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Re: Quantum Commutator

• To: mathgroup at smc.vnet.net
• Subject: [mg85325] Re: Quantum Commutator
• From: "Eric R. Bittner" <bittner at uh.edu>
• Date: Wed, 6 Feb 2008 01:25:49 -0500 (EST)

Dear Dick,
I'm sorry this is sort of late in the post, but I just came across your query looking for similar software using Google.   As it turns out, a bit of
digging through my own quantum mechanics course notes yields a nice solution.

There's a nice relation you my wish to exploit in calculating quantum commutators. I believe it's originally due to Dirac.
In essence, if you have quantum operators A(p,q) and B(p,q)  as functions of the quantum p & q operators, then
[A,B] = i hbar {A,B}
where {A,B} is the classical poisson bracket.  This is easy to code in Mathematica! In addition you can generalize this to any set of
canonical variables, for example the boson operators a & a^\dagger.
For example, if you have operators A(a,a^\dagger) and B(a,a^\dagger)
then
[A(a,a^\dagger), B(a,a^\dagger)] = i hbar (\partial_a A \partial_{a^\dagger}B-
\partial_{a^\dagger}A\partial_a B
For fermions, you need to modifiy this to take in to account Pauli exclusion,  but that's simple since you can relate the commutator to the
anti-commutator.

Cheers!
Eric B.



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