# Canonically conjugate operators

by Shyan
Tags: canonically, conjugate, operators
 P: 737 I've searched for this but found nothing,so I ask it here. What are canonically conjugate operators? Is $[A,B]=cI$ a definition for A and B being canonically conjugate? Thanks
P: 5,307
 Quote by Shyan Is $[A,B]=cI$ a definition for A and B being canonically conjugate?
Yes

Canonically conjugate operators A, B follow from canonically conjugate variables A, B in classical mechanics; their Poisson bracket is {A,B} = 1; they span the phase space of the system, can be used to formulate the Hamilton function H(A,B) and therefore their Hamilton e.o.m. fully define the dynamics of the theory.

In QM (canonical quantization) the variables on phase space are replaced by operators acting on Hilbert space; the commutators are defined as

$$\{A,B\}_\text{Poisson} = c \;\to\; [\hat{A},\hat{B}] = ic;\;c = \text{const.}$$
P: 737
http://en.wikipedia.org/wiki/Canonic...ation_relation
 Quote by Wikipedia In quantum mechanics (physics), the canonical commutation relation is the relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another), for example: $[x,p_x] = i\hbar$
You see,looks like its related to the fourier transform too.
In the things I've read,such relationship exists in classical mechanics too.
Just extending it to QM is a little hard for me.

Thanks tom

P: 5,307

## Canonically conjugate operators

It's correct, that iff one uses a representation like a wave function in x, then p acts as a derivative and this is related to the Fourier transform; and it's correct that the relation between x- and p-space wave functions is just the Fourier transform. But the defining operator equation [x,p] = i is more general than a specific representation and therefore does not require a Fourier transform in its definition.
P: 201
 Quote by tom.stoer Yes Canonically conjugate operators A, B follow from canonically conjugate variables A, B in classical mechanics; their Poisson bracket is {A,B} = 1; they span the phase space of the system, can be used to formulate the Hamilton function H(A,B) and therefore their Hamilton e.o.m. fully define the dynamics of the theory. In QM (canonical quantization) the variables on phase space are replaced by operators acting on Hilbert space; the commutators are defined as $$\{A,B\}_\text{Poisson} = c \;\to\; [\hat{A},\hat{B}] = ic;\;c = \text{const.}$$
Does the prescription of turning canonically conjugate variables into operators whose commutator is "i" hold for any pair of classically conjugate variables? I vaguely recall that this is not true for all pairs of classically conjugate variables (it's certainly true for Cartesian variables), but I don't remember exactly what the issue was.