2.7 Additional Points

This subsection describes a few further issues of importance for this book.

2.7.1 Dirac notation

Physicists like to write inner products such as
in Dirac notation

:

since this conforms more closely to how you would think of it in linear algebra:

The various advanced ideas of linear algebra can be extended to operators in this way, but they will not be needed in this book.

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
complex conjugate,

etcetera. Applying a dagger a second time gives the original back. Also, if you work out the dagger on a product, you need to reverse the order of the factors. For example

In words, putting into the left side of an inner product gives .

The second point will be illustrated for the case of vectors in
three dimensions. Such a vector can be written as

Here , , and are the three unit vectors in the axial directions. The components , and can be found using dot products:

Symbolically, you can write this as

In fact, the operator in parentheses can be defined by saying that for any vector , it gives the exact same vector back. Such an operator is called an “identity operator.”

The relation

is called the “completeness relation.” To see why, suppose you leave off the third part of the operator. Then

The -component is gone! Now the vector gets projected onto the -plane. The operator has become a “projection operator” instead of an identity operator by not suming over the complete set of unit vectors.

You will almost always find these things in terms of bras and kets.
To see how that looks, define

Then

so the completeness relation looks like

If you do not sum over the complete set of kets, you get a projection operator instead of an identity one.

2.7.2 Additional independent variables

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.