D.10 The curl is Hermitian
For later reference, it will be shown that the curl operator,
is Hermitian. In other words,
The rules of engagement are as follows:
- The Cartesian axes are numbered using an index , with
1, 2, and 3 for , , and
- Also, indicates the coordinate in the direction,
, , or .
- Derivatives with respect to a coordinate are indicated by
a simple subscript .
- If the quantity being differentiated is a vector, a comma is
used to separate the vector index from differentiation ones.
- Index is the number immediately following in the
cyclic sequence ...123123...and is the number
immediately preceding .
- A bare integral sign is assumed to be an integration over
all space, or over the entire box for particles in a box. The
is normally omitted for brevity and to be understood.
- A superscript indicates a complex conjugate.
In index notation, the integral in the left hand side above reads:
which is the same as
as can be checked by differentiating out the first two terms. Now the
third and fourth terms in the integral are
, as you can see from moving all indices in
the third term one unit forward in the cyclic sequence, and those in
the fourth term one unit back. (Such a shift does not change the sum;
the same terms are simply added in a different order.)
So, if the integral of the first two terms is zero, the fact that curl
is Hermitian has been verified. Note that the terms can be
integrated. Then, if the system is in a periodic box, the integral is
indeed zero because the upper and lower limits of integration are
equal. An infinite domain will need to be truncated at some large
distance from the origin. Then shift indices and apply the
divergence theorem to get
where is the surface of the sphere and the unit
vector normal to the sphere surface. It follows that the integral is
zero if and go to zero at infinity quickly enough.
Or at least their cross product has to go to zero quickly enough.