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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 quite Werner Heisenberg's route to obtaining quantum mechanics, but Paul Dirac introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization, and detailed it in his classic text Principles of Quantum Mechanics. The word canonical arises from the Hamiltonian approach to classical mechanics, in which a system's dynamics is generated via canonical Poisson brackets, a structure which is only partially preserved in canonical quantization. This method was further used by Paul Dirac in the context of quantum field theory, in his construction of quantum electrodynamics. In the field theory context, it is also called the second quantization of fields, in contrast to the semi-classical first quantization of single particles. History When it was first developed, quantum physics dealt only with the quantization of the motion of particles, leaving the electromagnetic field classical, hence the name quantum mechanics. Later the electromagnetic field was also quantized, and even the particles themselves became represented through quantized fields, resulting in the development of quantum electrodynamics (QED) and quantum field theory in general. Thus, by convention, the original form of particle quantum mechanics is denoted first quantization, while quantum field theory is formulated in the language of second quantization. First quantization Single particle systems The following exposition is based on Dirac's treatise on quantum mechanics. In the classical mechanics of a particle, there are dynamic variables which are called coordinates (x) and momenta (p). These specify the state of a classical system. The canonical structure (also known as the symplectic structure) of classical mechanics consists of Poisson brackets enclosing these variables, such as {x, p} = 1. All transformations of variables which preserve these brackets are allowed as canonical transformations in classical mechanics. Motion itself is such a canonical transformation. By contrast, in quantum mechanics, all significant features of a particle are contained in a state | ψ ⟩ {\displaystyle |\psi \rangle } , called a quantum state. Observables are represented by operators acting on a Hilbert space of such quantum states. The eigenvalue of an operator acting on one of its eigenstates represents the value of a measurement on the particle thus represented. For example, the energy is read off by the Hamiltonian operator H ^ {\displaystyle {\hat {H}}} acting on a state | ψ n ⟩ {\displaystyle |\psi _{n}\rangle } , yielding where En is the characteristic energy associated to this | ψ n ⟩ {\displaystyle |\psi _{n}\rangle } eigenstate. Any state could be represented as a linear combination of eigenstates of energy; for example, where an are constant coefficients. As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, X ^ {\displaystyle {\hat {X}}} and P ^ {\displaystyle {\hat {P}}} , respectively. The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator X ^ {\displaystyle {\hat {X}}} representing a particle at position x {\displaystyle x} , which is denoted by an element | x ⟩ {\displaystyle |x\rangle } in the Hilbert space, and which satisfies X ^ | x ⟩ = x | x ⟩ {\displaystyle {\hat {X}}|x\rangle =x|x\rangle } . Then, ψ ( x ) = ⟨ x | ψ ⟩ {\displaystyle \psi (x)=\langle x|\psi \rangle } . Likewise, the eigenstates | p ⟩ {\displaystyle |p\rangle } of the momentum operator P ^ {\displaystyle {\hat {P}}} specify the momentum representation: ψ ( p ) = ⟨ p | ψ ⟩ {\displaystyle \psi (p)=\langle p|\psi \rangle } . The central relation between these operators is a quantum analog of the above Poisson bracket of classical mechanics, the canonical commutation relation, This relation encodes (and formally leads to) the uncertainty principle, in the form Δx Δp ≥ ħ/2. This algebraic structure may be thus considered as the quantum analog of the canonical structure of classical mechanics. Many-particle systems When turning to N-particle systems, i.e., systems containing N identical particles (particles characterized by the same quantum numbers such as mass, charge and spin), it is necessary to extend the single-particle state function ψ ( r ) {\displaystyle \psi (\mathbf {r} )} to the N-particle state function ψ ( r 1 , r 2 , … , r N ) {\displaystyle \psi (\mathbf {r} _{1},\mathbf {r} _{2},\dots ,\mathbf {r} _{N})} . A fundamental difference between classical and quantum mechanics concerns the concept of indi.... Discover the Franz Schwabl popular books. Find the top 100 most popular Franz Schwabl books.