Tensors for Physics

106 7 Fields, Spatial Differential Operators

the equation (7.78) can be written as

∇μ=r̂μ

∂

∂r

−r−^1 εμνλr̂νLλ. (7.81)

Notice: the differential operatorL =r×∇only acts on the angular part of a

function depending on the vectorr. In particular, one hasLf(r)=0 when fis

only a function ofr=|r|.

The Cartesian components of the differential operator do not commute. More

specifically, one finds the commutation relation

LμLν−LνLμ=−εμνλLλ, (7.82)

or equivalently,

ελμνLμLν=Lλ. (7.83)

The differential operatorLis closely related to the quantum mechanical angular

momentum operator, cf. Sect.7.6.2.

7.4 Exercise: Radial and Angular Parts of the Nabla Operator, Compare
Equation(7.81)with(7.78)

7.5 Exercise: Prove the Relations(7.82)and(7.83)for the Angular Nabla
Operator

Hint:use(4.10) and observe that the names of summation indices can be changed
conveniently, as long as none appears more than twice.

7.6.2 Application: Orbital Angular Momentum Operator

The quantum mechanical angular momentum operatorLop, in spatial representation,

is given by

Lopμ=



i

Lμ=



i

εμνλrν∇λ. (7.84)

Hereis the Planck constanth, divided by 2π, andiis the imaginary unit, with the

propertyi^2 =−1. The expression (7.84) follows the definitionr×pfor the orbital

angular momentum, cf. Sect.3.4.1, when the linear momentumpis replaced by the

operator

pop=



i

∇, (7.85)

in spatial representation.