This subsection describes a few further issues of importance for this book.
Physicists like to write inner products such as
One thing will be needed in some more advanced addenda, however. That
is the case that operator is not Hermitian. In that case,
if you want to take to the other side of the inner product, you
need to change it into a different operator. That operator is called
the “Hermitian conjugate” of . In physics, it is almost
always indicated as . So, simply by definition,
Then there are some more things that this book will not use. However, you will almost surely encounter these when you read other books on quantum mechanics.
First, the dagger is used much like a generalization of
The second point will be illustrated for the case of vectors in
three dimensions. Such a vector can be written as
You will almost always find these things in terms of bras and kets.
To see how that looks, define
In many cases, the functions involved in an inner product may depend on more than a single variable . For example, they might depend on the position in three-dimensional space.
The rule to deal with that is to ensure that the inner product
integrations are over all independent variables. For example,
in three spatial dimensions:
Note that the time is a somewhat different variable from the rest, and time is not included in the inner product integrations.