Tensors for Physics

108 7 Fields, Spatial Differential Operators

7.6.3 Radial and Angular Parts of the Laplace Operator.

The Laplace operator can be decomposed into a radial and angular parts, by analogy

to the decomposition (7.81) of the nabla operator. With the help of the angular

differential operatorL, one has

Δ=Δr+r−^2 LμLμ. (7.90)

The radial partΔris given by

Δr=r−^2

∂

∂r

(

r^2

∂

∂r

)

=r−^1

∂^2

∂r^2

r=

∂^2

∂r^2

+ 2 r−^1

∂

∂r

. (7.91)

To prove the relation (7.90) with (7.91), one can computeLμLμ, starting from the

definition (7.80). One finds

LμLμ=εμνλεμαβrν∇λrα∇β=εμνλεμαβ(rνrα∇λ∇β+rνδλα∇β).

Nowuseofεμνλεμαβ=δναδλβ−δνβδλα, cf. Sect.4.1.2, leads toLμLμ=r^2 ∇λ∇λ−

r^2 ∂

2

∂r^2 +r

∂

∂r−^3 r

∂

∂r. Thus one obtains

LμLμ=r^2

(

Δ−

∂^2

∂r^2

− 2 r−^1

∂

∂r

)

.

Forr>0, this relation is equivalent to (7.90).

Notice thatΔr−^1 =0, forr =0. This result applies just for 3D, the three-

dimensional space we live in. In D dimensions one hasΔr(^2 −D)=0, see the next

exercise.

7.6 Exercise: Determine the Radial Part of the Laplace Operator in

DDimensions, ProveΔr(^2 −D)= 0

Hint: Compute∇μ∇μf=∇μ(∇μf), where the functionf=f(r)has no angular
dependence, and use∇μrμ=D.
Furthermore, make the ansatzf =rnand determine for which exponentnthe
equationΔrn=0 holds true.

7.6.4 Application: Kinetic Energy Operator

in Wave Mechanics

The kinetic energy of a particle with massmand with linear momentumpis

p·p/( 2 m). In Schrödinger’s wave mechanics, in spatial representation, the linear