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 that are compatible with the Euclidean coordinates of standard classical mechanics.
[2] 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) being controlled by laboratory apparatuses that are governed by classical mechanics, for example an old fashioned 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.
Published in 1901, Max Planck deduced the existence and value of the constant now bearing his name from considering only Wien's displacement law, statistical mechanics, and electromagnetic theory.
[3] Four years later in 1905, Albert Einstein went further to elucidate this constant and its deep connection to the stopping potential of electrons emitted in the photoelectric effect.
[5]) It can then be concluded that this was a key onset of quantization, that is the discretization of matter into fundamental constituents.
In the original presentation, the orbital angular momentum of the electron was named
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
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", 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.
In fact, the observation of the Balmer series of hydrogen in the history of spectroscopy dates as far back as 1885.
[9] Nonetheless, the watershed events that would come to denote the era of first quantization took place in the vital years spanning 1925–1928.
Simultaneously the authors Max Born and Pascual Jordan in December 1925,[10] together with Paul Dirac also in December 1925,[11] then Erwin Schrödinger in January 1926,[12] following that, Werner Heisenberg together with Born and Jordan in August 1926,[13] and finally Dirac in 1928.
[14] The results of these publications were three theoretical formalisms, two of which proved to be equivalent; that of Born, Heisenberg and Jordan was equivalent to that of Schrödinger, 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,[15] which can be regarded as the first attempt at "second quantization", a term used verbatim by Pauli in a 1943 publication of the American Physical Society.
[16] 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 Schrödinger in 1926.
The collected and expanded works of John von Neumann showed that the two theories were mathematically equivalent,[17] and it is this realization that is today understood as first quantization.
The classical theory of Newton is a second order nonlinear differential equation that gives the deterministic trajectory of a system of mass,
Therefore, it is natural to seek solutions of the Newton equation that are at least second order differentiable.
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 squared of the solution to the Schrödinger equation.
The vector space of infinite sequences, whose square summed up is a convergent series, is known as
It is in one-to-one correspondence with the infinite dimensional vector space of square-integrable functions,
In general, the one-particle state could be described by a complete set of quantum numbers denoted by
associated to an electron in a coulomb potential, like the hydrogen atom, form a complete set (ignoring spin).
All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state
obtaining the completeness relation: 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.
[19] When turning to N-particle systems, i.e., systems containing N identical particles i.e. particles characterized by the same physical parameters such as mass, charge and spin, an extension of the single-particle state function
[20] 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 and fermions which obey the rules: Where we have interchanged two coordinates
The usual wave function is obtained using the Slater determinant and the identical particles theory.
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.