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Canonically conjugate operators 
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#1
Dec1712, 01:43 PM

P: 868

I've searched for this but found nothing,so I ask it here.
What are canonically conjugate operators? Is [itex] [A,B]=cI [/itex] a definition for A and B being canonically conjugate? Thanks 


#2
Dec1712, 03:21 PM

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P: 5,451

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 [tex]\{A,B\}_\text{Poisson} = c \;\to\; [\hat{A},\hat{B}] = ic;\;c = \text{const.}[/tex] 


#3
Dec1712, 09:34 PM

P: 868

http://en.wikipedia.org/wiki/Canonic...ation_relation
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 


#4
Dec1812, 12:59 AM

Sci Advisor
P: 5,451

Canonically conjugate operators
I would not start with the Fourier transform.
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 pspace 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. 


#5
Dec1812, 04:06 PM

P: 230




#6
Dec1812, 05:27 PM

Sci Advisor
P: 5,451

Not knowing (globally) cartesian coordinates is a difficult starting point. On a manifold (with nontrivial metric) one may define "covariant derivatives" instead of standard ones; usually this results in a reasonable quantum theory. If you start with polar coordinates on the sphere using ∂_{Ω} does not makes sense, but when using covariant derivatives one e.g. arives at the generalized LaplaceBeltrami operator Δ_{g} (g is the metric on the manifold) which is equivalent to the standard 3dim. Laplacian expressed in polar coordinates plus ∂_{r} set to zero (fixed radius). Have a look at http://en.wikipedia.org/wiki/Canonical_quantization as a starting point 


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