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A first quantization of a physical system is a possibly semiclassical treatment of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, in which particles or physical objects are treated using quantum
wave functions A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
but the surrounding environment (for example a
potential well A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is ca ...
or a bulk
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
or
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
) is treated classically. However, this need not be the case. In particular, a fully quantum version of the theory can be created by interpreting the interacting fields and their associated potentials as operators of multiplication, provided the potential is written in the
canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
that are compatible with the Euclidean coordinates of standard
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
. First quantization is appropriate for studying a single quantum-mechanical system (not to be confused with a single particle system, since a single quantum wave function describes the state of a single quantum system, which may have arbitrarily many complicated constituent parts, and whose evolution is given by just one uncoupled
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
) being controlled by
laboratory A laboratory (; ; colloquially lab) is a facility that provides controlled conditions in which scientific or technological research, experiments, and measurement may be performed. Laboratory services are provided in a variety of settings: physi ...
apparatuses that are governed by
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, for example an old fashion voltmeter (one devoid of modern semiconductor devices, which rely on quantum theory-- however though this is sufficient, it is not necessary), a simple thermometer, a magnetic field generator, and so on.


History

Published in 1901,
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...
deduced the existence and value of the constant now bearing his name from considering only Wien's displacement law,
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
, and electromagnetic theory. Four years later in 1905, Albert Einstein went further to elucidate this constant and its deep connection to the stopping potential of photons emitted in the photoelectric effect. The energy in the photoelectric effect depended not only on the number of incident photons (the intensity of light) but also the frequency of light, a novel phenomena at the time, which would earn Einstein the 1921 Nobel Prize in Physics. It can then be concluded that this was a key onset of quantization, that is the discretization of matter into fundamental constituents. About eight years later
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
in 1913, published his famous three part series where, essentially by fiat, he posits the quantization of the angular momentum in hydrogen and hydrogen like metals. Where in effect, the orbital angular momentum L of the (valence) electron, takes the form L = l \hbar, where l is presumed a whole number 0,\,1,\,2,\,3,\,\ldots\,. In the original presentation, the orbital angular momentum of the electron was named M, the Planck constant divided by two pi as M_0, and the quantum number or "counting of number of passes between stationary points", as stated by Bohr originally as, \tau. See references above for more detail. While it would be later shown that this assumption is not entirely correct, it in fact ends up being rather close to the correct expression for the orbital angular momentum operator's (eigenvalue) quantum number for large values of the quantum number l, and indeed this was part of Bohr's own assumption. Regard the consequence of Bohr's assumption L^2 = l^2 \hbar^2, and compare it with the correct version known today as L^2 = l(l+1)\hbar^2. Clearly for large l, there is little difference, just as well as for l=0, the equivalence is exact. Without going into further historical detail, it suffices to stop here and regard this era of the history of quantization to be the "
old quantum theory The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory ...
", meaning a period in the history of physics where the corpuscular nature of subatomic particles began to play an increasingly important role in understanding the results of physical experiments, whose mandatory conclusion was the discretization of key physical observable quantities. However, unlike the era below described as the era of first quantization, this era was based solely on purely classical arguments such as Wien's displacement law,
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
,
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
, and the electromagnetic theory. In fact, the observation of the Balmer series of hydrogen in the history of spectroscopy dates as far back as 1885. Nonetheless, the watershed events, which would come to denote the era of first quantization, took place in the vital years spanning 1925-1928. Simultaneously the authors Born and Jordan in December of 1925, together with Dirac also in December of 1925, then Schrodinger in January 1926, following that, Born, Heisenberg and Jordan in August 1926, and finally Dirac in 1928. The results of these publications were 3 theoretical formalisms 2 of which proved to be equivalent, that of Born, Heisenberg and Jordan was equivalent to that of Schrodinger, while Dirac's 1928 theory came to be regarded as the relativistic version of the prior two. Lastly, it is worth mentioning the publication of Heisenberg and Pauli in 1929, which can be regarded as the first attempt at "
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
", a term used verbatim by Pauli in a 1943 publication of the
American Physical Society The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of k ...
. For purposes of clarification and understanding of the terminology as it evolved over history, it suffices to end with the major publication that helped recognize the equivalence of the matrix mechanics of Born, Heisenberg, and Jordan 1925-1926 with the wave equation of Schrodinger in 1926. The collected and expanded works of
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
showed that the two theories were mathematically equivalent, and it is this realization that is today understood as first quantization.This statement is not unique since it can be argued that the mathematically imprecise notation of Dirac, even still today, can elucidate the equivalence. Just as well, the "testing ground" of hydrogen can also be seen as strong evidence for a conclusion of equivalence.


