First quantization

A first quantization of a physical system is a possibly semiclassical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well or a bulk electromagnetic field or gravitational field) is treated classically.

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Quantum mechanics
${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }$ Schrödinger equation
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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 that are compatible with the Euclidean coordinates of standard classical mechanics. 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 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.