canonical quantization

canonical quantization

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

(

http://planetmath.org/HamiltonianOperatorOfAQuantumSystem

). 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

(

x

i

,

p

i

)

be a set of

Darboux coordinates

on

T

*

⁢

X

. Then we may obtain from each coordinate function an operator on the

Hilbert space

ℋ

=

L

2

⁢

(

X

,

μ

)

, consisting of functions on

X

that are square-integrable with respect to some measure

μ

, by the

operator substitution

rule:

x

i

↦

x

^

i

=

x

i

⋅

,

(1)

p

i

↦

p

^

i

=

-

i

⁢

ℏ

⁢

∂

∂

⁡

x

i

,

(2)

where

x

i

⋅

is the “multiplication by

x

i

” operator. Using this rule, we may obtain operators from a larger class of functions. For example,

1.

x

i

x

j

↦

x

^

i

x

^

j

=

x

i

x

j

⋅

,

2.

p

i

⁢

p

j

↦

p

^

i

⁢

p

^

j

=

-

ℏ

2

⁢

∂

2

∂

⁡

x

i

⁢

x

j

,

3.

if

i

≠

j

then

x

i

⁢

p

j

↦

x

^

i

⁢

p

^

j

=

-

i

⁢

ℏ

⁢

x

i

⁢

∂

∂

⁡

x

j

.

Remark

.

The substitution rule creates an ambiguity for the function

x

i

⁢

p

j

when

i

=

j

, since

x

i

⁢

p

j

=

p

j

⁢

x

i

, whereas

x

^

i

⁢

p

^

j

≠

p

^

j

⁢

x

^

i

. This is the

operator ordering

problem. One possible solution is to choose

x

i

⁢

p

j

↦

1

2

⁢

(

x

^

i

⁢

p

^

j

+

p

^

j

⁢

x

^

i

)

,

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 (

1

)-(

2

) to a map

C

∞

⁢

(

T

*

⁢

X

)

→

Op

⁡

(

ℋ

)

f

↦

f

^

.

Remark

.

This procedure is called “canonical” because it preserves the canonical

Poisson brackets

. In particular, we have that

-

i

ℏ

⁢

[

x

^

i

,

p

^

j

]

:=

-

i

ℏ

⁢

(

x

^

i

⁢

p

^

j

-

p

^

j

⁢

x

^

i

)

=

δ

j

i

,

which agrees with the Poisson bracket

{

x

i

,

p

j

}

=

δ

j

i

.

Example 1

.

Let

X

=

ℝ

. The

Hamiltonian

function for a one-dimensional point particle with mass

m

is

H

=

p

2

2

⁢

m

+

V

⁢

(

x

)

,

where

V

⁢

(

x

)

is the potential energy. Then, by operator substitution, we obtain the Hamiltonian operator

H

^

=

-

ℏ

2

2

⁢

m

⁢

d

2

d

⁢

x

2

+

V

⁢

(

x

)

.

Title

canonical quantization

Canonical name

CanonicalQuantization

Date of creation

2013-03-22 15:53:34

Last modified on

2013-03-22 15:53:34

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

18

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81S10

Classification

msc 53D50

Classification

msc 46L65

Related topic

Quantization

Related topic

PoissonBracket

Related topic

HamiltonianOperatorOfAQuantumSystem

Related topic

SchrodingerOperator

Related topic

AsymptoticMorphismsAndWignerWeylMoyalQuantizationProcedures

Defines

operator substitution rule

Defines

operator ordering problem

Generated on Sat Feb 10 13:06:27 2018 by

LaTeXML