Qualitative mathematical preliminaries

To understand the term first quantization one must first understand what it means for something to be quantum in the first place. The classical theory of Newton is a second order
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
that gives the deterministic trajectory of a system of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
, m. The
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
, a, in
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...
of motion, F=ma, is the second derivative of the system's position as a function of time. Therefore, it is natural to seek solutions of the Newton equation that are at least second order differentiable. Quantum theory differs dramatically in that it replaces physical observables such as the position of the system, the time at which that observation is made, the mass, and the velocity of the system at the instant of observation with the notion of operator observables. Operators as observables change the notion of what is measurable and brings to the table the unavoidable conclusion of the Max Born probability theory. In this framework of nondeterminism, the probability of finding the system in a particular observable state is given by a dynamic probability density that is defined as the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
squared of the solution to the Schrodinger equation. The fact that probability densities are integrable and normalizable to unity imply that the solutions to the Schrodinger equation must be square integrable. The vector space of infinite sequences, whose square summed up is a convergent series, is known as \ell^2 (pronounced "little ell two"). It is in one-to-one correspondence with the infinite dimensional vector space of square-integrable functions, L^2(\mathbb^d), from the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
\mathbb^d to the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, \mathbb. For this reason, \ell^2 and L^2(\mathbb^d) are often referred to indiscriminately as "the" Hilbert space. This is rather misleading because \mathbb^d is also a Hilbert space when equipped and completed under the Euclidean
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, albeit a finite dimensional space.


Types of systems

Both the Newton theory and the Schrodinger theory have a mass parameter in them and they can thus describe the evolution of a collection of masses or a single constituent system with a single total mass, as well as an idealized single particle with idealized single mass system. Below are examples of different types of systems.


One-particle systems

In general, the one-particle state could be described by a complete set of quantum numbers denoted by \nu. For example, the three quantum numbers n,l,m associated to an electron in a
coulomb potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
, like the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...
, form a complete set (ignoring spin). Hence, the state is called , \nu\rangle and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using \psi_\nu(\mathbf)= \langle \mathbf, \nu\rangle. All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state , \psi\rangle=\sum_\nu, \nu\rangle\langle \nu, \psi\rangle obtaining the completeness relation: :\sum_\nu, \nu\rangle\langle \nu, =\mathbf Many have felt that all the properties of the particle could be known using this vector basis, which is expressed here using the Dirac
Bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathem ...
. However this need not be true.


Many-particle systems

When turning to ''N''-particle systems, i.e., systems containing ''N''
identical particles In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
i.e. particles characterized by the same physical parameters such as
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
, charge and
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
, an extension of the single-particle state function \psi(\mathbf) to the ''N''-particle state function \psi(\mathbf_1,\mathbf_2,...,\mathbf_N) is necessary. A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-called
bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
and fermions which obey the rules: :\psi(\mathbf_1,...,\mathbf_j,...,\mathbf_k,...,\mathbf)=+\psi(\mathbf_1,...,\mathbf_k,...,\mathbf_j,...,\mathbf_N) (bosons), :\psi(\mathbf_1,...,\mathbf_j,...,\mathbf_k,...,\mathbf)=-\psi(\mathbf_1,...,\mathbf_k,...,\mathbf_j,...,\mathbf_N) (fermions). Where we have interchanged two coordinates (\mathbf_j, \mathbf_k) of the state function. The usual wave function is obtained using the
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electro ...
and the
identical particles In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
theory. Using this basis, it is possible to solve any many-particle problem that can be clearly and accurately described by a single wave function single system-wide diagonalizable state. From this perspective, first quantization is not a truly multi-particle theory but the notion of "system" need not consist of a single particle either.


See also

*
Canonical quantization 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. Historically, this was not quit ...
* Geometric quantization * Quantization *
Second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...


Notes


References

{{DEFAULTSORT:First Quantization Quantum mechanics