Momentum operator's relation to commutative algebra

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how is the quantum momentum operator (being a linear differential op.) related to commutative algebra?
 

strangerep

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how is the quantum momentum operator (being a linear differential op.) related to commutative algebra?
(Linear) momentum operators corresponding to independent directions commute.
 
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(Linear) momentum operators corresponding to independent directions commute.
Does the momentum operator by itself carry information about direction?
Or does the operator obtain it from the wavefunction?
 
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dextercioby

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It's customary to call them Abelian algebras.
 

strangerep

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Does the momentum operator by itself carry information about direction?
A (linear) momentum operator generates translations along a particular direction in space.
(I'm not sure what your background is. Normally I recommend Ballentine's textbook.)
 

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