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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
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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