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canonical quantization∗ bci1† 2013-03-21 20:29:15 Canonical quantization is a method of relating, or associating, a classical system of the form (T ∗ X, ω, H), where X is a manifold, ω is the canonical symplectic form on T ∗ X, with a (more complex) quantum system represented by H ∈ C ∞ (X), where H is the Hamiltonian operator. Some of the early formulations of quantum mechanics used such quantization methods under the umbrella of the correspondence principle or postulate. The latter states that a correspondence exists between certain classical and quantum operators, (such as the Hamiltonian operators) or algebras (such as Lie or Poisson (brackets)), with the classical ones being in the real (R) domain, and the quantum ones being in the complex (C) domain. Whereas all classical observables and states are specified only by real numbers, the ’wave’ amplitudes in quantum theories are represented by complex functions. Let (xi , pi ) be a set of Darboux coordinates on T ∗ X. Then we may obtain from each coordinate function an operator on the Hilbert space H = L2 (X, µ), consisting of functions on X that are square-integrable with respect to some measure µ, by the operator substitution rule: xi 7→ x̂i = xi ·, (1) ∂ pi 7→ p̂i = −i~ i , ∂x (2) where xi · is the “multiplication by xi ” operator. Using this rule, we may obtain operators from a larger class of functions. For example, 1. xi xj 7→ x̂i x̂j = xi xj ·, 2 2. pi pj 7→ p̂i p̂j = −~2 ∂x∂i xj , ∂ 3. if i 6= j then xi pj 7→ x̂i p̂j = −i~xi ∂x j. ∗ hCanonicalQuantizationi created: h2013-03-21i by: hbci1i version: h37894i Privacy setting: h1i hDefinitioni h81S10i h53D50i h46L65i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 Remark. The substitution rule creates an ambiguity for the function xi pj when i = j, since xi pj = pj xi , whereas x̂i p̂j 6= p̂j x̂i . This is the operator ordering problem. One possible solution is to choose xi pj 7→ 1 i x̂ p̂j + p̂j x̂i , 2 since this choice produces an operator that is self-adjoint and therefore corresponds to a physical observable. More generally, there is a construction known as Weyl quantization that uses Fourier transforms to extend the substitution rules (??)-(??) to a map C ∞ (T ∗ X) → Op(H) f 7→ fˆ. Remark. This procedure is called “canonical” because it preserves the canonical Poisson brackets. In particular, we have that −i i −i i [x̂ , p̂j ] := x̂ p̂j − p̂j x̂i = δji , ~ ~ which agrees with the Poisson bracket {xi , pj } = δji . Example 1. Let X = R. The Hamiltonian function for a one-dimensional point particle with mass m is p2 H= + V (x), 2m where V (x) is the potential energy. Then, by operator substitution, we obtain the Hamiltonian operator Ĥ = −~2 d2 + V (x). 2m dx2 2