same way as in canonical quantization.

In quantum field theory, there is also a way to quantize actions with gauge "flows". It involves the Batalin–Vilkovisky formalism, an extension of the BRST formalism.

Deformation quantization

One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. [6] Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, H. J. Groenewold [7] considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to discover the phase-space star-product of a pair of functions. More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's map is not satisfactory.

For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ2/2. (This extra term offset is pedagogically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom, even though the standard QM ground state of the atom has vanishing l.) [8]

As a mere representation change, however, Weyl's map is useful and important, as it underlies the alternate equivalent phase space formulation of conventional quantum mechanics.

Geometric quantization

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

A more geometric approach to quantization, in which the classical phase space can be a general symplectic manifold, was developed in the 1970s by Bertram Kostant and Jean-Marie Souriau. The method proceeds in two stages. [9] First, once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space. Here one can construct operators satisfying commutation relations corresponding exactly to the classical Poisson-bracket relations. On the other hand, this prequantum Hilbert space is too big to be physically meaningful. One then restricts to functions (or sections) depending on half the variables on the phase space, yielding the quantum Hilbert space.

Path integral quantization

A classical mechanical theory is given by an action with the permissible configurations being the ones which are extremal with respect to functional variations of the action. A quantum-mechanical description of the classical system can also be constructed from the action of the system by means of the path integral formulation.

Other types

See also

Related Research Articles

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<span class="mw-page-title-main">Phase space</span> Space of all possible states that a system can take

In dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the direct product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.

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Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum jumps supplanted the Bohr model's electron orbits. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation.

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<span class="mw-page-title-main">Canonical quantization</span> Process of converting a classical physical theory into one compatible with quantum mechanics

In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible.

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

<span class="mw-page-title-main">Wigner quasiprobability distribution</span> Wigner distribution function in physics as opposed to in signal processing

The Wigner quasiprobability distribution is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space.

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In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture.

First quantization is a procedure for converting equations of classical particle equations into quantum wave equations. The companion concept of second quantization converts classical field equations in to quantum field equations.

In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.

Quantum characteristics are phase-space trajectories that arise in the phase space formulation of quantum mechanics through the Wigner transform of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton equations in quantum form and play the role of characteristics in terms of which time-dependent Weyl's symbols of quantum operators can be expressed. In the classical limit, quantum characteristics reduce to classical trajectories. The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics.

The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position or momentum representations. The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution and operator multiplication is replaced by a star product.

References

Notes

  1. ↑ Folsing, Albrecht (1997), Albert Einstein: A Biography, trans. Ewald Osers, Viking
  2. ↑ McCormmach, Russell (Spring 1967). "Henri Poincaré and the Quantum Theory". Isis. 58 (1): 37–55. doi:10.1086/350182. S2CID   120934561.
  3. ↑ Irons, F.E. (August 2001). "Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms". American Journal of Physics. 69 (8): 879–84. Bibcode:2001AmJPh..69..879I. doi:10.1119/1.1356056.
  4. ↑ Hall 2013 Chapter 13
  5. ↑ Hall 2013 Theorem 13.13
  6. ↑ Weyl, H. (1927). "Quantenmechanik und Gruppentheorie". Zeitschrift für Physik . 46 (1–2): 1–46. Bibcode:1927ZPhy...46....1W. doi:10.1007/BF02055756. S2CID   121036548.
  7. ↑ Groenewold, H.J. (1946). "On the principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4. ISSN   0031-8914.
  8. ↑ Dahl, Jens Peder; Schleich, Wolfgang P. (2002). "Concepts of radial and angular kinetic energies". Physical Review A. 65 (2): 022109. arXiv: quant-ph/0110134 . Bibcode:2002PhRvA..65b2109D. doi:10.1103/PhysRevA.65.022109. ISSN   1050-2947. S2CID   39409789.
  9. ↑ Hall 2013 Chapters 22 and 23
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