Mathematical Methods for Physics and Engineering - Matematica.NET

QUANTUM OPERATORSRHS gives(−i) 2 ∂ ∂2m ∂x ∂x + (−i)2 ∂ ∂2m ∂y ∂y + (−i)2 ∂ ∂2m ∂z ∂z .The potential energy V , being a function of position only, becomes a purelymultiplicative operator, thus creating the full expression for the Hamiltonian,( ∂2H = − 22m∂x 2 + ∂2∂y 2 + ∂2∂z 2 )+ V (x, y, z),and giving the corresponding Schrödinger equation as(Hψ n = − 2 ∂ 2 )ψ n2m ∂x 2 + ∂2 ψ n∂y 2 + ∂2 ψ n∂z 2 + V (x, y, z)ψ n = E n ψ n .We are not so much concerned in this section with solving such differentialequations, but with the commutation properties of the operators from which theyare constructed. To this end, we now turn our attention to the topic of angularmomentum, the operators for which can be constructed in a straightforwardmanner from the two basic sets.19.2.1 Angular momentum operatorsAs required by the substitution rules, we start by expressing angular momentumin terms of the classical quantities r and p, namely L = r × p with CartesiancomponentsL z = xp y − yp x , L x = yp z − zp y , L y = zp x − xp z .Making the substitutions (19.22) yields as the corresponding quantum-mechanicaloperators(L z = −i x ∂∂y − y ∂ ),∂x(L x = −i y ∂ ∂z − z ∂ ), (19.25)∂y(L y = −i z ∂∂x − x ∂ ).∂zIt should be noted that for xp y , say, x and ∂/∂y commute, and there is noambiguity about the way it is to be carried into its quantum form. Further, sincethe operators corresponding to each of its factors commute and are Hermitian,the operator corresponding to the product is Hermitian. This was shown directlyfor matrices in exercise 8.7, and can be verified using equation (17.16).The first question that arises is whether or not these three operators commute.658