Principles - Methods of Molecular Quantum Mechanics - page 8

Now, let F(x,y,z) be a scalar function of the space point P(r). Then:

r F ¼ i @ F

@ x þ j @ F @ y þ k @ F

ð 1 : 31 Þ

@ z

is a vector, the gradient of F.
If F is a vector of components F x , F y , F z , we then have for the scalar
product

r F ¼ @ F x

@ x þ @ F y @ y þ @ F z @ z ¼ div F

ð 1 : 32 Þ

a scalar quantity, the divergence of F. As a particular case:

2 2 2

¼ @

@ x 2 þ @ @ y 2 þ @

2

rr¼r ð 1 : 33 Þ

@ z 2

is the Laplacian operator.
Fromthe vector product of r by the vector Fwe obtain a newvector, the
curl or rotation of F (written curl F or rot F):

i j k

@
@ x

@
@ y

@
@ z

r F ¼ ¼ curl F ¼ i curl x F þ j curl y F þ k curl z F ð 1 : 34 Þ

F x F y F z

a vector operator with components:

8
<
:

curl x F ¼ @ F z

@ y @ F y @ z

curl y F ¼ @ F x

@ z @ F z

ð 1 : 35 Þ

@ x

curl z F ¼ @ F y

@ x @ F x @ y

In quantummechanics, the vector product of the position vector r by the
linearmomentumvector operator i h r (see Section 1.3) gives the angular

momentum vector operator L:

i j k
xyz
@
@ x

L ¼ i

¼ iL x þ jL y þ kL z hr r¼ i h ð 1 : 36 Þ

@
@ y

@
@ z