Mathematical Methods for Physics and Engineering : A Comprehensive Guide

19.1 OPERATOR FORMALISM

whilstthatforBA|ψ〉is simply

x

∂ψ

∂x

,

which is not the same.

If the result

AB|ψ〉=BA|ψ〉

is true forallket vectors|ψ〉,thenAandBare said tocommute; otherwise they

are non-commuting operators.

A convenient way to express the commutation properties of two linear operators

is to define theircommutator,[A, B],by

[A, B]|ψ〉≡AB|ψ〉−BA|ψ〉. (19.14)

Clearly two operators that commute have a zero commutator. But, for the example

given above we have that

[

∂

∂x

,x

]

ψ(x)=

(

ψ(x)+x

∂ψ

∂x

)

−

(

x

∂ψ

∂x

)

=ψ(x)=1×ψ

or, more simply, that

[

∂

∂x

,x

]

= 1; (19.15)

in words, the commutator of the differential operator∂/∂xand the multiplicative

operatorxis the multiplicative operator 1. It should be noted that the order of

the linear operators is important and that

[A, B]=−[B, A]. (19.16)

Clearly any linear operator commutes with itself and some other obvious zero

commutators (when operating on wavefunctions with ‘reasonable’ properties) are:

[A, I],whereIis the identity operator;

[An,Am],for any positive integersnandm;

[A, p(A)],wherep(x) is any polynomial inx;

[A, c],whereAis any linear operator andcis any constant;

[f(x),g(x)],where the functions are mutiplicative;

[A(x),B(y)],where the operators act on different variables, with

[

∂

∂x

,

∂

∂y

]

as a specific example.