Tensors for Physics

7.6 Rules for Nabla and Laplace Operators 107

Side Remark:
Where Does the Expression for the Linear Momentum Operator Come From?

A plane waveψwith the wave vectorkand the (circular) frequencyωis proportional
to exp(ik·r−iωt). Application of the nabla operator yields∇ψ=ikψ, which is
just a mathematical identity. By analogy to the Einstein relationE=ωbetween
the energyEand the frequency, de Broglie suggested that the linear momentumpof
a particle should be associated with a wave vector according top=k. Schrödinger
took up this idea and inventedwave mechanics. Later it became clear that the wave
functionψis a probability amplitude, its absolute square characterizes the probability
to find a particle in a volume element at a specific positionr. For a free particle
with linear momentump,theψ-function is proportional to exp(ip·r/), hence
∇ψ=ipψ. This corresponds to (7.85). The expression for the linear momentum
operator derived for the special case of a plane wave holds true in general, in spatial
representation.

Dimensionless Angular Momentum Operator

It is convenient to introduce angular momentum operators in units ofand to denote
them by the same symbolLas the usual angular momentum, as long as no danger
of confusion exists. Then one has

Lμ=

1

i

Lμ=−iεμνλrν∇λ. (7.86)

Thanks to the imaginary unitiwhich is introduced in the definitions (7.84) and (7.86),
the angular momentum operator is ahermitian operatorwith real eigenvalues.
The commutation relations (7.82) for the components of the differential operator
Lnow lead to the angular momentum commutation relations

LμLν−LνLμ=iεμνλLλ. (7.87)

Similarly, relation (7.83) implies

ελμνLμLν=iLλ. (7.88)

The commutation relations for the orbital angular momentum hold true in general,
not only in the space representation which was used here to derive them.
Notice that (7.88) is equivalent to

L×L=iL. (7.89)

This reflects the fact that the components of the quantum mechanical angular momen-
tum do not commute, in contradistinction to the components of the classical angular
momentum for which the corresponding cross product vanishes.