Mathematical Methods for Physics and Engineering - Matematica.NET

19.1 OPERATOR FORMALISMwhilstthatforBA| ψ〉 is simplywhich is not the same.If the resultx ∂ψ∂x ,AB| ψ〉 = BA| ψ〉is true for all ket vectors | ψ〉, thenA and B are said to commute; otherwise theyare non-commuting operators.A convenient way to express the commutation properties of two linear operatorsis to define their commutator, [ A, B ], by[ A, B ] | ψ〉 ≡AB| ψ〉−BA| ψ〉. (19.14)Clearly two operators that commute have a zero commutator. But, for the examplegiven above we have that[ ∂∂x ,x ]ψ(x) =(ψ(x)+x ∂ψ ) (− x ∂ψ )= ψ(x) =1× ψ∂x ∂xor, more simply, that[ ∂∂x ,x ]= 1; (19.15)in words, the commutator of the differential operator ∂/∂x and the multiplicativeoperator x is the multiplicative operator 1. It should be noted that the order ofthe linear operators is important and that[ A, B ] = − [ B,A] . (19.16)Clearly any linear operator commutes with itself and some other obvious zerocommutators (when operating on wavefunctions with ‘reasonable’ properties) are:[ A, I ] , where I is the identity operator;[ A n ,A m ] , for any positive integers n and m;[ A, p(A) ] , where p(x) is any polynomial in x;[ A, c ] , where A is any linear operator and c is any constant;[ f(x),g(x) ] , where the functions are mutiplicative;[ A(x),B(y) ] , where the operators act on different variables, with[ ] ∂∂x , ∂as a specific example.∂y653