General commutation relations for quantum operators

[cdot]

(This is not a homework problem). I'm an undergrad physics student taking my second course in quantum. My question is about operator methods. Most of the proofs for different commutation relations for qm operators involve referring to specific forms of the operators given some basis. For example, to derive [x,p] = i hbar , you can use the representation of x and p in coordinate basis (multiplication by x and differential operator with respect to x) and consider the action of the commutator on some function of x. However, some of the material I've been reading seems to imply that we can understand the properties of operators without making explicit reference to a particular representation of an operator in some basis. My question is this: If you derive a commutation relation for 2 operators using a particular representation, is it valid for any representation? If so, is it generally easier to figure out a commutation relation by picking a representation or are there easier and possibly more general methods? Thank you

 

[andrewkirk]

My question is this: If you derive a commutation relation for 2 operators using a particular representation, is it valid for any representation?

It depends on how the relation is expressed. In general, if it is expressed without using any representation-specific items, then it will be representation-independent. That is, you can use representation-specific working to prove a non-representation-specific result, as long as the result is expressed in non-representation-specific symbols.