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 between 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 
, called quantum state. Observables are represented by operators acting on a Hilbert space of such quantum states. The (eigen)value 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 
acting on a state 
, yielding 
, where 
is the characteristic energy associated to this eigenstate.
, called quantum state. Observables are represented by operators acting on a Hilbert space of such quantum states. The (eigen)value 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 acting on a state , yielding , where is the characteristic energy associated to this eigenstate.
Any state could be represented as a linear combination of eigenstates of energy; for example, 
, where an are constant coefficients.
, where an are constant coefficients.
As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, 
and 
, respectively. The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator representing a particle at position x, which is denoted by an element 
in the Hilbert space, and which satisfies 
. Then, 
.
and , respectively. The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator representing a particle at position x, which is denoted by an element in the Hilbert space, and which satisfies . Then, .
Likewise, the eigenstates 
of the momentum operator specify the momentum representation: 
.
of the momentum operator specify the momentum representation: .
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.
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